Convergence of a semi-Lagrangian scheme for the ellipsoidal BGK model of the Boltzmann equation
aa r X i v : . [ m a t h . A P ] J u l CONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THEELLIPSOIDAL BGK MODEL OF THE BOLTZMANN EQUATION
GIOVANNI RUSSO AND SEOK-BAE YUN
Abstract.
The ellipsoidal BGK model is a generalized version of the original BGK model designedto reproduce the physical Prandtl number in the Navier-Stokes limit. In this paper, we propose anew implicit semi-Lagrangian scheme for the ellipsoidal BGK model, which, by exploiting specialstructures of the ellipsoidal Gaussian, can be transformed into a semi-explicit form, guaranteeingthe stability of the implicit methods and the efficiency of the explicit methods at the same time.We then derive an error estimate of this scheme in a weighted L ∞ norm. Our convergence estimateholds uniformly in the whole range of relaxation parameter ν including ν = 0, which corresponds tothe original BGK model. Introduction
The BGK model [8, 49] has been widely used as an efficient model for the Boltzmann equationbecause the BGK model is much more amenable to numerical treatment, and still maintains many ofthe important qualitative properties of the Boltzmann equation. But several short-commings are alsoreported where this model fails to reproduce the correct physical data of the Boltzmann equation. Onesuch example is the Prandtl number, which is defined as the ratio between the viscosity and the heatconductivity. The Prandtl number computed via the BGK model does not match with the one derivedfrom the Boltzmann equation, resulting in the incorrect hydrodynamic limit at the Navier-Stokes level.To overcome this, Holway [28] suggested a generalized version of the BGK model by replacing thelocal Maxwellian with an ellipsoidal Gaussian parametrized by a free parameter − / < ν <
1. Thismodel is called the ellipsoidal BGK model (ES-BGK model), whose initial value problem reads ∂ t f + v · ∇ f = 1 κ A ν ( M ν ( f ) − f ) ,f ( x, v,
0) = f ( x, v ) . (1)The velocity distribution function f ( x, v, t ) is the number density of the particle system on the phasepoint ( x, v ) ∈ T d × R d ( d ≤ d ) at time t ∈ R + . Here, T denotes the unit interval with periodicboundary condition and R is the whole real line. The Knudsen number κ is a dimensionless numberdefined by the ratio between the mean free path and the characteristic length. For later convenience,we allowed a slight abuse of notation so that the convection term v · ∇ x f is understood as v · ∇ x f = X ≤ i ≤ d v i ∂ x i f. The collision frequency A ν takes various forms depending on modeling assumptions. In this paper, weonly consider the fixed collision frequency: A ν = (1 − ν ) − . The ellipsoidal Gaussian M ν ( f ) reads: M ν ( f ) = ρ p det(2 π T ν ) exp (cid:18) −
12 ( v − U ) ⊤ T − ν ( v − U ) (cid:19) , Key words and phrases.
BGK model, ellipsoidal BGK model, Boltzmann equation, semi-Lagrangian scheme, errorestimate.S.-B. Yun was supported by Basic Science Research Program through the National Research Foundation of Ko-rea(NRF) funded by the Ministry of Education(NRF-2016R1D1A1B03935955). where the macroscopic density, velocity, temperature and the stress tensor are defined by ρ ( x, t ) = Z R d f ( x, v, t ) dv,ρ ( x, t ) U ( x, t ) = Z R d f ( x, v, t ) vdv,d ρ ( x, t ) T ( x, t ) = Z R d f ( x, v, t ) | v − U ( x, t ) | dv,ρ ( x, t )Θ( x, t ) = Z R d f ( x, v, t )( v − U ) ⊗ ( v − U ) dv. The temperature tensor T ν is given by a convex combination of T and Θ: T ν = (1 − ν ) T Id + ν Θ , where Id is the d × d identity matrix. The ellipsoidal relaxation operator satisfies the followingcancellation property: Z R d (cid:0) M ν ( f ) − f (cid:1) (cid:0) , v, | v | (cid:1) dv = 0 , which leads to the conservation of mass, momentum and energy: ddt Z R d × R d f dxdv = ddt Z R d × R d vf dxdv = ddt Z R d × R d f | v | dxdv = 0 . When Holway first suggested this model, H -theorem was not verified, which was the main reasonwhy the ES-BGK model has been neglected in the literature until very recently. It was resolved in [2](See also [10, 53]): ddt Z R d × R d f ln f dxdv ≤ , and ignited the interest on this model [1, 10, 22, 25, 31, 32, 44, 51, 52, 53, 54].It can be verified via the Chapman-Enskog expansion that the Prandtl number computed usingthe ES-BGK model is 1 / (1 − ν ). Therefore, the correct physical Prandtl number can be recoveredby choosing appropriate ν , namely, ν = 1 − /P r ≈ − /
2, where
P r denotes the correct Prandtlnumber. When ν = 0, the ES-BGK model reduces to the original BGK model. Hence, any results forthe ES-BGK model automatically hold for the original BGK model either. We also mention that, inthe range − / < ν <
1, the only possible equilibrium state of the ellipsoidal relaxation operator isthe usual Maxwellian, not the ellipsoidal Gaussian. That is, the only solution satisfying the relation M ν ( f ) = f is the local Maxwellian (See [51, 53] for the proof.): f = M ( f ) = ρ p (2 πT ) d exp (cid:18) − | v − U | T (cid:19) . Therefore, the ES-BGK model correctly captures the two most important asymptotic behavior of theBoltzmann equation, namely, the time asymptotic limit ( t → ∞ ) and the hydrodynamic limit at theEuler level ( κ → ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 3 naturally constructed in non-conservative form. Several papers have been devoted to the constructionof conservative SL schemes (see, for example, [19, 34, 36, 50]). SL schemes have also been used tosolve specific problems where accurate solutions are needed. In [41], the authors adopt a SL methodto study the decay of an oscillating plate in a rarefied gas described by the BGK model and comparethe results with those obtained by a scheme specifically designed for the problem. For a survey onnumerical schemes on BGK model, we refer to the review paper [21, 30] and references therein.In view of the increasing interest in the subject, our aim is to introduce a new family of semi-Lagrangian scheme for the ES-BGK model and to investigate their convergence properties. Numericalresults of the schemes will be presented in a separate paper.1.1.
Implicit semi-Lagrangian scheme.
First, we propose an implicit semi-Lagrangian scheme. Inthe numerical computation of the collisional or relaxational kinetic equations, it is common to employthe so called splitting method, which amounts to computing the transport part: ∂ t f + v · ∇ x f = 0 , (2)and the relaxational time evolution: dfdt = 1 κ A ν {M ν ( f ) − f } , (3)separately. The most naive way for this is to use the forward in time methods and the explicit Eulertype method respectively for (2) and (3). It is, however, well-known that the first procedure leads tothe restriction on the temporal grid size due to the CFL condition: △ t < △ x/ max j | v j | , while thesecond procedure entails a stability condition: △ t < Cκ for some constant C >
0, resulting in thefollowing two scale restriction on the size of time step: △ t < min (cid:26) Cκ, △ x/ max j | v j | (cid:27) . Since the two parameters of the right-hand side are independent of each other, the discrepancy be-tween these two scale can get arbitrary large, and the scheme becomes severely resource-consumingaccordingly. Such a stiffness problem has been one of the key difficulties in developing efficient stableschemes for kinetic equations. In this paper, we propose a new semi-Lagrangian scheme, which com-bines two numerical methods known to guarantee stable performances, namely the semi-Lagrangiantreatment for the transport part (2), and the implicit Euler for the evolution part (3), to overcomethe CFL restriction and secure the stability of the scheme over the large range of Knudsen number atthe same time: f n +1 i,j − ˜ f ni,j △ t = 1 κ A ν (cid:8) M ν,j ( f n +1 i ) − f n +1 i,j (cid:9) . Here e f ni,j denotes the linear reconstruction. (See Definition 2.1.) At first sight, the scheme seems verytime consuming due to the implicit implementation of the relaxation part. In the case of the originalBGK model ( ν = 0), such difficulties can be circumvented by a clever trick using the fact that (1)the local Maxwellian depends on the distribution function only through the conservative macroscopicfields, and (2) the macroscopic fields satisfy the following identities:(4) ρ n +1 i = e ρ ni , U n +1 i = e U ni , T n +1 i = e T ni , with small error, enabling one to explicitly solve for the numerical solutions [23, 33, 37]. Here, themacroscopic variables with tilde are those constructed from ˜ f ni,j . (see Section 2.) In this surprisingturn of events, the two seemingly contradicting properties: the stable performance of the implicitscheme and the efficiency of the explicit scheme, are reconciled. Such nice feature, of course, cannever be expected for the Boltzmann type collision operators.In the case of the ES-BGK model, however, the conservation laws are not sufficient to make thistrick work, since the ellipsoidal Gaussian contains the stress tensor, which is not a conserved quantity. GIOVANNI RUSSO AND SEOK-BAE YUN
Even though we cannot expect the stress tensor to satisfy similar conservation identities as (4), weobserve that the following approximation holds with small error (See (12) in Section 2):Θ n +1 i ≈ △ tκ + △ t e T ni Id + κκ + △ t e Θ ni , which enables us to rewrite the implicit ellipsoidal Gaussian in a semi-explicit manner. The resultantscheme can now be written in an explicit manner as (See Section 2.2) f n +1 i,j = κκ + A ν △ t e f ni,j + A ν △ tκ + A ν △ t M e ν,j ( ˜ f ni ) . We remark that the implementation of the scheme and its check on several numerical tests will bereported in an independent paper [9].1.2. L ∞ convergence theory. We then develop a convergence theory for this scheme that willguarantee the credibility of the method. In this regard, our main result is the following error estimatestated in Theorem 3.2 in Section 3: k f ( T f ) − f N t k L ∞ q ≤ C n △ x + △ v + △ t + △ x △ t o , for some constant C >
0. Compared to our previous result [37] where the convergence of a semi-Lagrangian scheme for the original BGK model was established in L space: k f ( T f ) − f N t k L ≤ C n △ x + △ v + △ t + △ x △ t o , we make four improvements. The error estimate in the spatial node is improved from △ x to ( △ x ) byassuming additional regularity on the initial data and refining the analysis of the interpolation part.This enables one to recover the first order error estimate by choosing △ = △ t . Secondly, we imposesize restriction only on the velocity nodes, whereas in [37], we needed to restrict the size of all thenode size: △ x , △ v and △ t . Thirdly, we develop a theory to measure the error in a weighted L ∞ norminstead of the weighted L norm, which provides a more clear and detailed picture on the convergenceof the scheme, since the error estimate in L ∞ norm gives a node-wise convergence estimate. Finally,the proof is greatly simplified, enabling one to extend the convergence theory to the whole range ofrelaxation parameter − / < ν <
1. As a result, we derive a convergence estimate that is uniform in − / ν <
1. Note that, since the above convergence estimate is valid for the whole range of relaxationparameter − / < ν < ν = 0.The most important step of the proof is the derivation of the following uniform stability estimate(For notations, see the next subsection): C , e − Aνκ T f e − C , | v j | α ≤ f ni,j ≤ e ( C M− Tfκ + Aν △ t k f k L ∞ q (1 + | v j | ) − q , which comes from the uniform-in- n control of the discrete ellipsoidal Gaussian in a weighted L ∞ norm(See Lemma 5.7): kM e ν ( f n ) k L ∞ q ≤ C M k f n k L ∞ q . Two technical issues arise in the process of obtaining the above estimates. First, we need to showthat the discrete temperature tensor e T n ˜ ν,i is strictly positive definite uniformly in n : k ⊤ (cid:8) e T n e ν,i (cid:9) k ≥ C ν,q,κ,T f > κ ∈ S . Otherwise, since the discrete ellipsoidal Gaussian involves the inverse of T n ˜ ν,i and det( T n ˜ ν,i ), it mayblow up as T n ˜ ν,i approaches arbitrarily close to zero. Second issue is more subtle. It turns out that weneed to show that ratios such as e ρ ni e T ni / k e f n k L ∞ q , k e f n k L ∞ q / e ρ ni ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 5 remain strictly larger than △ v , to guarantee the existence of proper decomposition of the macroscopicfields to derive necessary moment estimates. (See Lemma 5.6.) The restriction on the velocity nodein Theorem 3.2 mostly comes from this subtle technical issue.1.3. Notation.
