Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities
aa r X i v : . [ m a t h . NA ] M a r ENTROPY-DISSIPATIVE DISCRETIZATIONOF NONLINEAR DIFFUSION EQUATIONSAND DISCRETE BECKNER INEQUALITIES
CLAIRE CHAINAIS-HILLAIRET, ANSGAR J ¨UNGEL, AND STEFAN SCHUCHNIGG
Abstract.
The time decay of fully discrete finite-volume approximations of porous-medium and fast-diffusion equations with Neumann or periodic boundary conditions isproved in the entropy sense. The algebraic or exponential decay rates are computed ex-plicitly. In particular, the numerical scheme dissipates all zeroth-order entropies whichare dissipated by the continuous equation. The proofs are based on novel continuous anddiscrete generalized Beckner inequalities. Furthermore, the exponential decay of somefirst-order entropies is proved in the continuous and discrete case using systematic inte-gration by parts. Numerical experiments in one and two space dimensions illustrate thetheoretical results and indicate that some restrictions on the parameters seem to be onlytechnical. Introduction
This paper is concerned with the time decay of fully discrete finite-volume solutions tothe nonlinear diffusion equation(1) u t = ∆( u β ) in Ω , t > , u ( · ,
0) = u in Ω , and with the relation to discrete generalized Beckner inequalities. Here, β > ⊂ R d ( d ≥
1) is a bounded domain. When β >
1, (1) is called the porous-medium equation,describing the flow of an isentropic gas through a porous medium [35]. Equation (1) with β < β = [5] or in semiconductor theory with 0 < β < ∇ ( u β ) · ν = 0 on ∂ Ω , t > , Date : October 9, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Porous-medium equation, fast-diffusion equation, finite-volume method, en-tropy dissipation, Beckner inequality, entropy construction method.The authors have been partially supported by the Austrian-French Project Amade´e of the Austrian Ex-change Service ( ¨OAD). The second and last author acknowledge partial support from the Austrian ScienceFund (FWF), grants P22108, P24304, I395, and W1245. Part of this work was written during the stay ofthe second author at the University of Technology, Munich (Germany), as a John von Neumann Professor.The second author thanks the Department of Mathematics for the hospitality and Jean Dolbeault (Paris)for very helpful discussions on Beckner inequalities. where ν denotes the unit normal exterior vector to ∂ Ω, or multiperiodic boundary con-ditions (i.e. Ω equals the torus T d ). Let us denote by m the Lebesgue measure in R d or R d − ; we assume for simplicity that m(Ω) = 1. For existence and uniqueness results forthe porous-medium equation in the whole space or under suitable boundary conditions, werefer to the monograph [35].In the literature, there exist many numerical schemes for nonlinear diffusion equationsrelated to (1). Numerical techniques include (mixed) finite-element methods [1, 15, 33],finite-volume approximations [19, 32], high-order relaxation ENO-WENO schemes [11],or particle methods [30]. In these references, also stability and numerical convergenceproperties are proved.The preservation of the structure of diffusion equations is a very important propertyof a numerical scheme. For instance, ideas employed for hyperbolic conservation lawswere extended to degenerate diffusion equations, like the porous-medium equation, whichmay behave like hyperbolic ones in the regions of degeneracy [31]. Positivity-preservingschemes for nonlinear fourth-order equations were thoroughly investigated in the context oflubrication-type equations [3, 37] and quantum diffusion equations [25]. Entropy-consistentfinite-volume finite-element schemes for the fourth-order thin-film equation were suggestedby Gr¨un and Rumpf [22]. For quantum diffusion models, an entropy-dissipative relaxation-type finite-difference discretization was investigated by Carrillo et al. [9]. Furthermore,entropy-dissipative schemes for electro-reaction-diffusion systems were derived by Glitzkyand G¨artner [20]. However, it seems that there does not exist any systematic study onentropy-dissipative discretizations for (1) and the time decay of their discrete solutions.Our first aim is to prove that the finite-volume scheme for (1)-(2), defined in (30),dissipates the discrete versions of the functionals E α [ u ] = 1 α + 1 (cid:18)Z Ω u α +1 dx − (cid:16) Z Ω udx (cid:17) α +1 (cid:19) , (3) F α [ u ] = 12 Z Ω |∇ u α/ | dx, α > . (4)In fact, we will prove (algebraic or exponential) convergence rates at which the discretefunctionals converge to zero as t → ∞ . We call E α a zeroth-order entropy and F α a first-order entropy. The functional F is known as the Fisher information, used in mathematicalstatistics and information theory [16]. Our analysis of the decay rates of the entropies willbe guided by the entropy-dissipation method. An essential ingredient of this technique is afunctional inequality relating the entropy to the entropy dissipation [2, 8]. For the diffusionequation (1), this relation is realized by the Beckner inequality [4].The entropy-dissipation method was applied to (1) in the whole space to prove the decayof the solutions to the asymptotic self-similar profile in, e.g., [10, 12]. The convergencetowards the constant steady state on the one-dimensional torus was proved in [7]. However,we are not aware of general entropy decay estimates for solutions to (1) to the constantsteady state, even in the continuous case. The reason might be that generalizations to theBeckner inequality, needed to relate the entropy dissipation to the entropy, are missing. As NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 3 our second aim, we prove these generalized Beckner inequalities and provide some decayestimates for E α and F α along trajectories of (1).This paper splits into two parts. The first part is concerned with the proof of generalizedBeckner inequalities and the decay rates for the continuous case. The second—and main—part is the “translation” of these results to an implicit Euler finite-volume discretizationof (1). In the following, we summarize our main results.The first result is the proof of the generalized Beckner inequality(5) Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq ≤ C B ( p, q ) k∇ f k qL (Ω) , where f ∈ H (Ω) and 0 < q ≤ pq ≥
1. In the case q = 2, we require that − d ≤ p ≤ C B ( p, q ) > p , q , and the constant of the Poincar´e-Wirtingerinequality (see Lemma 2 for details). The usual Beckner inequality [4] is recovered for q = 2; see Remark 3 for a comparison of related Beckner inequalities in the literature.The proof is elementary and only employs the Poincar´e-Wirtinger inequality. By using adiscrete version of this inequality (see [6]), the proof can be easily “translated” to derivethe discrete generalized Beckner inequality Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq ≤ C b ( p, q ) | f | q , , T , where f is a function which is constant on each cell of the finite-volume triangulation T ofΩ and | · | , , T is the discrete H -seminorm; see Section 3.1 and Lemma 12 for details.The second result is the time decay of the entropies E α and F α along trajectories of (1).Differentiating E α [ u ( t )] with respect to time and employing the Beckner inequality (5), weshow for β > dE α dt [ u ( t )] ≤ CE α [ u ( t )] ( α + β ) / ( α +1) , t > , where C > α , β , and C B ( p, q ). By a nonlinear Gronwall inequality, thisimplies the algebraic decay of u ( t ) to equilibrium in the entropy sense; see Theorem 5. Ifthe solution is positive and 0 < α ≤
1, the above inequality becomes dE α dt [ u ( t )] ≤ C ( u ) E α [ u ( t )] , t > , which results in an exponential decay rate; see Theorem 6.The first-order entropies F α [ u ( t )] decay exponentially fast (for positive solutions) for all( α, β ) lying in the strip − ≤ α − β ≤ M d ,which is illustrated in Figure 1 below (multi-dimensional case); see Theorems 7 and 8. Theproof is based on systematic integration by parts [23]. In order to avoid boundary integralsarising from the iterated integrations by parts, these results are valid only if Ω = T d .Notice that all these results are new.The third—and main—result is the “translation” of the continuous decay rates to thefinite-volume approximation. We obtain the same results for a discrete version of E α inTheorems 14 (algebraic decay) and 15 (exponential decay). The situation is different for C. CHAINAIS-HILLAIRET, A. J ¨UNGEL, AND S. SCHUCHNIGG the first-order entropies F α . The reason is that it is very difficult to “translate” the iteratedintegrations by parts to iterated summations by parts since there is no discrete nonlinearchain rule. For the zeroth-order entropies, this is done by exploiting the convexity of themapping x x α +1 . For the first-order entropies, we employ the concavity of the discreteversion of dF α /dt with respect to the time approximation parameter. We prove in Theorem16 that for 1 ≤ α ≤ β = α/
2, the discrete first-order entropy is monotone (multi-dimensional case) and decays exponentially fast (one-dimensional case). We stress the factthat this is the first result in the literature on the decay of discrete first-order entropies.Throughout this paper, we assume that the solutions to (1) are smooth and positivesuch that we can perform all the computations and integrations by parts. In particular,we avoid any technicalities due to the degeneracy ( β >
1) or singularity ( β <
1) in (1).Most of our results can be generalized to nonnegative weak solutions by using a suitableapproximation scheme but details are left to the reader.The paper is organized as follows. In Section 2, we investigate the continuous case.We prove two novel generalized Beckner inequalities in Section 2.1, the algebraic andexponential decay of E α [ u ] in Section 2.2, and the exponential decay of F α [ u ] in Section2.3. The discrete situation is analyzed in Section 3. After introducing the finite-volumescheme in Section 3.1, the algebraic and exponential decay rates for the discrete version of E α [ u ] is shown in Section 3.3, and the exponential decay of the discrete version of F α [ u ]is proved in Section 3.4. In Section 4, we illustrate the theoretical results by numericalexperiments in one and two space dimensions. They indicate that some of the restrictionson the parameters ( α, β ) seem to be only technical. In the appendix, a discrete nonlinearGronwall lemma and some auxiliary inequalities are proved.2. The continuous case
It is convenient to analyze first the continuous case before extending the ideas to thediscrete situation. We prove new convex Sobolev inequalities and algebraic and exponentialdecay rates of the solutions to (1).2.1.
