Rigorous modelling of nonlocal interactions determines a macroscale advection-diffusion PDE
aa r X i v : . [ m a t h . D S ] J a n Rigorous modelling of nonlocal interactionsdetermines a macroscale advection-diffusionPDE
Prof A.J. RobertsSchool of Mathematical Sciences, University of Adelaide http://orcid.org/0000-0001-8930-1552
21 January 2020
Abstract
A slowly-varying or thin-layer multiscale assumption empowersmacroscale understanding of many physical scenarios from dispersionin pipes and rivers, including beams, shells, and the modulation ofnonlinear waves, to homogenisation of micro-structures. Here we be-gin a new exploration of the scenario where the given physics hasnon-local microscale interactions. We rigorously analyse the dynam-ics of a basic example of shear dispersion. Near each cross-section, thedynamics is expressed in the local moments of the microscale non-localeffects. Centre manifold theory then supports the local modelling ofthe system’s dynamics with coupling to neighbouring cross-sectionsas a non-autonomous forcing. The union over all cross-sections thenprovides powerful new support for the existence and emergence of amacroscale model advection-diffusion pde global in the large, finite-sized, domain. The approach quantifies the accuracy of macroscaleadvection-diffusion approximations, and has the potential to open pre-viously intractable multiscale issues to new insights.
Contents Many kernels generate local models 6
This paper introduces a new rigorous approach to the multiscale challengeof systematically modelling by macroscale pde s the dynamics of microscale, spatially nonlocal , systems. This approach provides a novel quantified errorformula. Previous research using this type of approach rigorously modelledsystems that were expressed as pde s on the microscale. This previous re-search encompassed both cylindrical multiscale domains (Roberts 2015 a ) andmore general multiscale domains (Roberts & Bunder 2017, Bunder & Roberts2018). But recall that pde s are themselves mathematical idealisations ofphysical processes that typically take place on microscale length scales. Hence,here we begin to address the challenges arising when the given mathemat-ical model of a system encodes microscale physical interactions over finitemicroscale lengths.Physical systems with nonlocal, microscale, spatial interactions arise in manyapplications. In neuroscience, a spatial convolution expresses the excitato-ry/inhibitory effects of a neurone on a nearby neurone, giving rise to non-local neural field equations, and “have been quite successful in explainingvarious experimental findings” (Ermentrout 2015, e.g.). Models of free crackpropagation in brittle materials invoke microscale nonlocal stress-strain laws,called peridynamics (Silling 2000, e.g.): one challenge is to derive the effec-tive mesoscale pde s from the nonlocal laws (Silling & Lehoucq 2008, Lipton2014, e.g.). Nonlocal dispersal and competition models arise in biology(Omelyan & Kozitsky 2018, Duncan et al. 2017, e.g.). Other examples arenon-local cell adhesion models (Buttensch¨on & Hillen 2020, e.g.). In this in-troduction we begin by exploring the specific example of a so-called ‘Zappa’dispersion in a channel (Section 2) in which material is transported by fi-nite jumps along the channel, and also is intermittently thoroughly mixed2cross the channel. General scenario