Before we finish this introduction, we summarize the notational convention keptthroughout this paper: • C denotes generic constants. The exact value may change in each line, but they are explicitlycomputable in principle. • We will use lower indices n , i , j exclusively for time, space and velocity variable respectively.For example x i , v j , t n . • We use upper indices for the components of vectors as v = ( v , v , v ), while the lower indicesare reserved for the spatial, velocity and temporal nodes. • T f will denote the final time, whereas T f represents for the local temperature constructedfrom the distribution function f . • We use the following notation for weighted L ∞ -Sobolev norm for continuous solution: k f k L ∞ q = sup x,v | f ( x, v )(1 + | v | ) q | , k f k W ℓ, ∞ q = X | α | + | β |≤ ℓ k ∂ αβ f k L ∞ q , where, α, β ∈ Z + and, the differential operator ∂ αβ stands for ∂ αβ = ∂ αx ∂ βv . • For any sequence a ni,j , we use the following notation for the weighted L ∞ -norm for of thesequence: k a n k L ∞ q = sup i,j | a ni,j (1 + | v j | ) q | . • In view of the above norms, k f ( t n ) − f n k L ∞ q is understood, with a slight abuse of notation, as k f ( t n ) − f n k L ∞ q = sup i,j (cid:12)(cid:12) (cid:8) f ( x i , v j , t n ) − f ni,j (cid:9) (1 + | v j | ) q (cid:12)(cid:12) , for simplicity.The paper is organized as follows. In the following Section 2, we derive a semi-Lagrangian scheme forthe ES-BGK model. The main convergence result of this paper is presented in Section 3. In section4, we establish some technical lemmas. Section 5 is devoted to the stability estimate of the scheme.In Section 6, we transform the ES-BGK model (1) to a form consistent with our scheme. In Section7, we estimate the discrepancy of discrete Gaussian and the continuous one. Finally, we prove theconvergence of our scheme in Section 7.2. Description of the numerical scheme
We fix d = 1 and d = 3 case with periodic boundary condition throughout this paper in order tostay in the simplest possible framework. We believe that the analysis of this paper can be extended tomore general conditions such as higher dimensions in x and/or different boundary conditions, althoughsuch extensions may give rise to unexpected difficulties. This will be a topic of future work. Notethat, in contrast to the original BGK model, the velocity domain must be at least 2-dimensional forthe ellipsoidal BGK to be meaningful. Otherwise the model reduces to the original BGK model. Wechoose a constant time step △ t with final time T f . The spatial domain and the velocity domain aredivided into uniform grids with mesh size △ x , △ v respectively: t n = n △ t, n = 0 , , ..., N t ,x i = i △ x, i = 0 , ± , ..., ± N x , ± ( N x + 1) , · · · where N t △ t = T f , N x △ x = 1, and v j = ( v j , v j , v j ) = ( j △ v, j △ v, j △ v ) , j = ( j , j , j ) ∈ Z . Note that the spatial node is defined on the whole line instead of unit interval, even though we areconsidering periodic problem. Periodicity will be imposed on the initial data f , which is defined on GIOVANNI RUSSO AND SEOK-BAE YUN the whole line with period 1. This facilitates the proof in several places. If not specified otherwise,we assume throughout this paper that n ≤ N t , to avoid unnecessary repetition. We denote theapproximate solution of f ( x i , v j , t n ) by f ni,j . To describe the numerical scheme more succinctly, weintroduce the following convenient notation. Definition 2.1. (1) Let x ( i, j ) = x i − △ t v j . Then define s = s ( i, j ) to be the spatial node such that x ( i, j ) lies in [ x s , x s +1 ).(2) The reconstructed distribution function e f ni,j is defined as e f ni,j = x ( i, j ) − x s △ x f ns +1 ,j + x s +1 − x ( i, j ) △ x f ns,j . Remark . ˜ f ni,j is the linear interpolation of f ns,j and f ns +1 ,j on x ( i, j ).We also need to define the discrete ellipsoidal Gaussian: M ν,j ( f ni ) = ρ ni q det(2 π T nν,i ) exp (cid:16) −
12 ( v j − U ni ) (cid:8) T nν,i (cid:9) − ( v j − U ni ) (cid:17) , and the discrete macroscopic field ρ ni , U ni , T ni , Θ ni and T nν,i : ρ ni = X j f ni,j ( △ v ) , ρ ni U ni = X j f ni,j v j ( △ v ) , ρ ni T ni = X j f ni,j (cid:12)(cid:12) v j − U ni (cid:12)(cid:12) ( △ v ) ,ρ ni Θ ni = X j f ni,j ( v j − U ni ) ⊗ ( v j − U ni )( △ v ) , T nν,i = (1 − ν ) T ni Id + ν Θ ni . Derivation of the scheme (14).
Let f j = f ( x, v j , t ). We rewrite the ES-BGK model (1) inthe characteristic formulation: df j dt = 1 κ A ν (cid:0) M ν ( f j ) − f j (cid:1) ,dxdt = v j . (5)Using implicit Euler scheme on (5), we obtain f n +1 j ( x i ) − f nj ( x i − v j △ t ) = △ tκ A ν (cid:8) M ν,j ( f n +1 ) − f n +1 j (cid:9) ( x i ) . We then approximate f nj ( x i ) by f ni,j , and f nj ( x i − v j △ t ) by e f ni,j to obtain f n +1 i,j − e f ni,j △ t = 1 κ A ν (cid:8) M ν,j ( f n +1 i ) − f n +1 i,j (cid:9) . (6)We attempt to convert (6) into a semi-explicit scheme keeping beneficial features of implicit schemessuch as the stability property. This idea seems to trace back to [18, 33], and successfully implementedin the semi-Lagrangian setting in [23, 27, 37]. We first impose the conservation of mass, momentumand energy at the discrete level (throughout this subsection ≈ means that they are identical up tonegligible error): P j f n +1 i,j φ ( v j )( △ v ) − P j e f ni,j φ ( v j )( △ v ) △ t = 1 κ A ν X j (cid:8) M ν,j ( f n +1 i ) φ ( v j ) − f n +1 i,j φ ( v j ) (cid:9) ( △ v ) ≈ , ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 7 for φ ( v j ) = 1 , v j , | v j | . Since r.h.s has a spectral accuracy for fast decaying functions, we canlegitimately assme that(7) X j f n +1 i,j φ ( v j ) ≈ X j e f ni,j φ ( v j ) . Therefore, if we define e ρ ni = X j e f ni,j ( △ v ) , e ρ ni e U ni = X j e f ni,j v j ( △ v ) , e ρ ni e T ni = X j e f ni,j (cid:12)(cid:12) v j − e U ni (cid:12)(cid:12) ( △ v ) , then (7) gives the following relation: ρ n +1 i ≈ ˜ ρ ni , U n +1 i ≈ ˜ U ni , T n +1 i ≈ ˜ T ni . (8)In the case of the original BGK mdoel ( ν = 0), identities in (8) are sufficient to conclude that M ,j ( f n +1 i ) ≈ M ,j ( ˜ f ni ). But this is not the case for the ellipsoidal case, since the ellipsoidal Gaussiancontains the temperature tensor T ν , which is not a conserved quantity. For this, we introduce φ ni,j ≡ ( v j − U ni ) ⊗ ( v j − U ni ) . Multiplying φ n +1 i,j ( △ v ) on both sides of (6) and summing over i and j , we get: P j f n +1 i,j φ n +1 i,j ( △ v ) − P j e f ni,j φ n +1 i,j ( △ v ) △ t = 1 κ A ν X j n M ν,j ( f n +1 i ) φ n +1 i,j − f n +1 i,j φ n +1 i,j o ( △ v ) . (9)Let’s denote the r.h.s of (9) by R and the l.h.s by L for simplicity. We then recall (8) to observe φ n +1 i,j = ( v j − U n +1 i ) ⊗ ( v j − U n +1 i ) ≈ ( v j − e U ni ) ⊗ ( v j − e U ni ) = e φ ni,j . Therefore, the second term of L becomes X j ˜ f ni,j φ n +1 i,j ( △ v ) ≈ X j ˜ f ni,j e φ ni,j ( △ v ) = e ρ ni e Θ ni , where e Θ ni is defined by e ρ ni e Θ ni = X j e f ni,j ( v j − e U ni ) ⊗ ( v j − e U ni )( △ v ) , so that L = ρ n +1 i Θ n +1 i − e ρ ni e Θ ni △ t . (10)On the other hand, we find for the right hand side, R ≈ κ A ν n ρ n +1 i T n +1 ν,i − ρ n +1 i Θ n +1 i o = 1 κ A ν h ρ n +1 i (cid:8) (1 − ν ) T n +1 i Id + ν Θ n +1 i (cid:9) − ρ n +1 i Θ n +1 i i = 1 κ A ν (1 − ν ) (cid:2) ρ n +1 i T n +1 i Id − ρ n +1 i Θ n +1 i (cid:3) = 1 κ (cid:8) ρ n +1 i T n +1 i Id − ρ n +1 i Θ n +1 i (cid:9) , and recall (8) to see that R = 1 κ (cid:8) ρ n +1 i T n +1 i Id − ρ n +1 i Θ n +1 i (cid:9) = 1 κ ne ρ ni e T ni Id − e ρ ni Θ n +1 i o . (11) GIOVANNI RUSSO AND SEOK-BAE YUN
Now, equating (10) and (11), we rewrite (9) as ρ n +1 i Θ n +1 i − e ρ ni e Θ ni △ t = 1 κ ne ρ ni e T ni Id − e ρ ni Θ n +1 i o . Dividing both sides by ρ n +1 i and gathering relevant terms, we getΘ n +1 i = △ tκ + △ t e T ni Id + κκ + △ t e Θ ni . (12)Therefore, T n +1 ν,i can be expressed as T n +1 ν,i = (1 − ν ) T n +1 i Id + ν Θ n +1 i ≈ (1 − ν ) e T ni Id + ν (cid:26) △ tκ + △ t e T ni Id + κκ + △ t e Θ ni (cid:27) = (cid:18) − κνκ + △ t (cid:19) e T ni Id + (cid:18) κνκ + △ t (cid:19) e Θ ni ≡ (1 − e ν ) e T ni Id + e ν e Θ ni ≡ e T n ˜ ν,i , (13)where we denoted e ν = κνκ + △ t . Using (8) and (13), the implicitly defined discrete ellipsoidal Gaussian can now be rewritten in anexplicit way as: M ν,j ( f n +1 i ) = M ν,j (cid:0) ρ n +1 i , U n +1 i , T n +1 ν,i (cid:1) ≈ M ν,j (cid:0)e ρ ni , e U ni , e T n ˜ ν,i (cid:1) = e ρ ni q det(2 π e T n ˜ ν,i ) exp (cid:18) −
12 ( v j − e U ni ) ⊤ (cid:8) e T n ˜ ν,i (cid:9) − ( v j − e U ni ) (cid:19) . With a slight abuse of notation, we now denote the r.h.s as M ˜ ν,j ( ˜ f ni ). We should note carefully that(for example in Lemma 7.3) e T n ˜ ν,i = (1 − ˜ ν ) e T ni + ˜ ν e Θ ni = (1 − ν ) e T ni + ν e Θ ni = e T nν,i throughout this paper. We then use this to rewrite the implicit scheme (6) as the following explicitform: f n +1 i,j = κκ + A ν △ t e f ni,j + A ν △ tκ + A ν △ t M e ν,j ( ˜ f ni ) . Implicit semi-Lagrangian scheme.