Generalized Beckner inequalities.
We assume in this subsection that Ω ⊂ R d ( d ≥
1) is a bounded domain such that the Poincar´e-Wirtinger inequality(6) k f − ¯ f k L (Ω) ≤ C P k∇ f k L (Ω) for all f ∈ H (Ω) holds, where ¯ f = m(Ω) − R Ω f dx and C P > d andΩ. This is the case if, for instance, Ω has the cone property [29, Theorem 8.11] or if ∂ Ω islocally Lipschitz continuous [36, Theorem 1.3.4]. Suppose that m(Ω) = 1 (to shorten theproof). Before stating our main result, we prove the following lemma.
Lemma 1 (Generalized Poincar´e-Wirtinger inequality) . Let < q ≤ and f ∈ H (Ω) .Then (7) k f k qL (Ω) ≤ C qP k∇ f k qL (Ω) + k f k qL q (Ω) holds, where C P > is the constant of the Poincar´e-Wirtinger inequality (6) . NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 5
Proof.
Let first 1 ≤ q ≤
2. The Poincar´e-Wirtinger inequality (6)(8) k f k L (Ω) − k f k L (Ω) = k f − ¯ f k L (Ω) ≤ C P k∇ f k L (Ω) together with the H¨older inequality leads to(9) k f k L (Ω) ≤ C P k∇ f k L (Ω) + k f k L q (Ω) . Here we use the assumption m(Ω) = 1. Since q/ ≤
1, it follows that k f k qL (Ω) ≤ (cid:0) C P k∇ f k L (Ω) + k f k L q (Ω) (cid:1) q/ ≤ C qP k∇ f k qL (Ω) + k f k qL q (Ω) , which equals (7).Next, let 0 < q <
1. We claim that(10) a q/ − a q − b − q/ ≤ ( a − b ) q/ for all a ≥ b > . Indeed, setting c = b/a , this inequality is equivalent to1 − c − q/ ≤ (1 − c ) q/ for all 0 < c ≤ . The function g ( c ) = 1 − c − q/ − (1 − c ) q/ for c ∈ [0 ,
1] satisfies g (0) = g (1) = 0 and g ′′ ( c ) = ( q/ − q/ c − − q/ + (1 − c ) q/ − ) ≥ c ∈ (0 , g ( c ) ≤
0, proving (10). Applying (10) to a = k f k L (Ω) and b = k f k L (Ω) and using (8), wefind that(11) k f k qL (Ω) − k f k q − L (Ω) k f k − qL (Ω) ≤ (cid:0) k f k L (Ω) − k f k L (Ω) (cid:1) q/ ≤ C qP k∇ f k qL (Ω) . In order to get rid of the L norm, we employ the interpolation inequality(12) k f k L (Ω) = Z Ω | f | θ | f | − θ dx ≤ k f k θL q (Ω) k f k − θL (Ω) , where θ = q/ (2 − q ) <
1. Since (2 − q ) θ = q and (2 − q )(1 − θ ) = 2(1 − q ), (11) becomes k f k qL (Ω) − k f k qL q (Ω) ≤ C qP k∇ f k qL (Ω) , which is the desired inequality. (cid:3) Lemma 2 (Generalized Beckner inequality I) . Let d ≥ , < q < , pq ≥ or q = 2 , − d ≤ p ≤ < p ≤ if d ≤ , and let f ∈ H (Ω) . Then the generalized Becknerinequality (13) Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq ≤ C B ( p, q ) k∇ f k qL (Ω) holds, where C B ( p, q ) = 2( pq − C qP − q if q < , C B ( p,
2) = C P if q = 2 , and C P > is the constant of the Poincar´e-Wirtinger inequality (6) . C. CHAINAIS-HILLAIRET, A. J ¨UNGEL, AND S. SCHUCHNIGG
Remark 3.
The case q = 2 corresponds to the usual Beckner inequality [4] Z Ω | f | dx − (cid:18)Z Ω | f | /r dx (cid:19) r ≤ C B ( p, k∇ f k L (Ω) , where 1 ≤ r = 2 p ≤
2. It is shown in [14] that the constant C B ( p,
2) can be related to thelowest positive eigenvalue of a Schr¨odinger operator if Ω is convex. On the one-dimensonaltorus, the generalized Beckner inequality (13) for p > < q < p = 2 /q was proved in [13]. Inthis work, it was also shown that (13) with q > p = 2 /q cannot be true unless theLebesgue measure dx is replaced by the Dirac measure. In the limit pq →
1, (13) leads toa generalized logarithmic Sobolev inequality (see (15) below). If q = 2 in this limit, theusual logarithmic Sobolev inequality [21] is obtained. (cid:3) Proof of Lemma 2.
Let first q = 2. Then the Beckner inequality is a consequence of thePoincar´e-Wirtinger inequality (6) and the Jensen inequality: C P k∇ f k L (Ω) ≥ k f − ¯ f k L (Ω) = k f k L (Ω) − k f k L (Ω) ≥ Z Ω f dx − (cid:18)Z Ω | f | /r dx (cid:19) r , where 1 − d ≤ r ≤ < r ≤ d ≤ r ensures that theembedding H (Ω) ֒ → L /r (Ω) is continuous. The choice p = r/ ∈ [ − d ,
1] yields theformulation (13).Next, let 0 < q <
2. The first part of the proof is inspired by the proof of Proposition2.2 in [13]. Taking the logarithm of the interpolation inequality k f k L r (Ω) ≤ k f k θ ( r ) L q (Ω) k f k − θ ( r ) L (Ω) , where q ≤ r ≤ θ ( r ) = q (2 − r ) / ( r (2 − q )), gives F ( r ) := 1 r log Z Ω | f | r dx − θ ( r ) q log Z Ω | f | q dx − − θ ( r )2 log Z Ω | f | dx ≤ . The function F : [ q, → R is nonpositive, differentiable and F ( q ) = 0. Therefore, F ′ ( q ) ≤
0, which equals − q log Z Ω | f | q dx + 1 q (cid:18)Z Ω | f | q dx (cid:19) − Z Ω | f | q log | f | dx + θ ′ ( q ) (cid:18)
12 log Z Ω | f | dx − q log Z Ω | f | q dx (cid:19) ≤ . We multiply this inequality by q R Ω | f | q dx to obtain(14) Z Ω | f | q log | f | q k f k qL q (Ω) dx ≤ − q k f k qL q (Ω) log k f k qL (Ω) k f k qL q (Ω) . NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 7
Then, we employ Lemma 1 and the inequality log( x + 1) ≤ x for x ≥ k f k qL q (Ω) log k f k qL (Ω) k f k qL q (Ω) ≤ k f k qL q (Ω) log C qP k∇ f k qL (Ω) k f k qL q (Ω) + 1 ! ≤ C qP k∇ f k qL (Ω) . Combining this inequality and (14), we conclude the generalized logarithmic Sobolev in-equality(15) Z Ω | f | q log | f | q k f k qL q (Ω) dx ≤ C qP − q k∇ f k qL (Ω) . The generalized Beckner inequality (13) is derived by extending slightly the proof of [27,Corollary 1]. Let G ( r ) = r log Z Ω | f | q/r dx, r ≥ . The function G is twice differentiable with G ′ ( r ) = (cid:18)Z Ω | f | q/r dx (cid:19) − (cid:18)Z Ω | f | q/r dx log Z Ω | f | q/r dx − qr Z Ω | f | q/r log | f | dx (cid:19) ,G ′′ ( r ) = q r (cid:18)Z Ω | f | q/r dx (cid:19) − Z Ω | f | q/r dx Z Ω | f | q/r (log | f | ) dx − (cid:18)Z Ω | f | q/r log | f | dx (cid:19) ! . The Cauchy-Schwarz inequality shows that G ′′ ( r ) ≥
0, i.e., G is convex. Consequently, r e G ( r ) is also convex and r H ( r ) = − ( e G ( r ) − e G (1) ) / ( r −
1) is nonincreasing on(1 , ∞ ), which implies that H ( r ) ≤ lim t → H ( t ) = − e G (1) G ′ (1) = Z Ω | f | q log | f | q k f k qL q (Ω) dx. This inequality is equivalent to(16) 1 r − (cid:18)Z Ω | f | q dx − (cid:18)Z Ω | f | q/r dx (cid:19) r (cid:19) ≤ Z Ω | f | q log | f | q k f k qL q (Ω) dx. Combining this inequality and the generalized logarithmic Sobolev inequality (15), it fol-lows that Z Ω | f | q dx − (cid:18)Z Ω | f | q/r dx (cid:19) r ≤ r − C qP − q k∇ f k qL (Ω) for all 0 < q < r ≥
1. Setting p := r/q , this proves (13) for all pq = r ≥ (cid:3) For the proof of exponential decay rates, we need the following variant of the Becknerinequality.