Zappa dispersion is a particular case of the followinggeneral scenario—a scenario that is the subject of ongoing research. In gen-erality we consider a field u ( x , y , t ) , on a ‘cylindrical’ spatial domain X × Y (where X ⊆ R and where Y denotes the cross-section). We suppose thefield u is governed by a given autonomous system in the form ∂u∂t = Z Y Z X k ( x , ξ , y , η ) u ( ξ , η , t ) dξ dη , (1)where the given kernel k ( x , ξ , y , η ) expresses both nonlocal and local physicaleffects at position ( x , y ) from the field at position ( ξ , η ) , both within thecylindrical domain X × Y . We allow the kernel to be a generalised functionso that local derivatives may be represented by derivatives of the Dirac deltafunction δ : for example, a component δ ′ ( x − ξ ) δ ( y − η ) in the kernel k encodesthe differential term − ∂u/∂x in the right-hand side of (1). In general thephysical effects encoded in the kernel k may be heterogeneous in space. But,as is common and apart from boundaries, Zappa dispersion is homogeneous inspace (translationally invariant) in which case some significant simplificationsensue.The nonlocal system (1) is linear for simplicity, but we invoke the frameworkof centre manifold theory so the approach should, with future development,apply to nonlinear generalisations as in previous work on such modellingwhere the system is expressed as pde s on the microscale (Roberts 2015 a ).Our aim is to rigorously establish that the emergent dynamics of the nonlocalsystem (1) are captured over the 1D spatial domain X by a mean/averaged/coarse/macroscale variable U ( x , t ) that satisfies a macroscale, second-order,advection-diffusion pde of the form ∂U∂t ≈ A ∂U∂x + A ∂ U∂x , x ∈ X , (2)for some derived coefficients A and A . This macroscale pde (2) is to modelthe dynamics of the microscale nonlocal (1) after transients have decayedexponentially quickly in time, and to the novel quantified error (6d). Ongoing research aims to generalise the approach here to certify the accuracy of pde struncated to N th-order for every N . Zappa shear dispersion
This section introduces a basic example system (non-dimensional) of nonlocalmicroscale jumps by a particle (inspired by W. R. Young, private communi-cation). Section 3 systematically derives an advection-diffusion pde (2) forthe particle that is valid over macroscale space-time. Consider a particlein a channel − < y < Y = (−
1, 1 ) , and of notionally infinite extentin x , X = R . Let u ( x , y , t ) be the probability density function ( pdf ) for theparticle’s location: equivalently, view u ( x , y , t ) as the concentration of somecontinuum material.The ‘Zappa’ dynamics of the particle’s pdf is encoded by ∂u∂t = h v ( y ) Z x − ∞ e −( x − ξ ) /v ( y ) u ( ξ , y , t ) dξ | {z } = e − x/v ( y ) ⋆ u , the convolution (5) − u i + (cid:20) Z − u dy − u (cid:21) (3)for some jump profile v ( y ) > v ( y ) is an effective velocity along the chan-nel. That is, the kernel of the Zappa system is the generalised function k ( x , ξ , y , η ) = h v ( y ) e −( x − ξ ) /v ( y ) H ( x − ξ ) − δ ( x − ξ ) i δ ( y − η )+ (cid:2) − δ ( y − η ) (cid:3) δ ( x − ξ ) , (4)where H ( x ) is the unit step function. The nonlocal equation (3) governs the pdf of the particle in Zappa dispersion through the following two physicalmechanisms. • We suppose that, at exponentially distributed time intervals with meanone, the particle gets ‘zapped’ across the channel (by a burst of inter-mittent turbulence for example) and lands at any cross channel po-sition y with uniform distribution. Consequently the Fokker–Planck pde (3) for the pdf contains the terms u t = (cid:2) R − u dy − u (cid:3) + · · · . • Further, suppose that, at exponentially distributed time intervals withmean one, the particle jumps in x a distance to the right, a distancewhich is exponentially distributed with some given mean v ( y ) . Con-sequently the Fokker–Planck pde (3) for the pdf contains the terms u t = (cid:2) v ( y ) e − x/v ( y ) ⋆ u − u (cid:3) + · · · , in terms of the upstream convolution e − x/v ( y ) ⋆ u = Z x − ∞ e −( x − ξ ) /v ( y ) u ( ξ , y , t ) dξ . (5)4e derive the macroscale model that the cross-sectional mean field U ( x , t ) evolves according to an advection-diffusion pde : U t ≈ A U x + A U xx . Thefield U ( x , t ) may be viewed as the marginal probability density of the par-ticle being at x , averaged