Summarizing, our semi-Lagrangian scheme for the ES-BGK model (1) reads: f n +1 i,j = κκ + A ν △ t e f ni,j + A ν △ tκ + A ν △ t M e ν,j ( e f ni ) , (14)where the discrete ellipsoidal Gaussian M e ν,j ( e f ni ) is defined as follows: M e ν,j ( e f ni ) = e ρ ni q det(2 π e T n e ν,i ) exp (cid:16) −
12 ( v j − e U ni ) (cid:8) e T n e ν,i (cid:9) − ( v j − e U ni ) (cid:17) , ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 9 and discrete macroscopic field e ρ ni , e U ni , e T ni , e θ ni,j and e T n e ν,i ( n ≥
1) are given by e ρ ni = X j e f ni,j ( △ v ) , e ρ ni e U ni = X j e f ni,j v j ( △ v ) , e ρ ni e T ni = X j e f ni,j (cid:12)(cid:12) v j − e U ni (cid:12)(cid:12) ( △ v ) , e ρ ni e Θ ni = X j e f ni,j (cid:8) ( v j − e U ni ) ⊗ ( v j − e U ni ) (cid:9) ( △ v ) , e T n e ν,i = (1 − e ν ) e T ni Id + e ν e Θ ni . (15)In the last line, ˜ ν denotes ˜ ν = κνκ + △ t . For the initial step ( n = 0), to ignore the error arising in the discretization of the initial data andsimplify the convergence proof, we sample values directly from continuous distribution function andmacroscopic fields at t = 0: f i,j = f ( x i , v j ) , e f i,j = e f ( x i , v j ) = f ( x i − v j △ t, v j ) , and e ρ i = e ρ ( x i ,
0) = Z R f ( x i − v △ t, v ) dv, e ρ i e U i = e ρ ( x i , e U ( x i ,
0) = Z R f ( x i − v △ t, v ) vdv, e ρ i e T i = 3 e ρ ( x i , e T ( x i ,
0) = Z R f ( x i − v △ t, v ) | v − e U i | dv. Main results
We are now ready to state our main result. We first record the existence result relevant to ourconvergence proof.
Theorem 3.1. [52]
Let − / < ν < and q > . Let f satisfy k f k W , ∞ q < ∞ . Suppose further thatthere exist positive constants C , C and α such that f ( x, v ) ≥ C e − C | v | α . Then, for any final time T f > , the ES-BGK model (1) has a unique solution f ∈ C ([0 , T f ] , k·k W , ∞ q ) such that (1) f is bounded in k · k W , ∞ q for [0 , T f ] : k f ( t ) k W , ∞ q ≤ C e C t n k f k W , ∞ q + 1 o , t ∈ [0 , T f ] , for some constants C and C . (2) The macroscopic fields satisfy the following lower and upper bounds: k ρ ( t ) k L ∞ x + k U ( t ) k L ∞ x + k T ( t ) k L ∞ x ≤ C q e C q T f ,ρ ( x, t ) ≥ C N,q e − C N,q t ,k ⊤ (cid:8) T ν ( x, t ) (cid:9) k ≥ C N,q e − C N,q t > , for any k ∈ S . Now, we state our main result.
Theorem 3.2.
Let − / < ν < . Let f be the unique smooth solution of (1) corresponding to anonnegative initial datum f satisfying the hypotheses of Theorem 3.1. Let f n be the approximatesolution constructed iteratively by (14) given in Section 2. Then, there exists a positive number r △ v ,which is explicitly determined in Theorem 5.5 in Section 4, such that, if △ v < r △ v , then we have k f ( T f ) − f N t k L ∞ q ≤ C n ( △ x ) + △ v + △ t + ( △ x ) △ t o , where N t is defined by T f = N t △ t and C = C ( T f , f , q, κ, ν ) > . Here, C is uniformly bounded in ν .Remark . (1) ν = 0 corresponds to the original BGK model. Therefore, our result holds for theoriginal BGK model too. (2) For the precise definition of r △ v , see Theorem 5.5. (3) When κ = 0,this error estimate breaks down, since the coefficients of the estimate contain κ − . Currently, it is notclear whether this is of inherent nature, or can be avoided by developing finer convergence analysis.(4) The bad term 1 / △ t is removed in [11]. But the argument cannot be implemented in our case sinceit depends heavily on the fact that the distribution function for the Vlaosv-Poisson equation remainscompactly supported, once it is so initially. (5) We believe the argument we develop in this work isrobust, and can be extended in many directions such as semi-Lagrangian scheme for polyatomic BGKmodels, high order semi-Lagrangian schemes, and semi-Lagrangian BDF methods, or Runge-Kuttamethod. We leave them for the future.4. Technical lemmas
Lemma 4.1.
Discrete solutions to (14) are periodic in the spatial nodes: f ni + N x ,j = f ni,j . Proof.
We use induction. We recall the definition of f i,j to get f i + N x ,j = f ( x i + N x , v j ) = f ( x i + N x △ x, v j ) = f ( x i + 1 , v j ) = f ( x i , v j ) = f i,j . Similarly, we have ˜ f i + N x ,j = e f ( x i + N x △ x, v j ) = e f ( x i + 1 , v j )= f ( x i − △ tv j + 1 , v j ) = f ( x i − △ tv j , v j )= e f ( x i , v j )= ˜ f i,j . Then, the periodicity of f i,j and ˜ f i,j implies the periodicity of ρ i , U i , T e ν,i and e ρ i , e U i , e T ν,i by definition.This completes the proof of the initial step of the induction. Now, assume that f ni,j , e f ni,j , ρ ni , U ni , T n e ν,i and e ρ ni , e U ni , e T n ˜ ν,i are all periodic in spatial variable. Then the periodicity of f n +1 i,j is immediate from(14). For the periodicity of e f n +1 i,j , we first observe x ( i + N x , j ) = x i + N x − △ tv j = x i + N x △ x − △ tv j = x ( i, j ) + N x △ x, so that s ( i + N x , j ) = s ( i, j ) + N x . Therefore, x ( i + N x , j ) − x s ( i + N x ,j ) = x ( i, j ) − x s ( i,j ) . Likewise, x s ( i + N x ,j )+1 − x ( i + N x , j ) = x s ( i,j )+1 − x ( i, j ) . ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 11
We then use these identities together with the periodicity of f n +1 i,j to derive e f n +1 i + N x ,j = x ( i + N x , j ) − x s ( i + N x ,j ) △ x f n +1 s ( i + N x ,j )+1 ,j + x s ( i + N x ,j )+1 − x ( i + N x , j ) △ x f n +1 s ( i + N x ,j ) ,j = x ( i, j ) − x s ( i,j ) △ x f n +1 s ( i,j )+ N x +1 ,j + x s ( i,j )+1 − x ( i, j ) △ x f n +1 s ( i,j )+ N x ,j = x ( i, j ) − x s ( i,j ) △ x f n +1 s ( i,j )+1 ,j + x s ( i,j )+1 − x ( i, j ) △ x f n +1 s ( i,j ) ,j = e f n +1 i,j , which gives the periodicity of e f n +1 i + N x ,j . Then the macroscopic fields associated with f n +1 i,j and e f n +1 i,j are periodic by construction. Therefore, the desired result follows from induction. (cid:3) Lemma 4.2. [52]
The reconstruction procedure does not increase the k · k L ∞ q -norm of the discretedistribution function: k e f n k L ∞ q ≤ k f n k L ∞ q . Proof.
We observe from the definition of e f n that, for n = 0 k e f n k L ∞ q = sup i,j (cid:12)(cid:12) e f ni,j (1 + | v j | ) q (cid:12)(cid:12) = sup i,j (cid:12)(cid:12)(cid:12)(cid:16) x ( i, j ) − x s,j △ x f ns +1 ,j + x s +1 ,j − x ( i, j ) △ x f ns,j (cid:17) (1 + | v j | ) q (cid:12)(cid:12)(cid:12) ≤ sup i,j (cid:12)(cid:12) max { f ns,j , f ns +1 ,j } (1 + | v j | ) q (cid:12)(cid:12) ≤ sup i,j (cid:12)(cid:12) f ni,j (1 + | v j | ) q (cid:12)(cid:12) = k f n k L ∞ q . Here, s denotes s ( i, j ). When n = 0, we have k e f k L ∞ q = sup i,j (cid:12)(cid:12) f ( x i − v j △ t, v j )(1 + | v j | ) q (cid:12)(cid:12) ≤ sup i,j (cid:12)(cid:12) f ( x i , v j )(1 + | v j | ) q (cid:12)(cid:12) = k f k L ∞ q . (cid:3) Lemma 4.3.