Lemma 4 (Generalized Beckner inequality II) . Let < q < , pq ≥ and f ∈ H (Ω) .Then (17) k f k − qL q (Ω) (cid:18)Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq (cid:19) ≤ C ′ B ( p, q ) k∇ f k L (Ω) , C. CHAINAIS-HILLAIRET, A. J ¨UNGEL, AND S. SCHUCHNIGG where C ′ B ( p, q ) = q ( pq − C P − q if ≤ q < , ( pq − C P if < q < . Proof.
By (14), it holds that for all 0 < q < Z Ω | f | q log | f | q k f k qL q (Ω) dx ≤ q − q k f k qL q (Ω) log k f k L (Ω) k f k L q (Ω) . Then, for q >
1, the Poincar´e-Wirtinger inequality in the version (9) and the inequalitylog( x + 1) ≤ x for x ≥ k f k qL q (Ω) log k f k L (Ω) k f k L q (Ω) ≤ k f k qL q (Ω) log C P k∇ f k L (Ω) k f k L q (Ω) + 1 ! ≤ C P k f k q − L q (Ω) k∇ f k L (Ω) . Taking into account (16), the conclusion follows for q > < q ≤
1. Suppose that the following inequality holds:(19) k f k L q (Ω) + 2 − qq C P k∇ f k L (Ω) − k f k L (Ω) ≥ . This implies that, by (16) and for r = pq , Z Ω | f | q dx − (cid:18)Z Ω | f | q/r dx (cid:19) r ≤ ( pq − q − q k f k qL q (Ω) log k f k L (Ω) k f k L q (Ω) ≤ ( pq − q − q k f k qL q (Ω) log (2 − q ) C P q k∇ f k L (Ω) k f k L q (Ω) + 1 ! ≤ ( pq − C P k∇ f k L (Ω) k f k q − L q (Ω) , which shows the desired Beckner inequality.It remains to prove (19). For this, we employ the Poincar´e-Wirtinger inequality (8) C P k∇ f k L (Ω) ≥ k f k L (Ω) − k f k L (Ω) and the interpolation inequality (12) in the form k f k L q (Ω) ≥ k f k /θL (Ω) k f k θ − /θL (Ω) , θ = q − q ≤ , to obtain k f k L q (Ω) + 2 − qq C P k∇ f k L (Ω) − k f k L (Ω) ≥ k f k /θL (Ω) k f k θ − /θL (Ω) + (cid:18) − qq − (cid:19) k f k L (Ω) − − qq k f k L (Ω) . NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 9
We interpret the right-hand side as a function G of k f k L (Ω) . Then, setting A = k f k L (Ω) , G ( y ) = y /θ A − /θ + 2(1 − q ) q A − − qq y,G ′ ( y ) = 1 θ y /θ − A − /θ − − qq ,G ′′ ( y ) = 1 θ (cid:18) θ − (cid:19) y /θ − A − /θ ≥ , Therefore, G is a convex function which satisfies G ( A ) = 0 and G ′ ( A ) = 0. This impliesthat G is a nonnegative function on R + and in particular, G ( k f k L (Ω) ) ≥
0. This proves(19), completing the proof. (cid:3)
Zeroth-order entropies.
Let u be a smooth solution to (1)-(2) and let u ∈ L ∞ (Ω),inf Ω u ≥ ≤ inf Ω u ≤ u ( t ) ≤ sup Ω u in Ω for t ≥ E α [ u ], defined in (3). Theorem 5 (Polynomial decay for E α ) . Let α > and β > . Let u be a smooth solutionto (1) - (2) and u ∈ L ∞ (Ω) with inf Ω u ≥ . Then E α [ u ( t )] ≤ C t + C ) ( α +1) / ( β − , t ≥ , where C = 4 αβ ( β − α + 1)( α + β ) (cid:18) α + 1 C B ( p, q ) (cid:19) ( α + β ) / ( α +1) , C = E [ u ] − ( β − / ( α +1) , and C B ( p, q ) > is the constant in the Beckner inequality for p = ( α + β ) / and q =2( α + 1) / ( α + β ) .Proof. We apply Lemma 2 with p = ( α + β ) / q = 2( α + 1) / ( α + β ). The assumptionson α and β guarantee that 0 < q < pq >
1. Then, with f = u ( α + β ) / , E α [ u ] = 1 α + 1 Z Ω u α +1 dx − (cid:18)Z Ω udx (cid:19) α +1 ! ≤ C B ( p, q ) α + 1 (cid:18)Z Ω |∇ u ( α + β ) / | dx (cid:19) ( α +1) / ( α + β ) . Now, computing the derivative, dE α dt = − Z Ω ∇ u α · ∇ u β dx = − αβ ( α + β ) Z Ω |∇ u ( α + β ) / | dx ≤ − αβ ( α + β ) (cid:18) α + 1 C B ( p, q ) (cid:19) ( α + β ) / ( α +1) E α [ u ] ( α + β ) / ( α +1) . An integration of this inequality gives the assertion. (cid:3)
Next, we show exponential decay rates.
Theorem 6 (Exponential decay for E α ) . Let u be a smooth solution to (1) - (2) , < α ≤ , β > , u ∈ L ∞ (Ω) with inf Ω u ≥ . Then E α [ u ( t )] ≤ E α [ u ] e − Λ t , t ≥ . The constant Λ is given by Λ = 4 αβC B ( ( α + 1) , α + 1) inf Ω ( u β − ) ≥ , for β > and Λ = 4 αβ ( α + 1) C ′ B ( p, q )( α + β ) k u k β − L (Ω) , for β > . Here, C B ( ( α + 1) , and C ′ B ( p, q ) are the constants in the Beckner inequalities (13) and (17) , respectively, with p = ( α + β ) / and q = 2( α + 1) / ( α + β ) .Proof. Let β >
0. We compute dE α dt = − αβ ( α + 1) Z Ω u β − |∇ u ( α +1) / | dx ≤ − αβ ( α + 1) inf Ω ( u β − ) Z Ω |∇ u ( α +1) / | dx. By the Beckner inequality (13) with p = ( α + 1) / q = 2, and f = u ( α +1) / , we find that dE α dt ≤ − αβC B ( p, α + 1) inf Ω ( u β − ) E α , and Gronwall’s lemma proves the claim. The restriction p ≤ α ≤ β >
1. By Lemma 4, with p = ( α + β ) / q = 2( α +1) / ( α + β ), and f = u ( α + β ) / ,it follows that k u k β − L α +1 (Ω) Z Ω u α +1 dx − (cid:18)Z Ω udx (cid:19) α +1 ! ≤ C ′ B ( p, q ) Z Ω |∇ u ( α + β ) / | dx. Hence, we can estimate dE α dt = − αβ ( α + β ) Z Ω |∇ u ( α + β ) / | dx ≤ − αβ ( α + 1)( α + β ) k u k β − L α +1 (Ω) C ′ B ( p, q ) E α [ u ] ≤ − αβ ( α + 1)( α + β ) k u k β − L (Ω) C ′ B ( p, q ) E α [ u ] , and Gronwall’s lemma gives the conclusion. Note that in the last step of the inequality weused that k u k L α +1 (Ω) ≥ k u k L (Ω) = k u k L (Ω) . (cid:3) NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 11
First-order entropies.