over the cross-section y . Innovatively, we putthe macroscale modelling on a rigorous basis that additionally quantifiesthe error.In particular, say we choose v ( y ) := − y then computer algebra (Appendix A)readily derives that over large space-time scales, and after transients decayroughly like e − t , from every initial condition the Zappa system (3) has thequasistationary distribution (Pollett & Roberts 1990, e.g.) u ( x , y , t ) ≈ U + ( y − ) ∂U∂x + ( y − y + ) ∂ U∂x , (6a)such that ∂U∂t = − ∂U∂x + ∂ U∂x + ρ , (6b)in terms of a macroscale variable here chosen to be the cross-sectional mean, U ( x , t ) := Z − u ( x , y , t ) dy . (6c)The macroscale pde (6b) is a precise equality because we include the errorterms in our analysis to find a precise, albeit complicated, expression for thefinal error ρ . The remainder error ρ in (6b) has the form ρ := r + h Z , W : B e B t ⋆ ~ r ′ i + h Z , W : ~ r ′ i − A h Z , W : e B t ⋆ ~ r ′ i − A h Z , W : e B t ⋆ ~ r ′ i (6d)where here the convolutions are over time, f ( t ) ⋆ g ( t ) = R t f ( t − s ) g ( s ) ds ,and other symbols are introduced in the next Section 3. We anticipate thiserror ρ is • ‘small’ in regions of slow variations in space, small gradients, and • ‘large’ in regions of relatively large gradients such as spatial boundarylayers.Then, simply, the macroscale pde model (6b) is valid whenever and whereverthe error ρ is small enough for the application purposes at hand. The nextsection includes deriving this error term and clarifies the notation.5 Many kernels generate local models
Inspired by earlier research (Roberts 2015 a , Proposition 1), this section’s aimis to rigorously derive and justify the model (6) that governs the emergentmacroscale evolution of Zappa dispersion. The algebra starts to ‘explode’—Section 4 discusses how to compactly do the algebra in physically meaningfulforms, and connect to other mathematical methodologies.To derive the advection-diffusion model (6b) we truncate the analysis tosecond order quadratic terms. Higher-orders appear to be similar in nature,but much more involved algebraically, and are left for later development. Let’s analyse the dynamics in the spatial locale about a generic longitudinalcross-section X ∈ X . Then invoke Lagrange’s Remainder Theorem—whichempowers us to track errors—to expand the pdf as u ( x , y , t ) = u ( X , y , t ) + u ( X , y , t )( x − X ) + u ( X , x , y , t ) ( x − X )
2! , (7)where u := u and u := ∂u/∂x both evaluated at the cross-section x = X ,and where u := ∂ u/∂x evaluated at some point x = ˆ x ( X , x , y , t ) which issome definite (but usually unknown) function of cross-section X , longitudinalposition x , cross-section position y , and time t . By the Lagrange RemainderTheorem, the location ˆ x satisfies X ≶ ˆ x ≶ x . The function ˆ x is implicit inour analysis because it is hidden in the dependency upon x of the secondderivative u ( X , x , y , t ) .Substitute (7) into the Zappa nonlocal equation (3) to obtain ∂u ∂t + ∂u ∂t ( x − X ) + ∂u ∂t ( x − X ) = Z Y (cid:20) Z X k ( x , ξ , y , η ) dξ (cid:21) u ( X , η , t ) dη + Z Y (cid:20) Z X k ( x , ξ , y , η )( ξ − X ) dξ (cid:21) u ( X , η , t ) dη + Z Y Z X k ( x , ξ , y , η ) ( ξ − X ) u ( X , ˆ ξ , η , t ) dξ dη . (8)The effect at cross-section x of the n th moment of the kernel at cross-section X is summarised in the integrals R X k ( x , ξ , y , η ) ( ξ − X ) n n ! dξ . So define6he local n th moment of the kernel to be, for every n > k n ( X , y , η ) := Z X k ( X , ξ , y , η ) ( ξ − X ) n n ! dξ = (cid:2) (− v ) n − δ n (cid:3) δ ( y − η ) + (cid:2) − δ ( y − η ) (cid:3) δ n (9)upon substituting the Zappa kernel (4). This Zappa problem is homogeneousin x , as are many problems, and so the kernel moments k n are independentof the cross-section X (except near the boundary inlet and outlet).The last integral term in the local expansion (8) requires special considera-tion: apply Lagrange’s Remainder Theorem to write u ( X , ξ , η , t ) = u ( X , X , η , t )+( ξ − X ) u x ( X , ˆ ξ , η , t ) for some uncertain function ˆ ξ ( X , ξ , η , t ) that satisfies X ≶ ˆ ξ ≶ ξ for every η , t , and where u x := ∂/∂x [ u ( X , x , η , t )] . Then thelast term distributes into two: Z Y Z X k ( x , ξ , y , η ) ( ξ − X ) u ( X , ˆ ξ , η , t ) dξ dη = Z Y Z X k ( x , ξ , y , η ) ( ξ − X ) dξ | {z } k ( X , y , η ) u ( X , X , η , t ) dη + Z Y Z X k ( x , ξ , y , η ) ( ξ − X ) u x ( X , ˆ ξ , η , t ) dξ dη | {z } a remainder, with a third x derivative in u x .Define u ( X , y , η ) := u ( X , X , y , η ) for notational consistency with lowermoments—see the definition (9).The local equation (8) is exact everywhere, but is most useful in the vicinityof the cross-section X , that is, for small ( x − X ) . Notionally we want to‘equate coefficients’ of powers of ( x − X ) in (8), but to be precise we mustcarefully evaluate lim x → X of various x -derivatives of (8). For example, let x → X in (8), then ∂u ∂t = Z Y k ( X , y , η ) u ( X , η , t ) dη + Z Y k ( X , y , η ) u ( X , η , t ) dη + Z Y k ( X , y , η ) u ( X , η , t ) dη + Z Y Z X k ( X , ξ , y , η ) ( ξ − X ) u x ( X , ˆ ξ , η , t ) dξ dη .Let’s rewrite this conveniently and compactly as the integro-differential equa-tion ( ide ) ∂u ∂t = L u + L u + L u + r , (10)7or y -operators defined to be, from the moments (9), L n u := Z Y k n ( X , y , η ) u | y = η dη = (cid:14) R − u dy − u , n = [− v ( y )] n u , n =
1, 2, . . . . (11)The ide (10) also has the remainder r which couples the local dynamics toneighbouring locales via u x and is the n = r n ( X , y , t ) := Z Y Z X ∂ n k∂x n (cid:12)(cid:12)(cid:12) x = X ( ξ − X ) u x ( X , ˆ ξ , η , t ) dξ dη . (12)Now we can see how this approach to modelling the spatial dynamics works:given that the y -operators (11) are evaluated at X , the spatially local powerseries with remainder, in ide s like (10), ‘pushes’ the coupling with neigh-bouring locales to a higher-order derivative term in r , here third-order viathe u x factor. Hence the local dynamics in u , u , u are essentially iso-lated from all other cross-sections whenever and wherever the coupling r is small enough for the purposes at hand—here when third derivatives aresmall—that is, when the solutions are, in space, slowly varying enough.The previous paragraph obtains the ide for u by simply taking the limitof (8) as x → X . We straightforwardly and similarly obtain ide s for u and u by finding the limits of spatial derivatives of (8):lim x → X ∂ (8) ∂x = ⇒ ∂u ∂t = L u + L u + r ; (13a)lim x → X ∂ (8) ∂x = ⇒ ∂u ∂t = L u + r ; (13b)for local coupling remainders r and r defined by (12). This section considers the collection of ‘local’ systems as one ‘global’ (inspace X ) system. Then theory establishes that the advection-diffusion pde (6b)arises as a globally valid, macroscale, model pde .Denote the vector of coefficients ~ u ( X , y , t ) := ( u , u , u ) , and similarly forthe local coupling remainder ~ r ( X , y , t ) := ( r , r , r ) . Then write the ide s (10)and (13), in the form of the ‘forced’ linear system d ~ udt = L L L L L L | {z } L ~ u + ~ r ( X , t ) . (14)8or upper triangular matrix/operator L . The system (14) might appearclosed, but it is coupled via the derivative u x , through the ‘forcing’ re-mainders ~ r , to the dynamics of cross-sections that neighbour X .At each locale X ∈ X , treat the remainder coupling ~ r (third-order) as a per-turbation (and if this was a nonlinear problem, then the nonlinearity wouldalso be part of the perturbation). Thus to a useful approximation the globalsystem satisfies the local linear ode s d ~ u/dt ≈ L ~ u for each X ∈ X . Hence,the linear operator L is crucial to modelling the dynamics: all solutions arecharacterised by the eigenvalues of L . Since L is block triangular, a