Let − / < ν < . Assume e f ni,j > and e ρ ni > . Then the discrete temperature tensor T n ˜ ν,i and its determinant det T n ˜ ν,i satisfy the following equivalence estimates: (1) min { − ˜ ν, ν } T ni Id ≤ e T n ˜ ν,i ≤ max { − ˜ ν, ν } T ni Id, (2) min { − ˜ ν, ν } ( T ni ) ≤ det { e T n ˜ ν,i } ≤ max { − ˜ ν, ν } ( T ni ) . In the first inequality, A ≥ B for × symmetric matrices A and B means A − B is positive definite.Proof. From (13), we see that e ρ ni e T n ˜ ν,i = (1 − ˜ ν ) e ρ ni e T ni Id + ˜ ν e ρ ni e Θ ni = (1 − ˜ ν )3 n X j ˜ f ni,j | v j − e U ni | ( △ v ) o + ˜ ν X j ˜ f ni,j ( v j − e U ni ) ⊗ ( v j − e U ni )( △ v ) . Then, in view of the identity: k T { U ⊗ U } k = ( k · U ) ( k, U ∈ R ), we have k ⊤ { e ρ ni e T n ˜ ν,i } k = (1 − ˜ ν )3 n X j ˜ f ni,j | v j − e U ni | ( △ v ) o | k | + ˜ ν X j ˜ f ni,j n ( v j − e U ni ) · k o ( △ v ) . We note from the definition of e ν that − / < ν < − / < e ν <
1, and divide our estimateinto the following two cases:(a) 0 < ˜ ν <
1: Since the second term is non-negative, we have k ⊤ { e ρ ni e T n ˜ ν,i } k ≥ (1 − ˜ ν )3 n X j ˜ f ni,j | v j − e U ni | ( △ v ) o | k | = (1 − ˜ ν ) e ρ ni e T ni | k | . (b) − < ˜ ν <
0: By Cauchy-Schwartz inequality, we see that k ⊤ { e ρ ni e T n ˜ ν,i } k ≥ (1 − ˜ ν )3 n X j ˜ f ni,j | v j − e U ni | ( △ v ) o | k | + ˜ ν n X j ˜ f ni,j | v j − e U ni | ( △ v ) o | k | = (1 + 2˜ ν )3 n X j ˜ f ni,j | v j − e U ni | ( △ v ) o | k | = (1 + 2˜ ν ) e ρ ni e T ni | k | . We then combine the above estimates to getmin { − ˜ ν, ν } e T ni ≤ k ⊤ { e T n ˜ ν,i } k. The r.h.s of the inequality follows in a similar manner.(2) Let λ i be the eigenvalues of e T ˜ ν . Then (1) implies that the values of these eigenvalues lie betweenmin { − ˜ ν, ν } e T ni and max { − ˜ ν, ν } e T ni . This givesmin { λ n , λ n , λ n } = min | k | =1 k ⊤ e T n ˜ ν,i k ≥ min { − ˜ ν, ν } e T ni , and max { λ n , λ n , λ n } = max | k | =1 k ⊤ e T n ˜ ν,i k ≤ max { − ˜ ν, ν } e T ni , so that min { − ν, ν } { e T ni } ≤ det e T n e ν,i = λ n λ n λ n ≤ max { − ν, ν } { e T ni } . (cid:3) In the following lemma, the symbol [ x ] denotes, as usual, the largest integer that does not exceed x . Lemma 4.4.
Fix velocity grid index j and the grid size △ x , △ v , △ t . We define s k inductively as s (1) = s ( i, j ) , s (2) = s ( s (1) , j ) , s (3) = s ( s (2) , j ) , · · · Using this notation, we define a s ( n ) j by a s ( n ) j = x s ( n ) − △ tv j − x s ( n +1) △ x . Then, a s ( n ) j is constant for all n > , that is a s ( n ) j = a s ( m ) j , for all positive integers m and n .Proof. We take a positive integer ℓ s ( n ) such that x s ( n ) = ℓ s ( n ) △ x. On the other hand, we can find a positive integer m j such that m j △ x ≤ v j △ t ≤ ( m j + 1) △ x. ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 13
That is,(16) m j = (cid:20) v j △ t △ x (cid:21) . From these, we immediately see that { ℓ s ( n ) − ( m j + 1) } △ x ≤ x s ( n ) − v j △ t ≤ { ℓ s ( n ) − m j } △ x. Since x s ( n +1) denotes the closest spatial node that lies before x s ( n ) − v j △ t , this gives x s ( n +1) = { ℓ s ( n ) − ( m j + 1) }△ x. Therefore, { x s ( n ) − △ tv j } − x s ( n +1) = { ℓ s ( n ) △ x − △ tv j } − { ℓ s ( n ) − ( m j + 1) }△ x = ( m j + 1) △ x − △ tv j . (17)Dividing both sides by △ xa s ( n ) j = x s ( n ) − △ tv j − x s ( n +1) △ x = ( m j + 1) − v j △ t △ x . (18)In view of (16), this can be rewritten as (cid:20) v j △ t △ x (cid:21) − v j △ t △ x + 1 , which is dependent on j but not on n . This completes the proof. (cid:3) Stability of the discrete distribution function
In this section, we derive uniform lower and upper bounds for the discrete distribution function˜ f ni,j and corresponding macroscopic fields. We start with series of definitions most of which wereintroduced for technical reasons. Definition 5.1. (1) We define C α , C q,α and C q − m ( q ≥ m ) by C α = Z R e − C , | v | α dv, C q,α = sup v (cid:8) (1 + | v | ) q e − C , | v | α (cid:9) , C q − m = Z R | v | ) q − m dv. (2) Throughout this section, we fix C M , C M as is defined in Lemma 5.6 and Lemma 5.7 respectively. Definition 5.2. (1) We say that f ni,j satisfies E n if the following two statements hold:( A n ) k e f ni,j k L ∞ q ≤ (cid:16) κ + C M A ν △ tκ + A ν △ t (cid:17) n k f k L ∞ q ≤ e ( C M− AνTfκ + Aν △ t k f k L ∞ q , ( B n ) e f ni,j ≥ C , (cid:18) κκ + A ν △ t (cid:19) n e − C , | v j | α ≥ C , e − Aνκ T f e − C , | v j | α . (2) We say that f ni,j satisfies E n if the following two statements hold:( C n ) e ρ ni ≥ C , C α e − Aνκ T f , e T ni ≥ C , C α C M k f k L ∞ q ! / e − (cid:16) κ + C M− κ + Aν △ t (cid:17) A ν T f , ( D n ) k e ρ n k L ∞ x , k e U n k L ∞ x , k e T n k L ∞ x ≤ C q n (cid:0) C , C α (cid:1) − o e (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f k f k L ∞ q . (3) We define E n = E n ∧ E n . Remark . In fact, the first inequaly in ( A n ) implies the second inequality due to the elementaryinequality (1 + x ) n ≤ e nx . We stated them in this seemingly redundant manner since both estimatesare interchangeably used in the following proofs. ( B n ) is stated in such a redundant manner for thesame reason. Definition 5.4. (1) We define constants a , a and a by a = (cid:18) (cid:19) (cid:18) π (cid:19) ( C , C α ) C M k f k L ∞ q e − (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f ,a = (cid:18) π q − (cid:19) q − ( C , C q,α ) q − C q − q n (cid:0) C , C α (cid:1) − o q − k f k q − L ∞ q e − q − (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f ,a = (cid:18) (cid:19) q +6)3( q +3) (cid:18) q π (cid:19) q +3 ( C , C α ) q +33( q +3) (cid:8) C M (cid:9) q q +3) k f k q − q +3) L ∞ q e − q +33( q +3) (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f . The main goal of this section is the following.
Theorem 5.5.
Choose ℓ > sufficiently small such that △ v < ℓ implies C α ≤ X j e − C , | v j | α ( △ v ) ≤ C α , C q,α ≤ sup j (cid:8) (1 + | v j | ) q e −| v j | C α (cid:9) ≤ C q,α , C q − m ≤ X j ( △ v ) (1 + | v j | ) q − m ≤ C q − m . Now, define r △ v by r △ v = min { a , a , a , ℓ, / } , and suppose △ v is sufficiently small in the following sense: △ v < r △ v . Then, f ni,j satisfies E n for all n ≥ . We postpone the proof to the end of this section, after establishing several preliminary results. Webegin with the discrete moment estimates:
Lemma 5.6.
Let q > . Suppose f ni,j satisfies E n and △ v satisfies the smallness condition stated inTheorem 5.5. Then there exists a positive constant C M which depends only on q , such that (1) e ρ in ( e T ni ) ≤ C M k f n k L ∞ q , (2) e ρ ni ( e T ni + | e U ni | ) q − ≤ C M k f n k L ∞ q , (3) e ρ n | e U ni | q +3 [( e T ni + | e U ni | ) e T ni ] ≤ C M k f n k L ∞ q , (19) for all n .Proof. (1) We split the macroscopic density into to the following two parts: e ρ ni = X | v j − e U ni |
Then, we apply the H¨older inequality to bound I as I ≤ X | v j − e U ni |≤ r + △ v e f ni,j ( △ v ) − /q X | v j − e U ni |≤ r + △ v e f ni,j | v j | q ( △ v ) /q ≤ π /q { e ρ ni } − q k f k /qL ∞ q ( r + 2 △ v ) /q ≤ (8 π ) /q { e ρ ni } − /q k f k /qL ∞ q ( r + △ v ) /q . For II , we employ the Schwartz inequality to see that II ≤ r + △ v X | v j − e U ni |≥ r +2 △ v e f ni,j | v j − e U ni || v j | ( △ v ) ≤ r + 2 △ v X j e f ni,j | v j | ( △ v ) / X j e f ni,j | v j − U ni | ( △ v ) / ≤ r + △ v ne ρ ni (3 e T ni + | e U ni | ) o / n e ρ ni e T ni o / = 3 / e ρ ni r + △ v n e T ni + | e U ni | o / n e T ni o / . Therefore, e ρ ni | e U ni | ≤ (8 π ) /q { e ρ ni } − /q k f k q L ∞ q ( r + △ v ) /q + 3 / e ρ ni r + △ v n e T ni + | e U ni | o / n e T ni o / . We then derive the desired result by optimizing the above estimate by setting r as r + △ v = / { e ρ ni } /q (3 e T ni + | e U ni | ) / (cid:8) e T ni (cid:9) / (8 π ) /q k f n k /qL ∞ q q/ ( q +3) . The fact that f ni,j satisfies E n guarantees that the r.h.s is greater than or equal to a , which guaranteesthe existence of r > (cid:3) We now show that the ellipsoidal Gaussian is controlled by the discrete distribution in L ∞ q . Lemma 5.7.
Suppose f ni,j satisfies E n , and △ v < r △ v . Then we have kM ˜ ν ( ˜ f n ) k L ∞ q ≤ C M k f n k L ∞ q , for some constant C M which depends only on q and ν . ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 17
Proof.