In this section, we consider the diffusion equation (1) on thetorus Ω = T d . We prove the exponential decay for the first-order entropies (4). Theorem 7 (Exponential decay of F α in one space dimension) . Let u be a smooth solutionto (1) on the one-dimensional torus Ω = T . Let u ∈ L ∞ (Ω) with inf Ω u ≥ and let α, β > satisfy − ≤ α − β < . Then F α [ u ( t )] ≤ F α [ u ] e − Λ t , ≤ t ≤ T, where Λ = 2 βC P inf Ω ( u α + β − γ − ) inf Ω ( u γ − α ) ≥ , γ = 23 ( α + β − , where C P > is the Poincar´e constant in (6) .Proof. We extend slightly the entropy construction method of [23]. The time derivative ofthe entropy reads as dF α dt = α Z Ω ( u α/ ) x ( u α/ − u t ) x dx = − α Z Ω ( u α/ ) xx u α/ − ( u β ) xx dx = − α β Z Ω u α + β − (cid:16)(cid:16) α − (cid:17) ( β − ξ G + (cid:16) α β − (cid:17) ξ G ξ L + ξ L (cid:17) dx, where we introduced ξ G = u x u , ξ L = u xx u . This integral is compared to Z Ω u α + β − γ − ( u γ/ ) xx dx = γ Z Ω u α + β − (cid:18)(cid:16) γ − (cid:17) ξ G + ( γ − ξ G ξ L + ξ L (cid:19) dx, where, compared to the method of [23], γ = 0 gives an additional degree of freedom in thecalculations. In the one-dimensional situation, there is only one relevant integration-by-parts rule: 0 = Z Ω ( u α + β − u x ) x dx = Z Ω u α + β − (cid:0) ( α + β − ξ G + 3 ξ G ξ L (cid:1) dx. We introduce the polynomials S ( ξ ) = (cid:16) α − (cid:17) ( β − ξ G + (cid:16) α β − (cid:17) ξ G ξ L + ξ L , (20) D ( ξ ) = (cid:16) γ − (cid:17) ξ G + ( γ − ξ G ξ L + ξ L , (21) T ( ξ ) = ( α + β − ξ G + 3 ξ G ξ L , where ξ = ( ξ G , ξ L ). We wish to show that there exist numbers c , γ ∈ R ( γ = 0) and κ > S ( ξ ) = S ( ξ ) + cT ( ξ ) − κD ( ξ ) ≥ ξ ∈ R . The polynomial S can be written as S ( ξ ) = a ξ G + a ξ G ξ L + (1 − κ ) ξ L , where a = −
14 ( γ − κ + ( α + β − c + 12 ( α − β − , a = − ( γ − κ + 3 c + 12 ( α + 2 β − . Therefore, the maximal value for κ is κ = 1. Let κ = 1. Then we need to eliminate themixed term ξ G ξ L . The solution of a = 0 is given by c = − ( α + 2 β − γ ), which leads to a = − (cid:18) γ −
23 ( α + β − (cid:19) −
118 ( α − β − α − β + 2) . Choosing γ = ( α + β −
1) to maximize a , we find that a ≥ S ( ξ ) ≥ − ≤ α − β ≤ dF α dt = − α β Z Ω u α + β − S ( ξ ) dx = − α β Z Ω u α + β − ( S ( ξ ) + cT ( ξ )) dx ≤ − α β Z Ω u α + β − D ( ξ ) dx = − α βγ Z Ω u α + β − γ − ( u γ/ ) xx dx ≤ − α βγ inf Ω × (0 , ∞ ) ( u α + β − γ − ) Z Ω ( u γ/ ) xx dx ≤ − α βγ C P inf Ω ( u α + β − γ − ) Z Ω ( u γ/ ) x dx ≤ − βC P inf Ω ( u α + β − γ − ) inf Ω ( u γ − α ) F α . For the last inequality, we use that ( u γ/ ) x = γα u ( γ − α ) / ( u α/ ) x , which cancels out the ratio α /γ . An application of the Gronwall’s lemma finishes the proof. (cid:3) We turn to the multi-dimensional case.
Theorem 8 (Exponential decay of F α in several space dimensions) . Let u be a smoothsolution to (1) on the torus Ω = T d . Let u ∈ L ∞ (Ω) with inf Ω u > and let ( α, β ) ∈ M d = (cid:8) ( α, β ) ∈ R : (2 − α + 2 β − d + αd )(4 − β − d + αd + 2 β + 2 βd ) > and ( α − β − α − β + 2) < (cid:9) (see Figure 1). Then there exists Λ > , depending on α , β , d , u , and Ω such that F α [ u ( t )] ≤ F α [ u ] e − Λ t , t ≥ . Proof.
The time derivative of the first-order entropy becomes(22) dF α dt = − α Z Ω u α/ − ∆( u α/ )∆( u β ) dx = − α β Z Ω u α + β − S dx, where S is defined in (20) with the (scalar) variables ξ G = |∇ u | /u , ξ L = ∆ u/u . Wecompare this integral to Z Ω u α + β − γ − (∆( u γ/ )) dx = γ Z Ω u α + β − D dx, NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 13
Figure 1.
Illustration of the set M d , defined in Theorem 8, for d = 9.where D is as in (21) and γ = 0. In contrast to the one-dimensional case, we employ twointegration-by-parts rules:0 = Z Ω div (cid:0) u α + β − |∇ u | ∇ u (cid:1) dx = Z Ω u α + β − T dx, Z Ω div (cid:0) u α + β − ( ∇ u − ∆ I ) · ∇ u (cid:1) dx = Z Ω u α + β − T dx, where T = ( α + β − ξ G + 2 ξ GHG + ξ G ξ L ,T = ( α + β − ξ GHG − ( α + β − ξ G ξ L + ξ H − ξ L , and ξ GHG = u − ∇ u ⊤ ∇ u ∇ u , ξ H = u − k∇ u k . Here, k∇ u k denotes the Frobenius normof the hessian.In order to compare ∇ u and ∆ u , we employ Lemma 2.1 of [24]: k∇ u k ≥ d (∆ u ) + dd − (cid:18) ∇ u ⊤ ∇ u ∇ u |∇ u | − ∆ ud (cid:19) . Therefore, there exists ξ R ∈ R such that ξ H = ξ L d + dd − (cid:18) ξ GHG ξ G − d ξ L (cid:19) + ξ R = ξ L d + dd − ξ S + ξ R , where we introduced ξ S = ξ GHG /ξ G − ξ L /d . Rewriting the polynomials T and T in termsof ξ = ( ξ G , ξ L , ξ S , ξ R ) ∈ R leads to: T ( ξ ) = ( α + β − ξ G + 2 + dd ξ G ξ L + 2 ξ G ξ S ,T ( ξ ) = 1 − dd ( α + β − ξ G ξ L + 1 − dd ξ L + ξ S ξ G ( α + β −
3) + dd − ξ S + ξ R . We wish to find c , c , γ ∈ R ( γ = 0) and κ > S ( ξ ) = S ( ξ ) + c T ( ξ ) + c T ( ξ ) − κD ( ξ ) ≥ ξ ∈ R . The polynomial S can be written as S ( ξ ) = a ξ G + a ξ G ξ L + a ξ L + a ξ G ξ S + a ξ S + c ξ R , where a = (cid:16) α − (cid:17) ( β −
1) + ( α + β − c − (cid:16) γ − (cid:17) κ,a = α β − (cid:18) d + 1 (cid:19) c − ( α + β − d − d c − ( γ − κ,a = 1 + 1 − dd c − κ,a = 2 c + ( α + β − c ,a = dd − c . We remove the variable ξ R by requiring that c ≥
0. The remaining polyomial can bereduced to a quadratic polynomial by setting x = ξ L /ξ G and y = ξ S /ξ G :(23) S ( x, y ) ≥ a + a x + a x + a y + a y ≥ x, y ∈ R . This quadratic decision problem can be solved by employing the computer algebra system
Mathematica . The result of the command
Resolve[ForAll[{x, y}, Exists[{C1, C2, kappa, gamma},a1 + a2*x + a3*x^2 + a4*y + a5*y^2 >= 0 && kappa > 0&& gamma != 0]], Reals] gives all ( α, β ) ∈ R such that there exist c , c , γ ∈ R ( γ = 0) and κ > M d , defined in the statement of the theorem.Similar to the one-dimensional case, we infer that dF α dt ≤ − α βκ Z Ω u α + β − D ( ξ ) dx = − α βκγ Z Ω u α + β − γ − (∆ u γ/ ) dx. Thus, proceeding as in the proof of Theorem 7 and using the identity Z Ω (∆ f ) dx = Z Ω k∇ f k dx for smooth functions f (which can be obtained by integration by parts twice), we obtain dF α dt ≤ − βκC P inf Ω ( u α + β − γ − ) inf Ω ( u γ − α ) F α . Gronwall’s lemma concludes the proof. (cid:3)
Remark 9.