struc-ture exploited previously (Roberts 2015 a , § L = R − u dy − u (definition (11)). Here it is straightforward to verifythat the y -operator L has: • one 0 eigenvalue corresponding to eigenfunctions constant across thechannel; and • an ‘infinity’ of eigenvalue − d ~ u/dt = L ~ u + ( perturbation ) at every X ∈ X ,and because of the ‘infinity’ of the continuum X , the linearised system has a‘thrice-infinity’ of the 0 eigenvalue, and a ‘double-infinity’ of eigenvalue − ( ∞ ) ’-D slow manifold—the quasistationary (6a);2. which is exponentially quickly attractive to all initial conditions, withtransients roughly e − t —it is emergent; and3. which we approximate by approximately solving the governing differ-ential equations (14)—done in encoded form by Appendix A.We obtain a useful approximation to the global slow manifold by neglectingthe ‘perturbing’ remainder ~ r . Because the remainder ~ r is the only couplingbetween different locales X this approximation may be constructed inde-pendently at each and every cross-section X . Further, because the Zappasystem is homogeneous in space, the construction is identical at each andevery X ∈ X . These two properties vastly simplify the construction of theattractive slow manifold.Neglecting the coupling remainder ~ r gives the linear problem d ~ u/dt = L ~ u .The approximate slow manifold is thus the zero eigenspace of L . We findthe zero eigenspace via (generalised) eigenvectors. With the (generalised)eigenvectors in the three columns of block-matrix V , in essence we seek9 u ( t ) = V ~ U ( t ) such that d ~ U/dt = A ~ U for 3 × A having all thezero eigenvalues. To be an eigenspace we need to solve LV = VA . Now let’sinvoke previously established results (Roberts 2015 a , § L , defined in (14), has the same block Toeplitz structure as previously(Roberts 2015 a , (7) on p.1496). Consequently (Roberts 2015 a , Lemma 4), abasis for the zero eigenspace of L is the collective columns of V = V V V V V V , and further, A = A A A .The hierarchy of equations to solve for the components of these has beenpreviously established (Roberts 2015 a , Lemma 3): the hierarchy is essentiallyequivalent to the hierarchy one would solve if using the method of multiplescales, but the theoretical framework here is more powerful. The upshot isthat for Zappa dispersion, in which overlines denote cross-channel averages, V = V = v − v , V = ( v − v − vv + v ) , A = − v , A = ( v − v ) + v . (15)In the specific case of v ( y ) = − y , these expressions reduce to the coeffi-cients and polynomials of the slowly varying, slow manifold, model (6).So now we know that the evolution on the zero eigenspace, the approx-imate slow manifold, is d ~ U/dt = A ~ U : let’s see how this translates intothe macroscale pde (6b). Now, the first line of d ~ U/dt = A ~ U is the ode dU /dt = A U + A U . Defining U = U ( X , t ) := u ( X , y , t ) , Proposition 6of Roberts (2015 a ) applies, and so generally U ( x , t ) satisfies the macroscaleeffective advection-diffusion pde (2)—a pde that reduces to the specific (6b)in the case v ( y ) = − y . Now we treat the exact ‘local’ system d ~ u/dt = L ~ u + ~ r as non-autonomously‘forced’ by coupling to all cross-sections in X through the remainder (akaMori–Zwanzig transformation, e.g., Venturi et al. 2015). There are two justi-fications, both a simple and a rigorous, for being able to project such ‘forcing’onto the local model. First, simply, the rational projection of initial condi-tions for low-dimensional dynamical models leads to a cognate projectionof any forcing (Roberts 1989, § X ∈ X . Keep clear the contrasting points of view that contribute:on the one hand we consider the relatively low-dimensional system at eachlocale X in space, a system that is weakly coupled to its neighbours; on theother-hand we consider the relatively high-dimensional system of all locales X coupled together