We divide the proof into the two cases: q = 0 and q = 0.(1) q = 0: Since the exponential part is less than or equal to 1, we see from Lemma 4.3 that M ˜ ν,j ( ˜ f ni ) ≤ e ρ ni q det(2 π e T n ˜ ν,i ) ≤ C ˜ ν e ρ ni { e T ni } / . Then Lemma 5.6 (1) gives the desired estimate.(2) q = 0: We split v j as: | v j | q M ˜ ν,j ( ˜ f ni ) ≤ C q n | e U ni | q + | v j − e U ni | q o M ˜ ν,j ( ˜ f ni ) ≡ I + I . (i) The estimate for I : We first bound the exponential part by 1 to get I ≤ C ˜ ν | e U ni | q e ρ ni { e T ni } / . (a) | e U ni | > { e T ni } : Lemma 5.6 (3) gives: I ≤ C | e U ni | q e ρ ni { e T ni } ≤ C | e U ni | q +3 e ρ ni | e U ni | { e T ni } ≤ C | e U ni | q +3 e ρ ni (cid:8) e T ni + | e U ni | (cid:9) { e T ni } ≤ C ν,q k ˜ f n k L ∞ q . (b) | e U ni | ≤ { e T ni } : In this case, we employ Lemma 5.6 (2) as I ≤ { e T ni } q e ρ ni { e T ni } / ≤ e ρ ni { e T ni } q − ≤ e ρ ni (cid:8) e T ni + | e U ni | (cid:9) q − ≤ C q k ˜ f n k L ∞ q . (ii) The estimate for I : By Lemma 4.3, we have I ≤ C ˜ ν | v j − e U ni | q e ρ ni { e T ni } / exp − C ν | v j − e U ni | e T ni ! = C ˜ ν e ρ ni { e T ni } / { e T ni } q | v j − e U ni | e T ni ! q exp − C ν | v j − e U ni | e T ni ! ≤ C ˜ ν e ρ ni { e T ni } q − . In the last line, we used the elementary inequality | x a e − bx | ≤ C a,b for some positive C a,b ( a, b, x > e ρ ni { e T ni } q − ≤ e ρ ni (cid:8) e T ni + | e U ni | (cid:9) q − ≤ C q k e f n k L ∞ q . This completes the proof. (cid:3)
Lemma 5.8.
Let △ v < r △ v . Assume that f satisfies the assumptions of Theorem 3.1. Then f satisfies E .Proof. • ( A ) Thanks to the assumption on the initial data, we have k f k L ∞ q < ∞ . Therefore, in viewof Lemma 4.2, we have k ˜ f k L ∞ q ≤ k f k L ∞ q ≤ k f k L ∞ q < ∞ . • ( B ) We recall the lower bound assumption imposed on the initial data in Theorem 3.1 to see that e f i,j = f ( x i − v j △ t, v j ) ≥ C , e − C , | v j | α . • ( C ) Using the lower bound on the initial data again, one finds e ρ i = Z R f ( x i − v △ t, v ) dv ≥ C , Z R e − C , | v | α dv = C , C α > . For the upper bound for e ρ i , we decompose the integral domain as e ρ i = Z R f ( x i − v △ t, v ) dv ≤ Z | v − e U i |≤ r f ( x i − v △ t, v ) dv + Z | v − e U i | >r f ( x i − v △ t, v ) dv ≤ π k f k L ∞ q r + 3 r e ρ i e T i and optimize r with r = e ρ i e T i π k f k L ∞ q ! / to get e T i ≥ (cid:18) π (cid:19) / e ρ i k f k L ∞ q ! / ≥ (cid:18) π (cid:19) / C , C α k f k L ∞ q ! / . If necessary, we can replace C M in Lemma 4.4 by max n C M , √ π o to get the desired result. • ( D ) Lemma 5.10 gives e ρ i = Z R f ( x i − v △ t, v ) dv ≤ k f k L ∞ q Z R | v | ) q = C q k f k L ∞ q . The estimate for e U i follows from (cid:12)(cid:12) e U i (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e ρ i Z R f ( x i − v △ t, v ) vdv (cid:12)(cid:12)(cid:12) ≤ { C α C , } − k f k L ∞ q Z R | v | ) q − dv = C q − { C α C , } − k f k L ∞ q . For the estimate of e T ni , we compute3 e T i = 1 e ρ i Z R f ( x i − v △ t, v ) | v | dv − | e U i | ≤ e ρ i Z R f ( x i − v △ t, v ) | v | dv ≤ e ρ i k f k L ∞ q Z R | v | ) q − = C q − { C , C α } − k f k L ∞ q . This completes the proof for E . (cid:3) Lemma 5.9.
Assume f n − i,j satisfies E n − . Then, f ni,j satisfies B n : e f ni,j ≥ C , (cid:18) κκ + A ν △ t (cid:19) n e − C , | v j | α ≥ C , e − Aνκ T f e − C , | v j | α , for all i, j . From this, we also have k e f n k L ∞ q ≥ C , C q,α e − Aνκ T f . ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 19
Proof.
Since f n − i,j satisfies E n − , M e ν,j ( e f ni ) is strictly positive. Therefore, we have from (14) f ni,j ≥ κκ + A ν △ t e f n − i,j , which, in view of the definition of B n − , gives f ni,j ≥ C , (cid:18) κκ + A ν △ t (cid:19) (cid:18) κκ + A ν △ t (cid:19) n − e − C | v j | α = C , (cid:18) κκ + A ν △ t (cid:19) e − C | v j | α . (21)This immediately leads to the same lower bound estimate for ˜ f ni,j :˜ f ni,j = a j f ns ( i,j ) ,j + (1 − a j ) f ns ( i,j )+1 ,j ≥ C , (cid:18) κκ + A ν △ t (cid:19) n e − C , | v j | α with a j = a s ( n ) j . We suppressed the dependence on n , which is justified by Lemma 4.4. Then weemploy the following elementary inequality (1 + x ) n ≤ e nx , ( x ≥
0) to derive (cid:18) κκ + A ν △ t (cid:19) n = (cid:18) A ν △ tκ (cid:19) − n ≥ e − nA ν △ tκ ≥ e − Aνκ T f , where we used n △ t ≤ N t △ t = T f . The second estimate follows directly from this: k e f ni,j k L ∞ q ≥ C , e − Aνκ T f sup j (cid:8) (1 + | v j | ) q e − C , | v j | α (cid:9) ≥ C , C q,α e − Aνκ T f This completes the proof. (cid:3)
Lemma 5.10.
Assume f n − i,j satisfies E n − . Then, f ni,j satisfies A n : k f n k L ∞ q ≤ (cid:16) κ + C M A ν △ tκ + A ν △ t (cid:17) n k f k L ∞ q ≤ e ( C M− AνTfκ + Aν △ t k f k L ∞ q . Proof.
Applying Lemma 5.7 and to (14), one finds k f n k L ∞ q ≤ κκ + A ν △ t k e f n − k L ∞ q + A ν △ tκ + A ν △ t kM ˜ ν ( e f n − ) k L ∞ q ≤ κκ + A ν △ t k e f n − k L ∞ q + A ν △ tκ + A ν △ t C M k e f n − k L ∞ q ≤ κ + C M A ν △ tκ + A ν △ t k e f n − k L ∞ q . We then recall A n − to bound this further by (cid:16) κ + C M A ν △ tκ + A ν △ t (cid:17)(cid:16) κ + C M A ν △ tκ + A ν △ t (cid:17) n − k f k L ∞ q ≤ (cid:16) κ + C M A ν △ tκ + A ν △ t (cid:17) n k f k L ∞ q . The second estimate follows from (cid:16) κ + C M A ν △ tκ + A ν △ t (cid:17) n ≤ (cid:16) C M − A ν △ tκ + A ν △ t (cid:17) n ≤ e ( C M− Aν n △ tκ + Aν △ t ≤ e ( C M− Aν Tfκ + Aν △ t , where we used (1 + x ) n ≤ e nx and n △ t ≤ N t △ t = T f . (cid:3) Using this, we can prove the uniform lower bound of the macroscopic fields:
Lemma 5.11.
Assume f ni,j satisfies A n ∧ B n . Then, f ni,j satisfies C n : e ρ ni ≥ C , C α e − Aνκ T f , e T ni ≥ C , C α C M k f k L ∞ q ! / e − (cid:18) κ + ( Cq,ν − Tfκ + Aν △ t (cid:19) A ν T f . Note also that Lemma 4.3 then immediately yields the lower bound for e T n ˜ ν : e T n ˜ ν ≥ C M C , C α min { − ν, ν } e − (cid:18) κ + ( C M− Tfκ + Aν △ t (cid:19) A ν T f Id.
Proof.
For lower bound control for the discrete local density, we multiply e f ni,j by ( △ v ) and sum over j to get e ρ ni = X j ˜ f ni,j ( △ v ) ≥ C , e − Aνκ T f X j e − C , | v j | α ( △ v ) ≥ C α C , e − Aνκ T f . This, together with Lemma 5.6 (1) and Lemma 5.10 gives the lower bound for e T ni : e T ni ≥ e ρ ni C M k ˜ f n k L ∞ q ! / ≥ C , C α C M k f k L ∞ q e − (cid:18) κ + ( Cq,ν − T fκ + Aν △ t (cid:19) A ν T f ! / . Then the lower bound for e T ni follows from the equivalence estimate in Lemma 4.3. (cid:3) Lemma 5.12.
Assume f ni,j satisfies A n ∧ B n . Then, f ni,j satisfies D n : k e ρ n k L ∞ x , k e U n k L ∞ x , k e T n k L ∞ x ≤ C q n (cid:0) C , C α (cid:1) − o e (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f k f k L ∞ q . Proof.
Lemma 5.10 gives e ρ ni = X j e f ni,j ( △ v ) ≤ k f n k L ∞ q X j ( △ v ) (1 + | v j | ) q ≤ C q e ( C M− AνTfκ + Aν △ t k f k L ∞ q . We combine this with the lower bound estimates of e ρ ni established in Lemma 5.11 and the upperbound of the discrete solution in Lemma 5.10 to obtain (cid:12)(cid:12) e U ni (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) e ρ ni X j ˜ f ni,j v j ( △ v ) (cid:12)(cid:12)(cid:12) ≤ { C α C , } − e Aνκ T f k f n k L ∞ q X j ( △ v ) (1 + | v j | ) q − ≤ C q − { C α C , } − e (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f k f k L ∞ q . The estimate for e T ni follows from3 e T ni = 1 e ρ ni X ˜ f ni,j | v j | ( △ v ) − | e U ni | ≤ e ρ ni X ˜ f ni,j | v j | ( △ v ) ≤ e ρ ni k f n k L ∞ q X j ( △ v ) (1 + | v j | ) q − ≤ C q − { C , C α } − e (cid:16) κ + ( C M− κ + Aν △ t (cid:17) A ν T f k f k L ∞ q by a similar manner. (cid:3) Proof of Theorem 5.5.
Due to Lemma 5.8 , E holds. Assume E n − is satisfied. Then Lemma5.9, 5.10, 5.11 and 5.12 respectively show that f ni,j satisfies A n , B n , C n , D n , that is E n . Therefore,we can conclude that f ni,j satisfies E n for all n ≥ ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 21 Consistent form
In this section, we transform the ES-BGK model (1) into a form which is consistent to our scheme(14). We use the following notation for continuous solutions: e f ( x, v, t ) = f ( x − v △ t, v, t ) . Theorem 6.1.