Under the additional constraints a = a = 0, we are able to solve the decisionproblem (23) without the help of the computer algebra system. The solution set, however, NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 15 is slightly smaller than M d which is obtained from Mathematica without these constraints.Indeed, we can compute c and c from the equations a = a = 0 giving c = dd + 2 (cid:16) α − κ (1 + γ − α − β ) (cid:17) , c = d (1 − κ ) d − . The decision problem (23) reduces to a + a y + a y ≥ y ∈ R . If κ <
1, it holds c > a >
0. Therefore, the above polynomial isnonnegative for all y ∈ R if it has no real roots, i.e., if0 ≤ a a − a = q + q γ + q γ for some γ = 0, where (for d > q = − d κ ( d + 2) ( d − (cid:0) d ( d − κ + ( d + 2) (cid:1) < , and q , q are polynomials depending on d , α , β , and κ . The above problem is solvable ifand only if there exist real roots, i.e. if0 ≤ q − q q = 4 κ (1 − κ )( d + 2) ( d − ( s + s κ + s κ ) , where s = − d (5 d −
8) + 6 d ( d − α + 2 d ( d + 2) β + 2( d + 2) αβ − (2 d + 1) α − ( d + 2) β ,s = 2 d (3 d − − d (4 d − α − d ( d + 1) β + 2 d (3 d − αβ + 2 d ( d + 1) α − d ( d − β ,s = − d ( α + β − . We set f ( κ ) = s + s κ + s κ . We have to find 0 < κ < f ( κ ) ≥
0. Since s ≤ f ( κ ) possesses two (not necessarily distinct) real roots κ and κ and ifat least one of these roots is between zero and one. Since f (1) = − ( d − ( α − β ) ≤ κ and κ : either κ ≤ ≤ κ ≤ ≤ κ ≤ κ ≤ f (0) = s ≥
0, the second one if f ′ (0) = s ≥ , f ′ (1) = s + 2 s ≤ , (24) s − s s = − d ( α − β + 2)( α − β − − d + dα + 2 dβ )(25) × (2 − d + ( d − α + 2 β ) ≥ . The set of all ( α, β ) ∈ R fulfilling these conditions is illustrated in Figure 2. (cid:3) The discrete case
We introduce the finite-volume scheme and prove discrete versions of the generalizedBeckner inequality as well as the discrete decay rates.
Figure 2.
Set of all ( α, β ) fulfilling s ≥
0, (24), and (25) for d = 9.3.1. Notations and finite-volume scheme.
Let Ω be an open bounded polyhedral sub-set of R d ( d ≥
2) with Lipschitz boundary and m(Ω) = 1. An admissible mesh of Ω isgiven by a family T of control volumes (open and convex polyhedral subsets of Ω withpositive measure); a family E of relatively open parts of hyperplanes in R d which representthe faces of the control volumes; and a family of points ( x K ) K ∈T which satisfy Definition9.1 in [17]. This definition implies that the straight line between two neighboring centersof cells ( x K , x L ) is orthogonal to the edge σ = K | L between the two control volume K and L . For instance, triangular meshes in R satisfy the admissibility condition if all anglesof the triangles are smaller than π/ R d are alsoadmissible meshes [17, Examples 9.2].We distinguish the interior faces σ ∈ E int and the boundary faces σ ∈ E ext . Then theunion E int ∪ E ext equals the set of all faces E . For a control volume K ∈ T , we denote by E K the set of its faces, by E int ,K the set of its interior faces, and by E ext ,K the set of edgesof K included in ∂ Ω.Furthermore, we denote by d the distance in R d . We assume that the family of meshessatisfies the following regularity requirement: There exists ξ > K ∈ T and all σ ∈ E int ,K with σ = K | L , it holds(26) d( x K , σ ) ≥ ξ d( x K , x L ) . This hypothesis is needed to apply a discrete Poincar´e inequality; see Lemma 11. Intro-ducing for σ ∈ E the notation d σ = (cid:26) d( x K , x L ) if σ ∈ E int , σ = K | L, d( x K , σ ) if σ ∈ E ext ,K , we define the transmissibility coefficient τ σ = m( σ ) d σ , σ ∈ E . NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 17
The size of the mesh is denoted by △ x = max K ∈T diam( K ). Let T > M T the number of time steps. Then the time step size and the time points are givenby, respectively, △ t = TM T , t k = k △ t for 0 ≤ k ≤ M T . We denote by D an admissiblespace-time discretization of Ω T = Ω × (0 , T ) composed of an admissible mesh T of Ω andthe values △ t and M T .A classical finite volume scheme provides an approximate solution which is constant oneach cell of the mesh and on each time interval. Let X ( T ) be the linear space of functionsΩ → R which are constant on each cell K ∈ T . We define on X ( T ) the discrete L p norm,discrete W ,p seminorm, and discrete W ,p norm by, respectively, k u k ,p, T = (cid:18)Z Ω | u | p dx (cid:19) /p = X K ∈T m( K ) | u K | p ! /p , | u | ,p, T = (cid:18) X σ ∈E int ,σ = K | L m( σ ) d p − σ | u K − u L | p (cid:19) /p , k u k ,p, T = k u k ,p, T + | u | ,p, T , where u ∈ X ( T ), u = u K in K ∈ T , and 1 ≤ p < ∞ . The discrete entropies for u ∈ X ( T )are defined analogously to the continuous case: E dα [ u ] = 1 α + 1 X K ∈T m( K ) u α +1 K − X K ∈T m( K ) u K ! α +1 , (27) F dα [ u ] = 12 | u α/ | , , T . (28)We are now in the position to define the finite-volume scheme of (1)-(2). Let D be afinite-volume discretization of Ω T . The initial datum is approximated by its L projectionon control volumes:(29) u = X K ∈ T u K K , where u K = 1m( K ) Z K u ( x ) dx, and K is the characteristic function on K . Then k u k , , T = k u k L (Ω) . The numericalscheme reads as follows:(30) m( K ) u k +1 K − u kK △ t + X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K ) β − ( u k +1 L ) β (cid:1) = 0 , for all K ∈ T and k = 0 , . . . , M T −
1. This scheme is based on a fully implicit Eulerdiscretization in time and a finite-volume approach for the volume variable. The Neumannboundary conditions (2) are taken into account as the sum in (30) applies only on theinterior edges. The implicit scheme allows us to establish discrete entropy-dissipationestimates which would not be possible with an explicit scheme.
We summarize in the next proposition the classical results of existence, uniqueness andstability of the solution to the finite volume scheme (29)-(30).
Proposition 10.
Let u ∈ L ∞ (Ω) , m ≥ , M ≥ such that m ≤ u ≤ M in Ω .Let T be an admissible mesh of Ω . Then the scheme (29) - (30) admits a unique solution ( u kK ) K ∈T , ≤ k ≤ M T satisfying m ≤ u kK ≤ M, for all K ∈ T , ≤ k ≤ M T , X K ∈T m( K ) u kK = k u k L (Ω) , for all ≤ k ≤ M T . We refer, for instance, to [17] and [18] for the proof of this proposition.3.2.
Discrete generalized Beckner inequalities.