Under the assumption of Theorem 3.1, (1) can be represented in the following form: f ( x, v, t + △ t ) = κκ + A ν △ t e f ( x, v, t ) + A ν △ tκ + A ν △ t M ˜ ν ( e f )( x, v, t )+ A ν κ + A ν △ t (cid:8) R + R (cid:9) , (22) where R = Z t + △ tt n M ν ( f )( x, v, t ) − M e ν ( ˜ f )( x, v, t ) o ds − Z t + △ tt (cid:8) ( t + △ t − s ) ∂ x M ( x θ , v, t θ ) ds + ( s − t ) ∂ t M ν ( f )( x θ , v, t θ ) (cid:9) ds,R = Z t + △ tt ( s − t − △ t ) (cid:0) M ( f ) − f (cid:1) ( x θ , v, t θ ) ds for some ( x θ i , v, t θ i ) ( i = 1 , .Proof. Along the characteristic line, (1) reads dfdt ( x + v t, v, t ) = 1 κ A ν ( M ν ( f ) − f )( x + v t, v, t ) . Integrating on [ t, t + △ t ], we get f ( x, v, t + △ t ) = f ( x − △ tv , v, t ) + 1 κ A ν Z t + △ tt ( M ν ( f ) − f )( x − ( t + △ t − s ) v , v, s ) ds ≡ f ( x − △ tv , v, t ) + 1 κ A ν ( I − I ) . By Taylor’s theorem around ( x − △ tv , v, t ), we see that there exist x θ which lies between x and x − ( t + △ t − s ) and t θ ∈ [ s, t ] such that M ν ( f )( x − ( t + △ t − s ) v , v, s )= M ν ( f )( x, v, t ) − ( t + △ t − s ) ∂ x M ν ( f )( x θ , v, t θ )+ ( s − t ) ∂ t M ν ( f )( x θ , v, t θ )= M ˜ ν ( ˜ f )( x, v, t ) + (cid:8) M ν ( f )( x, v, t ) − M ˜ ν ( ˜ f )( x, v, t ) (cid:9) − ( t + △ t − s ) ∂ x M ν ( f )( x θ , v, t θ )+ ( s − t ) ∂ t M ν ( f )( x θ , v, t θ ) . Therefore, we have(23) I = △ t M e ν ( x − △ tv , v, t ) + R . On the other hand, by Taylor expansion around ( x, v, t + △ t ), we get f ( x − ( t + △ t − s ) v, v, s )= f ( x, v, t + △ t ) + ( s − t − △ t ) (cid:0) ∂ t + v · ∇ x (cid:1) f ( x θ , v, t θ )= f ( x, v, t + △ t ) + 1 κ ( s − t − △ t ) {M ν ( f ) − f } ( x θ , v, t θ ) for some x θ lies between x and x − ( t + △ t − s ) and t θ ∈ [ s, t ]. Therefore, I can be rewritten as I = △ tf ( x, v, t + △ t ) + R . (24)Substituting (23) and (24) into (22), we get f ( x, v, t + △ t ) = e f ( x, v, t ) + A ν △ tκ M ˜ ν ( ˜ f )( x, v, t ) − A ν △ tκ f ( x, v, t + △ t ) + A ν κ (cid:8) R − R (cid:9) . We then collect relevant terms to derive the desired result. (cid:3)
We now estimate the remainder terms. First, we need the following estimates.
Lemma 6.2. [51]
Let f be solution in Theorem 3.1 corresponding to the initial data f . Then wehave for q ≥ kM ν ( f ) − M ν ( g ) k L ∞ q ≤ C q k f − g k L ∞ q , X ≤| α | + | β |≤ k ∂ αβ M ν ( f ) k L ∞ ≤ C T f (cid:8) k f k W , ∞ q + 1 (cid:9) . Lemma 6.3. R , R satisfy the following estimate: k R k L ∞ q + k R k L ∞ q ≤ C T f ,f ( △ t ) . (25) Proof.
We start with R . We decompose M ˜ ν ( ˜ f ) − M ν ( f ) = n M ˜ ν ( ˜ f ) − M ν ( ˜ f ) o + n M ν ( ˜ f ) − M ν ( f ) o ≡ I + II.
For I , we compute I = (˜ ν − ν ) ∂ T ν ∂ν ∂ M ν ∂ T ν = − νA ν △ tκ + A ν △ t (cid:8) e T Id − e Θ (cid:9) ∂ M ν ∂ T ν . Then, since | T Id − Θ | ≤ ρ Z R f (cid:12)(cid:12) | v − U | Id + ( v − U ) ⊗ ( v − U ) (cid:12)(cid:12) dv ≤ Cρ Z R f | v − U | dv = Cρ (cid:26)Z R f | v | dv + ρ | U | (cid:27) ≤ Cρ − k f k L ∞ q + | U | , the lower and upper bound estimates of the macroscropic field given in Theorem 3.1 yield | T Id − Θ | ≤ C T f . On the other hand, it was derived in [51] that (cid:13)(cid:13)(cid:13) ∂ M ν ∂ T ν (cid:13)(cid:13)(cid:13) L ∞ q ≤ C T f . Therefore we can estimate I as k I k L ∞ q ≤ C T f ,κ,ν △ t. For II , we first recall Lemma 6.2 to deduce k II k L ∞ q ≤ C k ˜ f − f k L ∞ q . Then we apply the mean value theorem to estimate k f − ˜ f k L ∞ q ≤ k△ tv · ∇ x f k L ∞ q ≤ C q k f k W , ∞ q +1 △ t ≤ C T f n k f k W , ∞ q +1 + 1 o △ t to get k II k L ∞ q ≤ C T f n k f k W , ∞ q +1 + 1 o △ t. ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 23
We combine these estimates to obtain kM ˜ ν ( ˜ f ) − M ν ( f ) k L ∞ q ≤ C T f ,f △ t, from which we can estimate (cid:13)(cid:13)(cid:13) Z t + △ tt M ˜ ν ( ˜ f ) − M ν ( f ) ds (cid:13)(cid:13)(cid:13) L ∞ q ≤ Z t + △ tt (cid:13)(cid:13) M ˜ ν ( ˜ f ) − M ν ( f ) (cid:13)(cid:13) L ∞ q ds ≤ C T f ,f ( △ t ) . On the other hand, again from Lemma 6.2, (cid:13)(cid:13)(cid:13) Z t + △ tt ( t + △ t − s ) ∂ x M ν ( f ) ds (cid:13)(cid:13)(cid:13) L ∞ q ≤ k ∂ x M ν ( f ) k L ∞ q Z t + △ tt ( t + △ t − s ) ds ≤ C f ,q ( △ t ) , and (cid:13)(cid:13)(cid:13) Z t + △ tt ( s − t ) ∂ t M ν ( f ) ds (cid:13)(cid:13)(cid:13) L ∞ q ≤ k ∂ t M ν ( f ) k L ∞ q Z t + △ tt ( s − t ) ds ≤ C f ,q ( △ t ) . This gives the desired remainder estimate for R . We compute R similarly as (cid:13)(cid:13)(cid:13) Z t + △ tt ( s − t − △ t ) (cid:0) M ν ( f ) − f (cid:1) ( x θ , v, t θ ) ds (cid:13)(cid:13)(cid:13) L ∞ q ≤ kM ν ( f ) − f k L ∞ q Z t + △ tt ( s − t − △ t ) ds ≤ C T f k f k L ∞ q ( △ t ) . (cid:3) Estimate of M e ν ( ˜ f ( t n )) − M e ν ( ˜ f n )The main purpose of this section is to establish the continuity estimate of the discrete ellipsoidalGaussian. (See Proposition 7.1). Lemma 7.1.
Assume q > and △ v < r △ v . Let f and f n denote the solution of (1) and (14)respectively. Then, the difference of ˜ f n and ˜ f ( t n ) satisfies k ˜ f n − ˜ f ( t n ) k L ∞ q ≤ k f n − f ( t n ) k L ∞ q + C T f k f k W , ∞ q ( △ x ) . Proof. (1) We expand ˜ f ( x i , v, t n ) = f ( x i −△ tv, v, t n ) in the following two ways using Taylor’s theorem: f ( x s , v j , t n ) = ˜ f ( x i , v j , t n ) + ( x s − x i + v j △ t ) g ∂ x f ( x i , v j , t n ) + 12 ( x s − x i + v j △ t ) g ∂ x f ( x θ , v j , t n ) ,f ( x s +1 , v j , t n ) = ˜ f ( x i , v j , t n ) + ( x s +1 − x i + v j △ t ) g ∂ x f ( x i , v j , t n ) + 12 ( x s +1 − x i + v j △ t ) g ∂ x f ( x θ , v j , t n ) , for some x s < x θ < x i + v j △ t and x i − v j △ t < x θ < x s +1 . Making a convex combination of abovetwo identities, we get (for the definition of a j , see Lemma 4.4.)˜ f ( x i , v, t n ) = (1 − a j ) f ( x s , v j , t n ) + a j f ( x s +1 , v j , t n ) + R θ where R θ is given by R θ = 12 ( x s +1 − x i + v j △ t ) g ∂ x f ( x θ , v j , t n ) + 12 ( x s +1 − x i + v j △ t ) g ∂ x f ( x θ , v j , t n ) . (26)Note that the first order derivitives cancelled each other due to the definition of a j :( x s − x i + v j △ t ) = − a j △ x, ( x s +1 − x i + v j △ t ) = (1 − a j ) △ x. (27)We then use | x s − ( x i − v j △ t ) | , | x s +1 − ( x i − v j △ t ) | ≤ △ x (28) together with | g ∂ x f ( x θ i , v j , t n ) | , ≤ k f k W , ∞ q (1 + | v j | ) q ≤ C T f k f k W , ∞ q + 1(1 + | v j | ) q , to estimate R θ as | R θ | ≤ C T f k f k W , ∞ q (1 + | v j | ) q ( △ x ) . (29)With these estimates, we can compute the difference of e f ni,j and e f ( x i , v, t n ) as follows: | e f ni,j − e f ( x i , v j , t n ) | ≤ | (1 − a j ) f ns,j + a j f ns +1 ,j − (1 − a j ) f ( x s , v j , t n ) − a j f ( x s +1 , v j , t n ) | + | R θ |≤ (1 − a j ) | f ns,j − f ( x s , v j , t n ) | + a j | f ns +1 ,j − f ( x s +1 , v j , t n ) | + | R θ | . We then multiply (1 + | v j | ) q on both sides and take supremum over i, j . The desired result followsfrom (26). (cid:3) Lemma 7.2.
Assume q > and |△ v | < r △ v . Let f and f n denote the solution of (1) and (14)respectively. Let φ ( v ) denote one of , v, | v | , (1 − ν ) | v | Id + νv ⊗ v for v ∈ R . Then we have (cid:12)(cid:12)(cid:12) X j e f ni,j φ ( v j )( △ v ) − Z R e f ( x i , v, t n ) φ ( v ) dv (cid:12)(cid:12)(cid:12) ≤ C T f k f n − f ( t n ) k L ∞ q + C T f k f k W , ∞ q (cid:8) ( △ x ) + △ v + △ v △ t (cid:9) . Proof.