The decay properties rely on discretegeneralized Beckner inequalities which follow from the discrete Poincar´e-Wirtinger inequal-ity [6, Theorem 5]:
Lemma 11 (Discrete Poincar´e-Wirtinger inequality) . Let Ω ⊂ R d be an open boundedpolyhedral set and let T be an admissible mesh satisfying the regularity constraint (26) .Then there exists a constant C p > only depending on d and Ω such that for all f ∈ X ( T ) , (31) k f − ¯ f k , , T ≤ C p ξ / | f | , , T where ¯ f = R Ω f dx (recall that m (Ω) = 1 ) and ξ is defined in (26) . Lemma 12 (Discrete generalized Beckner inequality I) . Let < q < , pq > or q = 2 and < p ≤ , and f ∈ X ( T ) . Then Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq ≤ C b ( p, q ) | f | q , , T holds, where C b ( p, q ) = 2( pq − C qp (2 − q ) ξ q/ if q < , C b ( p,
2) = C p ξ if q = 2 .C p is the constant in the discrete Poincar´e-Wirtinger inequality, and ξ is defined in (26) .Proof. The proof follows the lines of the proof of Lemma 2 noting that in the discrete(finite-dimensional) setting, we do not need anymore the lower bound on p . Indeed, if q = 2, the conclusion results from the discrete Poincar´e-Wirtinger inequality (31) and theJensen inequality. If q <
2, let f ∈ X ( T ). Then we have from (16) and (14) Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq ≤ ( pq − Z Ω | f | q log | f | q k f k q ,q, T dx ≤ pq − − q k f k q ,q, T log k f k q , , T k f k q ,q, T . (32) NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 19
We employ the discrete Poincar´e-Wirtinger inequality (31), k f k , , T − k f k , , T = k f − ¯ f k , , T ≤ C p ξ − | f | , , T , which implies, as in the proof of Lemma 1 (see (9)), that k f k q , , T ≤ C qp ξ − q/ | f | q , , T + k f k q ,q, T . After inserting this inequality into (32) to replace k f k , , T and using log( x + 1) ≤ x for x ≥
0, the lemma follows. (cid:3)
The following result is proved exactly as in Lemma 4 using the discrete Poincar´e-Wirtinger inequality (31) instead of (6).
Lemma 13 (Discrete generalized Beckner inequality II) . Let < q < , pq ≥ , and f ∈ X ( T ) . Then k f k − q ,q, T (cid:18)Z Ω | f | q dx − (cid:18)Z Ω | f | /p dx (cid:19) pq (cid:19) ≤ C ′ b ( p, q ) | f | , , T holds, where C ′ b ( p, q ) = q ( pq − C p (2 − q ) ξ if ≤ q < , ( pq − C p ξ if < q < ,C p is the constant in the discrete Poincar´e-Wirtinger inequality, and ξ is defined in (26) . Zeroth-order entropies.
We prove a result which is the discrete analogue of Theo-rem 5. Recall that the discrete entropies E dα [ u k ] are defined in (27). Theorem 14 (Polynomial decay of E dα ) . Let α > and β > . Let ( u kK ) K ∈T ,k ≥ be asolution to the finite-volume scheme (30) with inf K ∈T u K ≥ . Then E dα [ u k ] ≤ c t k + c ) ( α +1) / ( β − , k ≥ , where c = ( β − ( α + 1)( α + β ) αβ (cid:18) C b ( p, q ) α + 1 (cid:19) ( α + β ) / ( α +1) + ( α + β ) △ tE dα [ u ] ( α +1) / ( β − ! − ,c = E dα [ u ] − ( β − / ( α +1) , and C b ( p, q ) for p = ( α + β ) / and q = 2( α + 1) / ( α + β ) is defined in Lemma 12.Proof. The idea is to “translate” the proof of Theorem 5 to the discrete case. To this end,we use the elementary inequality y α +1 − x α +1 ≤ ( α + 1) y α ( y − x ), which follows from theconvexity of the mapping x x α +1 , and scheme (30): E dα [ u k +1 ] − E dα [ u k ] = 1 α + 1 X K ∈T m( K ) (cid:0) ( u k +1 K ) α +1 − ( u kK ) α +1 (cid:1) ≤ X K ∈T m( K )( u k +1 K ) α ( u k +1 K − u kK ) ≤ −△ t X K ∈T X σ ∈E int ,σ = K | L τ σ ( u k +1 K ) α (cid:0) ( u k +1 K ) β − ( u k +1 L ) β (cid:1) . Rearranging the sum leads to(33) E dα [ u k +1 ] − E dα [ u k ] ≤ −△ t X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K ) α − ( u k +1 L ) α (cid:1)(cid:0) ( u k +1 K ) β − ( u k +1 L ) β (cid:1) . Then, employing the inequality in Lemma 19 (see the appendix), it follows that E dα [ u k +1 ] − E dα [ u k ] ≤ − αβ △ t ( α + β ) X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K ) ( α + β ) / − ( u k +1 L ) ( α + β ) / (cid:1) ≤ − αβ △ t ( α + β ) | ( u k +1 ) ( α + β ) / | , , T , and applying Lemma 12 with p = ( α + β ) / q = 2( α + 1) / ( α + β ), and f = ( u k +1 ) ( α + β ) / , E dα [ u k +1 ] − E dα [ u k ] ≤ − αβ △ t ( α + β ) (cid:18) α + 1 C b ( p, q ) (cid:19) ( α + β ) / ( α +1) E dα [ u k +1 ] ( α + β ) / ( α +1) . The discrete nonlinear Gronwall lemma (see Corollary 18 in the appendix) with τ = 4 αβ △ t ( α + β ) (cid:18) α + 1 C b ( p, q ) (cid:19) ( α + β ) / ( α +1) , γ = α + βα + 1 > , implies that E dα [ u k ] ≤ E dα [ u ] − γ + c t k ) / ( γ − , k ≥ , where c = ( γ − / (1 + γτ E dα [ u ] γ − ). Finally, computing c shows the result. (cid:3) The discrete analogue to Theorem 6 is as follows.
Theorem 15 (Exponential decay for E dα ) . Let ( u kK ) K ∈T ,k ≥ be a solution to the finite-volume scheme (30) and let < α ≤ , β > , inf K ∈T u K ≥ . Then E dα [ u k ] ≤ E dα [ u ] e − λt k , k ≥ . The constant λ is given by λ = 4 αβC b ( ( α + 1) , α + 1) inf K ∈T (cid:0) ( u K ) β − (cid:1) ≥ , for β > , and λ = 4 αβ ( α + 1) C ′ b ( p, q )( α + β ) k u k β − , , T NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 21 for β > . Here C ′ b ( p, q ) > is the constant from Lemma 13 with p = ( α + β ) / and q = 2( α + 1) / ( α + β ) .Proof. Let α ≤ β >
0. As in the proof of Theorem 14, we find that (see (33)) E dα [ u k +1 ] − E dα [ u k ] ≤ −△ t X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K ) α − ( u k +1 L ) α (cid:1)(cid:0) ( u k +1 K ) β − ( u k +1 L ) β (cid:1) . Employing Corollary 20 (see the appendix), we obtain E dα [ u k +1 ] − E dα [ u k ] ≤ − αβ △ t ( α + 1) X σ ∈E int ,σ = K | L τ σ min (cid:8) ( u k +1 K ) β − , ( u k +1 L ) β − (cid:9) × (cid:0) ( u k +1 K ) ( α +1) / − ( u k +1 L ) ( α +1) / (cid:1) ≤ − αβ △ t ( α + 1) inf K ∈T ( u k +1 K ) β − | ( u k +1 ) ( α +1) / | , , T ≤ − αβ △ tC b ( ( α + 1) , α + 1) inf K ∈T ( u K ) β − E dα [ u k +1 ] , where we have used Lemma 12 with p = ( α + 1) / q = 2, and f = u ( α +1) / . Now, theGronwall lemma shows the claim.Next, let β >
1. As in the proof of Theorem 14, we find that E dα [ u k +1 ] − E dα [ u k ] ≤ − αβ △ t ( α + β ) | ( u k +1 ) ( α +1) / | , , T . We apply Lemma 13 with p = ( α + β ) / q = 2( α + 1) / ( α + β ), and f = u ( α + β ) / to obtain E dα [ u k +1 ] − E dα [ u k ] ≤ − αβ ( α + 1) △ t ( α + β ) k u k +1 k β − ,α +1 , T C ′ b ( p, q ) E dα [ u k +1 ] ≤ − αβ ( α + 1) △ t ( α + β ) k u k β − , , T C ′ b ( p, q ) E dα [ u k +1 ] . Then Gronwall’s lemma finishes the proof. (cid:3)
First-order entropies.
We consider the diffusion equation (1) on the half open unitcube [0 , d ⊂ R d with multiperiodic boundary conditions (this is topologically equivalentto the torus T d ). By identifying “opposite” faces on ∂ Ω, we can construct a family ofcontrol volumes and a family of edges in such a way that every face is an interior face.Then cells with such identified faces are neighboring cells.