For simplicity, we define △ j = [ v j , v j +1 ] × [ v j , v j +1 ] × [ v j , v j +1 ] , so that Z R e f ( x i , v, t n ) φ ( v ) dv = X j Z △ j e f ( x i , v, t n ) φ ( v ) dv. Therefore, X j e f ni,j φ ( v j )( △ v ) − Z R e f ( x i , v, t n ) φ ( v ) dv = X j e f ni,j φ ( v j )( △ v ) − X j Z △ j e f ( x i , v, t n ) φ ( v ) dv = n X j e f ni,j φ ( v j )( △ v ) − X j Z △ j e f ( x i , v, t n ) φ ( v j ) dv o + X j Z △ j e f ( x i , v, t n ) (cid:8) φ ( v j ) − φ ( v ) (cid:9) dv ≡ I + II. • (The estimate of I ) : Assume v ∈ △ j and expand ˜ f ( x i , v, t n ) = f ( x i − v △ t, v, t n ) in thefollowing two ways: f ( x s , v j , t n ) = ˜ f ( x i , v, t n ) + ( x s − x i + v △ t ) g ∂ x f ( x i , v, t n ) + 12 ( x s − x i + v △ t ) g ∂ x f ( z θ s, )+ ( v − v j ) · ∇ v ˜ f ( z θ s, ) , ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 25 and f ( x s +1 , v j , t n ) = ˜ f ( x i , v, t n ) + ( x s +1 − x i + v △ t ) g ∂ x f ( x i , v, t n ) + 12 ( x s +1 − x i + v △ t ) g ∂ x f ( z θ s +1 , )+ ( v − v j ) · g ∇ v f ( z θ s +1 , ) , where z θ s,i and z θ s +1 ,i ( i = 1 ,
2) denote the mean value points defined similarly as in the previouscase. We rewrite f ( x s , v j , t n ) and f ( x s +1 , v j , t n ) as f ( x s , v j , t n ) = ˜ f ( x i , v, t n ) + ( x s − x i + v j △ t ) g ∂ x f ( x i , v, t n ) + 12 ( x s − x i + v △ t ) g ∂ x f ( z θ s, )+ ( v − v j ) △ t g ∂ x f ( x i , v, t n ) + ( v − v j ) · g ∇ v f ( z θ s, ) ,f ( x s +1 , v j , t n ) = ˜ f ( x i , v, t n ) + ( x s +1 − x i + v j △ t ) g ∂ x f ( x i , v, t n ) + 12 ( x s +1 − x i + v △ t ) g ∂ x f ( z θ s +1 , )+ ( v − v j ) △ t g ∂ x f ( x i , v, t n ) + ( v − v j ) · f ∇ v ˜ f ( z θ s +1 , ) , and make a linear combination and cancel out the first order derivatives in x using (27) as in the proofof the previous lemma, to get˜ f ( x i , v, t n ) = (1 − a j ) f ( x s , v j , t n ) + a j f ( x s +1 , v j , t n ) + R θ (30)where R θ = − ( v − v j ) △ t g ∂ x f ( x i , v, t n )+ 12 (1 − a j )( x s − x i + v △ t ) ˜ ∂ x f ( z θ s, ) + 12 a j ( x s − x i + v △ t ) g ∂ x f ( z θ s +1 , )+ (1 − a j )( v − v j ) · g ∇ v f ( z θ s, ) + a j ( v − v j ) · g ∇ v f ( z θ s +1 , )From this, we see that X j Z △ j e f ( x i , v, t n ) φ ( v j ) dv = X j Z △ j { (1 − a j ) f ( x s , v j , t n ) + a j f ( x s +1 , v j , t n ) + R θ } φ ( v j ) dv = X j { (1 − a j ) f ( x s , v j , t n ) + a j f ( x s +1 , v j , t n ) } ( △ v ) φ ( v j )+ X j Z △ j R θ φ ( v j ) dv, so that (cid:12)(cid:12)(cid:12) X j e f ni,j φ ( v j )( △ v ) − Z R e f ( x i , v, t n ) φ ( v ) dv (cid:12)(cid:12)(cid:12) ≤ X j (cid:12)(cid:12) e f ni,j − (cid:8) (1 − a j ) f ( x s , v j , t n ) + a j f ( x s +1 , v j , t n ) (cid:9)(cid:12)(cid:12) | φ ( v j ) | ( △ v ) + X j Z △ j | R θ || φ ( v j ) | dv = I + I . We first estimate I since we have from the definition of e f i,j , | e f ni,j − (1 − a j ) f ( x s , v j , t n ) − a j f ( x s +1 , v j , t n ) | = | (1 − a j ) f ns,j + a j f ns +1 ,j − (1 − a j ) f ( x s , v j , t n ) − a j f ( x s +1 , v j , t n ) |≤ (1 − a j ) | f ns,j − f ( x s , v j , t n ) | + a j | f ns +1 ,j − f ( x s +1 , v j , t n ) | , I can be estimated as follows: I = X j (cid:12)(cid:12) e f ni,j − (cid:8) (1 − a j ) f ( x s , v j , t n ) + a j f ( x s +1 , v j , t n ) (cid:12)(cid:12) | φ ( v j ) | ( △ v ) ≤ X j n (1 − a j ) k f n − f ( t n ) k L ∞ q + a j k f n − f ( t n ) k L ∞ q o | φ ( v j ) | ( △ v ) (1 + | v j | ) q ≤ n X j φ ( v j )( △ v ) (1 + | v j | ) q o k f n − f ( t n ) k L ∞ q ≤ C k f n − f ( t n ) k L ∞ q , where we used φ ( v j ) ≤ C (1 + | v j | ) p , p = 0 , , q − p > I , we first observe from the definition of x s and the fact that v ∈ △ j , | x i − v △ t − x s | ≤ | x i − v j △ t − x s | + 2 | ( v − v j ) △ t | ≤ (cid:8) ( △ x ) + ( △ v ) ( △ t ) (cid:9) . Similarly, | x s +1 − ( x i − v △ t ) | ≤ (cid:8) ( △ x ) + ( △ v ) ( △ t ) (cid:9) . On the other hand, since △ v < /
2, we have ( i = 1 , X | α | + | β |≤ (cid:12)(cid:12) g ∂ αx f ( z θ i ) (cid:12)(cid:12) + (cid:12)(cid:12) g ∂ βv f ( z θ i ) (cid:12)(cid:12) ≤ k f k W , ∞ q (1 + | v j + θ i △ v | ) q ≤ k f k W , ∞ q (1 − |△ v | + | v j | ) q ≤ C T f k f k W , ∞ q + 1(1 + | v j | ) q , so that | R θ | ≤ C T f { ( △ x ) + ( △ v ) ( △ t ) + △ v + △ v △ t } k f k W , ∞ q + 1(1 + | v j | ) q ≤ C T f { ( △ x ) + △ v + △ v △ t } k f k W , ∞ q + 1(1 + | v j | ) q . (31)Hence we have I ≤ C T f { ( △ x ) + △ v + △ v △ t }k f k W , ∞ q X j | φ ( v j ) | ( △ v ) (1 + | v j | ) q ≤ C T f ,f { ( △ x ) + △ v + △ v △ t } , where we used R △ j dv = ( △ v ) . Therefore, we have the following estimate for II ≤ C k f n − f ( t n ) k L ∞ q + C T f ,f { ( △ x ) + △ v + △ v △ t } . • (The estimate of II ) : Since | v j | = | v + θ △ v | ≤ (1 + | v | ) in △ j , we have for v ∈ △ j | φ ( v ) − φ ( v j ) (cid:12)(cid:12) ≤ C | v − v j | (cid:8) | v | p + | v j | p (cid:9) ≤ C △ v (1 + | v | ) p . ( p = 0 , , ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 27
With this, we can estimate II as II ≤ C △ v X j Z △ j e f ( x i , v, t n )(1 + | v | ) p dv ≤ C △ v k f k L ∞ q X j Z △ j dv (1 + | v | ) q − p +1 ≤ C △ v k f k L ∞ q Z R dv (1 + | v | ) q − p +1 ≤ C q − p +1 k f k L ∞ q △ v, which, upon applying Lemma 6.2 yields II ≤ C T f k f k L ∞ q △ v. (32)Finally, we combine the estimates for I and II to get the desired result. (cid:3) Lemma 7.3.
Let △ v < r △ . Suppose f satisfies the assumptions in Theorem 3.1, then we have | e ρ ni − e ρ ( x i , t n ) | , | e U ni − e U ( x i , t n ) | , | e T nν,i − e T e ν ( x i , t n ) |≤ C k f n − f ( t n ) k L ∞ q + C T f ,f { ( △ x ) + △ v + △ v △ t } . Proof.
Note that e ρ ni − e ρ ( x i , t n ) = X j e f ni,j ( △ v ) − Z R e f ( x i , v, t n ) dv. Therefore, it’s a direct consequence of the Lemma 7.2. For the second estimate, we observe that e U ni − e U ( x i , t n ) = 1 e ρ ni (cid:8)e ρ ni e U ni − (cid:8)e ρ e U (cid:9) ( x i , t n ) (cid:9) − e U e ρ ni (cid:8)e ρ ni − e ρ ( x i , t n ) (cid:9) . We then recall the lower bound of the discrete density in Lemma 5.11: e ρ ni ≥ C C α e − Aνκ T f , to compute (cid:12)(cid:12)(cid:12)(cid:12) e U ni e ρ ni (cid:12)(cid:12)(cid:12)(cid:12) = 1( e ρ ni (cid:1) (cid:12)(cid:12)e ρ ni e U ni (cid:12)(cid:12) ≤ C α,T f X j | e f ni,j || v j | ( △ v ) ≤ C α,T f k f n k L ∞ q X j ( △ v ) (1 + | v j | ) q − ≤ C α,T f e ( C M− Tfκ + Aν △ t k f k L ∞ q . Here, we used Lemma 5.10. Hence we have | e U ni − e U ( x i , t n ) | ≤ C T f ,κ,f n | (cid:8)e ρ ni e U ni − (cid:8)e ρ e U (cid:9) ( x i , t n ) (cid:9) | + | e ρ ni − e ρ ( x i , t n ) | o ≤ C T f ,κ,f (cid:12)(cid:12)(cid:12) X j e f ni,j ( △ v ) − Z R e f ( x i , v, t n ) dv (cid:12)(cid:12)(cid:12) + C T f ,κ,f (cid:12)(cid:12)(cid:12) X j e f ni,j v j ( △ v ) − Z R e f ( x i , v, t n ) vdv (cid:12)(cid:12)(cid:12) , which, in view of Lemma 7.2, gives the desired result. For the estimate of the temperature tensor, werecall that e T n ˜ ν,i contains ˜ ν , and decompose it as e T n e ν,i − e T e ν ( x i , t n ) = (1 − e ν ) e T ni Id + e ν e Θ ni − n (1 − ν ) e T ( x i , t n ) Id + ν e Θ( x i , t n ) o = ( ν − e ν ) n e T ni Id − e Θ ni o + hn (1 − ν ) e T ni Id + ν e Θ ni o − n (1 − ν ) e T ( x i , t n ) Id + ν e Θ( x i , t n ) oi ≡ I + II.