Theorem 16 (Exponential decay of F dα ) . Let ( u kK ) K ∈T , k ≥ be a solution to the finite-volumescheme (30) with Ω = T d and inf K ∈T u K ≥ . Then, for all ≤ α ≤ and β = α/ , F dα [ u k +1 ] ≤ F dα [ u k ] , k ∈ N . Furthermore, if d = 1 and the grid is uniform with N subintervals, F dα [ u k ] ≤ F dα [ u ] e − λt k , where λ = 4 β sin ( π/N ) min i (( u i ) β − ) ≥ .Proof. The difference G α = F dα [ u k +1 ] − F dα [ u k ] can be written as G α = 12 X σ ∈E int ,σ = K | L τ σ (cid:16)(cid:0) ( u k +1 K ) α/ − ( u k +1 L ) α/ (cid:1) − (cid:0) ( u kK ) α/ − ( u kL ) α/ (cid:1) (cid:17) . Introducing a K = ( u k +1 K − u kK ) /τ , we find that G α = 12 X σ ∈E int ,σ = K | L τ σ (cid:16)(cid:0) ( u k +1 K ) α/ − ( u k +1 L ) α/ (cid:1) − (cid:0) ( u k +1 K − τ a K ) α/ − ( u k +1 L − τ a L ) α/ (cid:1) (cid:17) . We claim that G α is concave with respect to τ . Indeed, we compute ∂G α ∂τ = α X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K − τ a K ) α/ − ( u k +1 L − τ a L ) α/ (cid:1) × (cid:0) ( u k +1 K − τ a K ) α/ − a K − ( u k +1 L − τ a L ) α/ − a L (cid:1) ,∂ G α ∂τ = − α X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K − τ a K ) α/ − a K − ( u k +1 L − τ a L ) α/ − a L (cid:1) − α (cid:16) α − (cid:17) X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K − τ a K ) α/ − ( u k +1 L − τ a L ) α/ (cid:1) × (cid:0) ( u k +1 K − τ a K ) α/ − a K − ( u k +1 L − τ a L ) α/ − a L (cid:1) . Replacing u k +1 K − τ a K , u k +1 L − τ a L by u kK , u kL , respectively, the second derivative becomes ∂ G α ∂τ = − α X σ ∈E int ,σ = K | L τ σ (cid:0) ( u kK ) α/ − a K − ( u kL ) α/ − a L (cid:1) − α (cid:16) α − (cid:17) X σ ∈E int ,σ = K | L τ σ (cid:0) ( u kK ) α/ − ( u kL ) α/ (cid:1)(cid:0) ( u kK ) α/ − a K − ( u kL ) α/ − a L (cid:1) = − α X σ ∈E int ,σ = K | L τ σ ( c a K + c a K a L + c a L ) , where c = ( α − (cid:0) ( u kK ) α/ − ( u kL ) α/ (cid:1) ( u kK ) α/ − + α ( u kK ) α − ,c = − α ( u kK ) α/ − ( u kL ) α/ − , NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 23 c = − ( α − (cid:0) ( u kK ) α/ − ( u kL ) α/ (cid:1) ( u kL ) α/ − + α ( u kL ) α − . We show that the quadratic polynomial in the variables a K and a L is nonnegative for all u kK and u kL . This is the case if and only if c ≥ c c − c ≥
0. The former conditionis equivalent to 2( α − u kK ) α − ≥ ( α − u kK ) α/ − ( u kL ) α/ , which is true for 1 ≤ α ≤
2. After an elementary computation, the latter conditionbecomes 4 c c − c = 8( α − − α )( u kK ) α/ − ( u kL ) α/ − (cid:0) ( u kK ) α/ − ( u kL ) α/ (cid:1) ≥ ≤ α ≤
2. This proves the concavity of τ G α ( τ ).A Taylor expansion and G α (0) = 0 leads to G α ( τ ) ≤ G α (0) + τ ∂G α ∂τ (0)= ατ X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K ) α/ − ( u k +1 L ) α/ (cid:1)(cid:0) ( u k +1 K ) α/ − a K − ( u k +1 L ) α/ − a L (cid:1) = ατ X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 K ) α/ − ( u k +1 L ) α/ (cid:1) ( u k +1 K ) α/ − a K + ατ X σ ∈E int ,σ = K | L τ σ (cid:0) ( u k +1 L ) α/ − ( u k +1 K ) α/ (cid:1) ( u k +1 L ) α/ − a L . Replacing a K and a L by scheme (30) and performing a summation by parts, we infer that G α ( △ t ) ≤ − α △ t X K ∈T X σ ∈E int ,σ = K | L τ σ X e σ ∈E int , e σ ′ = K | M τ e σ ( u k +1 K ) α/ − × (cid:0) ( u k +1 K ) β − ( u k +1 M ) β (cid:1)(cid:0) ( u k +1 K ) α/ − ( u k +1 L ) α/ (cid:1) . (34)Note that the expression on the right-hand side is the discrete counterpart of the integral − α Z Ω u α/ − ( u β ) xx ( u α/ ) xx dx, appearing in (22). The condition α = 2 β implies immediately the monotonicity of k F dα [ u k ].For the proof of the second statement, let d = 1 and decompose the interval Ω in N subintervals K , . . . , K N of length h >
0. Because of the periodic boundary conditions,we may set u kN +1 = u k and u k − = u kN , where u ki is the approximation of the mean valueof u ( · , t k ) on the subinterval K i , i = 1 , . . . , N . We rewrite (34) for α = 2 β in one spacedimension: G β ( τ ) ≤ − βτ h N X i =1 (cid:18) X j ∈{ i − ,i +1 } ( u k +1 i ) β − (cid:0) ( u k +1 i ) β − ( u jk +1 ) β (cid:1)(cid:19) ≤ − βτ h min i =1 ,...,N (cid:0) ( u k +1 i ) β − (cid:1) N X i =1 ( z i − z i − ) , where z i = ( u k +1 i ) β − ( u k +1 i +1 ) β . The periodic boundary conditions imply that P Ni =1 z i = 0.Hence, we can employ the discrete Wirtinger inequality in [34, Theorem 1] to obtain G β ( τ ) ≤ − βτh sin πN min i =1 ,...,N (cid:0) ( u ki ) β − (cid:1) N X i =1 z i = − βτh sin πN min i =1 ,...,N (cid:0) ( u ki ) β − (cid:1) F dα [ u k +1 ] . By the discrete maximum principle, max i ( u k +1 i ) − β ) ≤ max i ( u i ) − β ) which is equivalentto min i ( u k +1 i ) β − ≥ min i ( u i ) β − . Therefore, F dα [ u k +1 ] − F dα [ u k ] = G β ( △ t ) ≤ − β △ th sin πN min i =1 ,...,N (cid:0) ( u i ) β − (cid:1) F dα [ u k +1 ] , and Gronwall’s lemma finishes the proof. (cid:3) Numerical experiments
We illustrate the time decay of the solutions to the discretized porous-medium ( β = 2)and fast-diffusion equation ( β = 1 /
2) in one and two space dimensions.First, let β = 2. We recall that the Barenblatt profile u B ( x, t ) = ( t + t ) − A (cid:16) C − B ( β − β | x − x | ( t + t ) B (cid:17) / ( β − is a special solution to the porous-medium equation in the whole space. (Here, z + denotesthe positive part of a function z + := max { , z } .) The constants are given by A = dd ( β −
1) + 2 , B = 1 d ( β −
1) + 2 , and C is typically determined by the initial datum via R Ω u ( x, t ) dx = R Ω u ( x, dx . Wechoose C = B ( β − β ) − ( t + t ) − B | x − x | , where t > u ( x , t ) = 0.In the one-dimensional situation, we choose Ω = (0 ,
1) with homogeneous Neumannboundary conditions and a uniform grid ( x i , t j ) ∈ [0 , × [0 , .
2] with 1 ≤ i ≤
50 and0 ≤ j ≤ u B ( · ,
0) with x = 0 . x = 1 and t = 0 .
01. The constant C is computed by using t = 0 .
1, which yields C ≈ . ≤ t ≤ .
1, the analytical solution corresponds to the Barenblatt profile.The time decay of the zeroth- and first-order entropies are depicted in Figure 3 in semi-logarithmic scale for various values of α . The decay rates are exponential for sufficientlylarge times, even for α > α = 2 β (see Theorem 16),which indicates that the conditions imposed in these theorems are technical. For smalltimes, the decay seems to be faster than the decay in the large-time regime. This fact NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 25 has been already observed in [7, Remark 4]. There is a significant change in the decayrate of the first-order entropies F dα for times around t = 0 .
1. Indeed, the positive part ofthe discrete solution, which approximates the Barenblatt profile u B for t < t , arrives theboundary and does not approximate u B anymore. The change is more apparent for α < α = 6 α = 2 α = 1 α = .
50 0 .
05 0 . .
15 0 . − − − − α = 6 α = 2 α = 1 α = .
50 0 .
05 0 . .
15 0 . − − Figure 3.
The natural logarithm of the entropies log( E dα [ u ]( t )) (left) andlog( F dα [ u ]( t )) (right) versus time for different values of α ( β = 2, d = 1).Next, we investigate the two-dimensional situation (still with β = 2). The domainΩ = (0 , is divided into 144 quadratic cells each of which consists of four control volumes(see Figure 4). Again we employ the Barenblatt profile as the initial datum, choosing t = 0 . t = 0 .
1, and x = (0 . , . △ t = 8 · − . Figure 4.
Four of the 144 cells used for the two-dimensional finite-volume scheme.In Figure 5, the time evolution of the (logarithmic) zeroth- and first-order entropies arepresented. Again, the decay seems to be exponential for large times, even for values of α not covered by the theoretical results. At time t = t , the profile reaches the boundaryof the domain. In contrast to the one-dimensional situation, since the radially symmetricprofile does not reach the boundary everywhere at the same time, the time decay rate of F dα does not change as distinct as in Figure 3. α = 6 α = 2 α = 1 α = .
50 0 . . . . − − − − − α = 6 α = 2 α = 1 α = .
50 0 . . . . − − − Figure 5.
The natural logarithm of the entropies log( E dα [ u ]( t )) (left) andlog( F dα [ u ]( t )) (right) versus time for different values of α ( β = 2, d = 2).Let β = 1 /
2. The one-dimensional interval Ω = (0 ,
1) is discretized as before using 51grid points and the time step size is △ t = 2 · − . We impose homogeneous Neumannboundary conditions. As initial datum, we choose the following truncated polynomial u ( x ) = C (( x − x )( x − x )) , where x = 0 . x = 0 .
7, and C = 3000. In the two-dimensional box Ω = (0 , , we employ the discretization described above and the initialdatum u ( x ) = C ( R − | x − x | ) , where R = 0 . x = (0 . , .
5) and again C = 3000.In the fast-diffusion case β <
1, we do not expect significant changes in the decay ratesince the initial values propagate with infinite speed. This expectation is supported bythe numerical results presented in Figures 6 and 7. For a large range of values of α , thedecay rate is exponential, at least for large times. Interestingly, the rate seems to approachalmost the same value for α ∈ { . , , } in Figure 7. Appendix A. Some technical lemmas
A.1.
Discrete Gronwall lemmas.
First, we prove a rather general discrete nonlinearGronwall lemma.
Lemma 17 (Discrete nonlinear Gronwall lemma) . Let f ∈ C ([0 , ∞ )) be a positive, non-decreasing, and convex function such that /f is locally integrable. Define w ( x ) = Z x dzf ( z ) , x ≥ . Let ( x n ) be a sequence of nonnegative numbers such that x n +1 − x n + f ( x n +1 ) ≤ for n ∈ N . Then x n ≤ w − (cid:18) w ( x ) − n f ′ ( x ) (cid:19) , n ∈ N . NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 27 α = 6 α = 2 α = 1 α = .
50 0 .
02 0 .
04 0 .
06 0 .
08 0 . − α = 6 α = 2 α = 1 α = .
50 0 .
02 0 .
04 0 .
06 0 .
08 0 . − Figure 6.
The natural logarithm of the entropies log( E dα [ u ]( t )) (left) andlog( F dα [ u ]( t )) (right) versus time for different values of α ( β = 1 / d = 1). α = 6 α = 2 α = 1 α = .
50 0 .
02 0 .
04 0 .
06 0 .
08 0 . − − − α = 6 α = 2 α = 1 α = .
50 0 .
02 0 .
04 0 .
06 0 .
08 0 . − − Figure 7.
The natural logarithm of the entropies log( E dα [ u ]( t )) (left) andlog( F dα [ u ]( t )) (right) versus time for different values of α ( β = 1 / d = 2).Notice that the function w is strictly increasing such that its inverse is well defined. Proof.
Since f is nondecreasing and ( x n ) is nonincreasing, we obtain w ( x n +1 ) − w ( x n ) = Z x n +1 x n dzf ( z ) ≤ x n +1 − x n f ( x n ) . The sequence ( x n ) satisfies f ( x n +1 ) / ( x n +1 − x n ) ≥ −
1. Therefore, w ( x n +1 ) − w ( x n ) ≤ (cid:16) f ( x n +1 ) x n +1 − x n + f ( x n ) − f ( x n +1 ) x n +1 − x n (cid:17) − ≤ (cid:16) − − f ( x n ) − f ( x n +1 ) x n − x n +1 (cid:17) − . By the convexity of f , f ( x n ) − f ( x n +1 ) ≤ f ′ ( x n )( x n − x n +1 ) ≤ f ′ ( x )( x n − x n +1 ), whichimplies that w ( x n +1 ) − w ( x n ) ≤ ( − − f ′ ( x )) − . Summing this inequality from n = 0 to N −
1, where N ∈ N , yields w ( x N ) ≤ w ( x ) − N f ′ ( x ) . Applying the inverse function of w shows the lemma. (cid:3) The choice f ( x ) = τ Kx γ for some γ > Corollary 18.
Let ( x n ) be a sequence of nonnegative numbers satisfying x n +1 − x n + τ x γn +1 ≤ , n ∈ N , where K > and γ > . Then x n ≤ (cid:0) x − γ + cτ n (cid:1) / ( γ − , n ∈ N , where c = ( γ − / (1 + γτ x γ − ) . A.2.
Some inequalities.
We show some inequalities in two variables.
Lemma 19.
Let α , β > . Then, for all x , y ≥ , (35) ( y α − x α )( y β − x β ) ≥ αβ ( α + β ) ( y ( α + β ) / − x ( α + β ) / ) . Proof. If y = 0, inequality (35) holds. Let y = 0 and set z = ( x/y ) β . Then the inequalityis proved if for all z ≥ f ( z ) = (1 − z α/β )(1 − z ) − αβ ( α + β ) (1 − z ( α + β ) / β ) ≥ . We differentiate f twice: f ′ ( z ) = − − αβ z α/β − + ( α − β ) β ( α + β ) z α/β + 4 αα + β z ( α + β ) / β ,f ′′ ( z ) = α ( α − β ) β z α/ β − / (cid:16) − β z α/ β − / + α − ββ ( α + β ) z α/ β +1 / + 2 α + β (cid:17) . NTROPY DISSIPATIVE FINITE-VOLUME SCHEME 29
Then f (1) = 0 and f ′ (1) = 0. Thus, if we show that f is convex, the assertion follows. Inorder to prove the convexity of f , we define g ( z ) = − β z α/ β − / + α − ββ ( α + β ) z α/ β +1 / + 2 α + β . Then g (1) = 0 and it holds g ′ ( z ) = α − β β z α/ β − / ( − z ) , and therefore, g ′ (1) = 0. Now, if α > β , g (0) = 2 / ( α + β ) >
0, and g is decreasing in [0 , , ∞ ). Thus, g ( z ) ≥ z ≥
0. If α < β then g (0+) = −∞ , and g is increasing in [0 ,
1] and decreasing in [1 , ∞ ). Hence, g ( z ) ≤ z ≥
0. Independentlyof the sign of α − β , we obtain f ′′ ( z ) = α ( α − β ) β z α/ β − / g ( z ) ≥ z ≥
0, which shows the convexity of f . (cid:3) Corollary 20.
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Laboratoire Paul Painlev´e, U.M.R. CNRS 8524, Universit´e Lille 1 Sciences et Tech-nologies, Cit´e Scientifique, 59655 Villeneuve d’Ascq Cedex and Project-Team SIMPAF,INRIA Lille-Nord-Europe, Villeneuve d’Ascq, France
E-mail address : [email protected]
Institute for Analysis and Scientific Computing, Vienna University of Technology,Wiedner Hauptstraße 8–10, 1040 Wien, Austria
E-mail address : [email protected] Institute for Analysis and Scientific Computing, Vienna University of Technology,Wiedner Hauptstraße 8–10, 1040 Wien, Austria
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