Since | e ν − ν | = ν △ tκ + △ t , I is bounded by C T f κ △ tκ + △ t . The estimate of II can be carried out in the exactlysame manner as in the previous case, through a tedious computation, using the following identity: T ν,f − T ν,g = ρ − f { ( ρ T ν + ρU ⊗ U ) f − ( ρ T ν + ρU ⊗ U ) g } + ( ρ T ν + ρU ⊗ U ) g ρ f ρ g ( ρ f − ρ g )+ 1 ρ f { ρ f U f ⊗ ρ f U f − ρ g U g ⊗ ρ g U g } − ( ρ f + ρ g ) ρ f ρ g ( ρ g U g ) ( ρ f − ρ g ) . We omit the computation. (cid:3)
Proposition . Assume that k f n k L ∞ q < ∞ with q >
5. Then we have kM e ν ( e f ( t n )) − M e ν ( e f n ) k L ∞ q ≤ C T f k f n − f ( t n ) k L ∞ q + C T f { ( △ x ) + △ v + △ v △ t } . The constants depend only on q, ν, T f and f . Proof.
We note that M e ν ( ˜ f ( x i , t n ))( v j ) − M e ν,j ( e f ni )= M e ν (cid:0)e ρ ( x i , t n )) , e U ( x i , t n ) , e T ( x i , t n ) (cid:1) ( v j ) − M e ν (cid:0)e ρ ni , e U ni , e T ni (cid:1) ( v j ) . Therefore, applying the Taylor series, we expand this as M e ν ( ˜ f ( x i , t n ))( v j ) − M e ν,j ( e f ni ) = (cid:8)e ρ ( x i , t n ) − e ρ ni (cid:9) Z ∂ M e ν ( θ ) ∂ρ dθ + (cid:8) e U ( x i , t n ) − e U ni (cid:9) Z ∂ M e ν ( θ ) ∂U dθ + (cid:8) e T e ν ( x i , t n ) − e T n e ν,i (cid:9) Z ∂ M e ν ( θ ) ∂ T ν dθ ≡ I + I + I , (33)where ∂ M e ν ( θ ) ∂X = ∂ M e ν ∂X (cid:12)(cid:12)(cid:12) ( e ρ ni,θ , e U ni,θ , e T ni,θ ) . For simplicity of notation, we define transitional macroscopic fields e ρ nθi , e U nθi and e T nθi by( e ρ nθi , e U nθi , e T nθi ) = (1 − θ ) (cid:16)e ρ ( x i , t n ) , e U ( x i , t n ) , e T ν ( x i , t n ) (cid:17) + θ (cid:16)e ρ ni , e U ni , e T n e ν,i (cid:17) . Since this is a linear combination, we can derive the following estimates for the transitional macroscopicfields: k e ρ nθi k L ∞ x , k e U nθi k L ∞ x , k e T nθi k L ∞ x ≤ C T f , e ρ nθi ≥ C T f e − CT f , e T nθi ≥ C T f e − CT f ,k ⊤ (cid:8) e T nθi (cid:9) k ≥ C T f e − CT f | k | , k ∈ R . (34) ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 29 from the lower and upper bounds of continuous and discrete macroscopic fields given in Theorem 3.1,Proposition 5.11 and Proposition 5.12. On the other hand, Brum-Minkowski inequality implies thatdet (cid:8) e T nθi (cid:9) = det (cid:8) (1 − θ ) e T e ν ( x i , t n ) + θ e T n e ν,i (cid:9) ≥ det (cid:8) e T e ν ( x i , t n ) (cid:9) − θ det (cid:8) e T n e ν,i (cid:9) θ ≥ (cid:8) C T f e − CT f (cid:9) − θ (cid:8) C T f e − CT f (cid:9) θ = C T f e − CT f . (35)From these observations we have M e ν ( θ ) = e ρ nθi q det(2 π e T nθi ) exp (cid:18) −
12 ( v j − e U nθi ) ⊤ (cid:8) e T nθi (cid:9) − ( v j − e U nθi ) (cid:19) ≤ C T f exp (cid:16) − C T f | v j − e U nθi | (cid:17) ≤ C T f exp (cid:16) C T f | e U nθi | (cid:17) exp (cid:0) − C T f | v j | (cid:1) ≤ C T f , e − C Tf , | v j | . (36)We now estimate each integral in I i ( i = 1 , , I comes straightforwardly from (36): (cid:12)(cid:12)(cid:12)(cid:12)Z ∂ M e ν ( θ ) ∂ e ρ nθi ds (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z e ρ nθi M e ν ( θ ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ C T f , e − C Tf , | v j | . (37)For the integral in I , we compute (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂ M e ν ( θ ) ∂ e U nθi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ( v j − e U nθi ) ⊤ { e T nθi } − + { e T nθi } − ( v j − e U nθi ) (cid:12)(cid:12)(cid:12) M ˜ ν ( θ ) ≤ C T f , (cid:12)(cid:12)(cid:12) ( v j − e U nθi ) ⊤ { e T nθi } − + { e T nθi } − ( v j − e U nθi ) (cid:12)(cid:12)(cid:12) e − C Tf , | v j | . (38)In the last line, we used (36). For simplicity, set X = v j − e U nθi . Then, (cid:12)(cid:12) X ⊤ (cid:8) e T nθi (cid:9) − (cid:12)(cid:12) = sup | Y | =1 (cid:12)(cid:12) X ⊤ { e T nθi } − Y (cid:12)(cid:12) = 12 sup | Y | =1 (cid:12)(cid:12)(cid:12) ( X + Y ) ⊤ { e T nθi } − ( X + Y ) − X ⊤ { e T nθi } − X − Y ⊤ { e T nθi } − Y (cid:12)(cid:12)(cid:12) ≤ C sup | Y | =1 | X + Y | + | X | + 1 e T nθi ! ≤ C | v j − e U nθi | e T nθi ! , which is, by (34), bounded by C T f (1 + | v j | ) . Similarly, we can derive |{ e T nθi } − ( v j − e U nθi ) | ≤ C T f (1 + | v j | ) , (39)so that from (38) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z ∂ M e ν ( θ ) ∂ e U nθi ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C T f , (1 + | v j | ) e − C Tf , | v j | . (40) We now turn to the integral in I . We first observe ∂ M e ν ( θ ) ∂ e T nαβθi = − ( { det e T nθi } / ∂ det e T nθ ∂ e T nαβθi ) M ˜ ν ( θ )+ 12 ( ( v j − e U nθi ) ⊤ { e T nθi } − ∂ e T nθi ∂ e T nαβθi ! { e T nθ } − ( v j − e U nθi ) ) M ˜ ν ( θ ) . To estimate the first term, we observe through an explicit computation thatdet e T nθi = e T n θi e T n θi e T θi − e T n θi e T n θi e T n θi − e T n θi n e T n θi o − e T n θi n e T n θi o − e T n θi n e T n θi o . Therefore, ∂ det e T θ ∂ e T nαβθi is a homogeneous polynomial of degree 2 having e T nαβθi (1 ≤ α, β ≤
3) as variables.Now, recalling e T nαβθi ≤ Ce T f from (34), together with the lower bound of det e T nθi in (35), we conclude (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { det e T nθi } / ∂ det e T nθi ∂ e T nαβθi (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C T f . On the other hand, since all the entries of ∂ T θ ∂ T αβθ are 0 except for the α, β entry, we see that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( v − e U nθi ) ⊤ (cid:8) e T nθi (cid:9) − ∂ e T nθi ∂ e T nαβθi !(cid:8) e T nθi (cid:9) − ( v − e U nθi ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) ( v − e U nθα ) ⊤ { e T nθα } − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) { e T nθβ } − ( v − e U nθβ ) (cid:12)(cid:12)(cid:12) , which can be shown to be bounded by C T f ,ν (1 + | v j | ) using the same argument as in the previouscase. Combining these two estimates, we bound the integral in I as Z (cid:12)(cid:12)(cid:12) ∂ M e ν ( θ ) ∂ e T θ (cid:12)(cid:12)(cid:12) dθ ≤ C T f e − C Tf | v j | . We now insert all the above estimates into (33) to get |M n e ν,j ( ˜ f ni ) − M e ν ( ˜ f ( x i , t n ))( v j ) |≤ C T f n | e ρ ni − e ρ ( x i , t n ) | + | e U ni − e U ( x i , t n ) | + | e T n e ν,i − e T e ν ( x i , t n ) | o e − C Tf | v j | . Then, Lemma 7.3 gives the desired result. (cid:3) Proof of Theorem 3.2
We are now ready to prove our main theorem. Subtracting (14) from (22) and taking L ∞ q norms,we get k f ( t n +1 ) − f n +1 k L ∞ q ≤ κκ + A ν △ t k e f ( t n ) − e f n k L ∞ q + A ν △ tκ + A ν △ t kM e ν ( e f ( t n )) − M e ν ( e f n ) k L ∞ q + A ν κ + A ν △ t k R k L ∞ q + A ν κ + A ν △ t k R k L ∞ q . We then recall the estimates in Lemma 6.3, Lemma 7.1 and Proposition 7.1: k e f ( t n ) − e f n k L ∞ q ≤ k f ( t n ) − f n k L ∞ q + C T f ( △ x ) , kM e ν ( e f ( t n )) − M e ν ( e f n ) k L ∞ q ≤ C T f k f ( t n ) − f n k L ∞ q + C T f (cid:8) ( △ x ) + △ v + △ v △ t (cid:9) , k R k L ∞ q + k R k L ∞ q ≤ C T f k f k W , ∞ q ( △ t ) , to derive the following recurrence inequality for the numerical error: k f ( t n +1 ) − f n +1 k L ∞ q = (cid:16) C T f △ tκ + A ν △ t (cid:17) k f ( t n ) − f n k L ∞ q + C T f P ( △ x, △ v, △ t ) , (41) ONVERGENCE OF A SEMI-LAGRANGIAN SCHEME FOR THE ES-BGK MODEL 31 where P ( △ x, △ v, △ t ) = ( △ x ) + △ t { ( △ x ) + △ v + △ v △ t } + ( △ t ) κ + A ν △ t . Put Γ = C Tf △ tκ + A ν △ t for simplicity of notation and iterate (41) until the initial step is reached, to derive: k f ( N t △ t ) − f N t k L ∞ q ≤ (1 + Γ) N t k f − f k L ∞ q + C ν N t X i =1 (1 + Γ) i − P. (42)We first note from the definition of f i,j = f ( x i , v j ) that k f − f k L ∞ q = sup i,j | f ( x i , v j ) − f i,j | (1 + | v j | ) q = 0 . Besides, we use (1 + x ) n ≤ e nx to see that(1 + Γ) N t ≤ e N t Γ ≤ e CTf Nt △ tκ + Aν △ t = e CTf Tfκ + △ t , so that N t X i =1 (1 + Γ) i − = (1 + Γ) N t − − ≤ C T f (cid:18) κ + A ν △ t △ t (cid:19) (cid:26) e CTf Tfκ + Aν △ t − (cid:27) , which gives N t X i =1 (1 + Γ) i − P ≤ C T f (cid:18) e CTf Tfκ + Aν △ t − (cid:19) (cid:26) △ x + △ v + △ t + △ x △ t (cid:27) . Substituting these estimates into (42), we find k f ( T f ) − f N t k L ∞ q ≤ C T f (cid:18) e CTf Tfκ + △ t − (cid:19) (cid:26) ( △ x ) + △ v + △ t + ( △ x ) △ t (cid:27) . This completes the proof.
Acknowledgement
The work of S.-B. Yun was supported by Samsung Science and Technology Foundation underProject Number SSTF-BA1801-02.
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