and then theory establishes global properties.The upshot is that here we need to project the coupling remainder ~ r ( t ) onto each local slow manifold. Fortunately, the structure of the linear localdynamics (14) is identical to that discussed by Roberts (2015 a ). Hence,many of the results reported there apply here. Linear algebra involvingadjoint eigenvectors Z and W n ( L † Z = L † W = WA , Roberts 2015 a , § e − t ⋆ ~ r , leads tothe error formula (6d) (equation (23) from Roberts 2015 a ). Then the generalmacroscale advection-diffusion model (2) becomes exact with the error term ρ included (here the error (6d) is third-order in spatial derivatives) ∂U∂t = A ∂U∂x + A ∂ U∂x + ρ .Then, simply, the macroscale effective advection-diffusion model pde (2) isvalid simply whenever and wherever the error term ρ is acceptably small.There is: no ǫ ; no limit; no required scaling; no ‘balancing’; no ad hochierarchy of space-time variables. It is very tedious to perform all the algebraic machinations of Section 3 onthe Taylor series coefficients. Instead, we may compactify the analysis bydefining the quadratic generating polynomial (Roberts 2015 a , § u ( X , ζ , y , t ) := u ( X , y , t ) + ζu ( X , Y , t ) + ζ u ( X , X , y , t ) (16)(or a higher-order polynomial if the analysis is to higher-order). This gener-ating polynomial then satisfies the exact differential equation (17). Consider11 ˜ u/∂t , at ( X , ζ , y , t ) , and substitute the equations (14) for the Taylor coef-ficients at ( X , y , t ) : ∂ ˜ u∂t = ∂u ∂t + ζ ∂u ∂t + ζ ∂u ∂t = L u + L u + L u + r + L ζu + L ζu + ζr + L ζ u + ζ r = L ˜ u + L ∂ ˜ u∂ζ + L ∂ ˜ u∂ζ + ˜ r = ⇒ ∂ ˜ u∂t = (cid:20) L + L ∂∂ζ + L ∂ ∂ζ (cid:21) ˜ u + ˜ r (17)for the generating polynomial of the coupling remainder, ˜ r := r + ζr + ζ r .Appropriate analysis of the ide (17) then reproduces the previous Section 3.But the algebra is done much more compactly as the separate components u , u , u are all encompassed in the one generating polynomial ˜ u . One important prop-erty of the analysis is that although we normally regard the derivative ∂/∂ζ as unbounded, in the analysis of ide (17) the space of functions is just thatof quadratic polynomials in ζ , and so here ∂/∂ζ is bounded, as well as pos-sessing other nice properties.Indeed, since we are only interested in the space of quadratic polynomialsin ζ , the analysis neglects any term O (cid:0) ζ (cid:1) . Equivalently, we would workto ‘errors’ O (cid:0) ∂ /∂ζ (cid:1) . This view empowers us to organise the necessaryalgebra in a framework where we imagine ∂/∂ζ is ‘small’. Note: in themethodology here ∂/∂ζ is not assumed small, as we track errors exactly inthe remainder ˜ r , it is just that we may organise the algebra as if ∂/∂ζ wassmall. Such organisation then leads to the same hierarchy of problems as inSection 3.2, just more compactly. Connect to extant methodology
Since the notionally small ∂/∂ζ iseffectively a small spatial derivative, we now connect to extant multiscalemethods that a priori assume slow variations in space. That is, we now showthat the non-remainder part of ide (17) appears in a conventional multiscaleapproximation of the governing microscale system (1).In conventional asymptotics we invoke restrictive scaling assumptions at thestart. Here one would assume that the solution field u ( x , y , t ) is slowly-varying in space x . Then the argument goes that the field may be usefully12ritten near any X ∈ X as the local Taylor quadratic approximation u ( ξ , y , t ) ≈ u | ξ = X + ( ξ − X ) u ξ | ξ = X + ( ξ − X ) u ξξ | ξ = X .Substituting into the nonlocal microscale (1) gives, at ( X , y , t ) and lettingdashes/primes denote derivatives with respect to the first argument, ∂u∂t = Z Y Z X k ( X , ξ , y , η ) u ( ξ , η , t ) dξ dη ≈ Z Y Z X k ( X , ξ , y , η ) (cid:20) u | ξ = X + ( ξ − X ) u ′ | ξ = X + ( ξ − X ) u ′′ | ξ = X (cid:21) dξ dη = Z Y Z X k ( X , ξ , y , η ) dξ u ( X , η , t ) + Z X k ( X , ξ , y , η )( ξ − X ) dξ u ′ ( X , η , t )+ Z X k ( X , ξ , y , η ) ( ξ − X ) dξ u ′′ ( X , η , t ) dη = Z Y k ( X , y , η ) u ( X , η , t ) + k ( X , y , η ) u ′ ( X , η , t )+ k ( X , y , η ) u ′′ ( X , η , t ) dη = L u + L u ′ + L u ′′ . (18)Now the generating polynomial ˜ u , defined by (16), is such that u ( X + ζ , y , t ) = ˜ u ( X , ζ , y , t ) + O (cid:0) ζ (cid:1) . Hence, rewriting the approximate pde (18)for u ( X + ζ , y , t ) at fixed X gives precisely the ide (17) except that the remain-der coupling ˜ r is omitted. Consequently, extant multiscale methodologiescontinuing on from pde (18) generate equivalent results to that of Section 3,but in a different framework—a framework without the error term (6d).Most extant multiscale analysis invokes, at the outset, balancing of scalingparameters, requires a small parameter, is only rigorous in the limit of infinitescale separation, and often invents heuristic multiple space-time variables.The approach developed herein connects with such analysis, but is consider-ably more flexible and, furthermore, justifies a more formal approach devel-oped 30 years ago (Roberts 1988), and implemented in Appendix A. Furtherthis approach derives the rigorous error expression (6d) at finite scale sepa-ration. I continue to conjecture that truncations to orders other than quadratic give corre-sponding analysis and results. Ongoing research will elucidate. Conclusion
This article initiates a new multiscale modelling approach applied to a spe-cific basic problem. This article considers the scenario where the givenphysical problem (1) has non-local microscale interactions, such as inter-particle forces or dynamics on a lattice. Many extant mathematical method-ologies derive, for such physical systems, an approximate macroscale pde ,such as the advection-diffusion (2). The novelty of our approach is thatit derives a precise expression for the error of the macroscale approximate pde , here (6d). Then, simply, and after microscale transients decay, themacroscale advection-diffusion pde (2) is valid wherever and whenever thequantified error (6d) is acceptable.Of course, in all such applications, we need the third moment of the mi-croscale interaction kernel k ( x , ξ , y , η ) to exist (see definition (12)) for theerror analysis of Section 3.1 to proceed and provide the error term. All mo-ments exist for the Zappa problem, see (9). If, in some application, the thirdmoment does not exist, but the second moment does, then the advection-diffusion pde (2) may be an appropriate macroscale model, but this workwould not provide a quantifiable error.Another important characteristic of our new approach is that the validityof the macroscale pde is not confined by a limit ‘ ǫ → X . Further, and in contrast to most extant methodologies, theapproach here should generalise in further research to arbitrary order modelsjust as it does when the microscale is expressed as pde s (Roberts 2015 a ).The developed scenario here is that of linear nonlocal systems (1). How-ever, key parts of the argument are justified with centre manifold theory(Aulbach & Wanner 2000, Potzsche & Rasmussen 2006, Haragus & Iooss 2011,Roberts 2015 b , e.g.). Consequently, further research should be able to showthat cognate results hold for nonlinear microscale systems.With further research, correct boundary conditions for the macroscale pde sshould be derivable by adapting earlier arguments to derive rigorous bound-ary conditions for approximate pde s (Roberts 1992, Chen et al. 2018).Interesting applications of this novel approach would arise whenever thereare microscale nonlocal interactions in the geometry of problems such as(e.g., Roberts 2015 b ) dispersion in channels and pipes, the lubrication flowof thin viscous fluids, shallow water approximations whether viscous or tur-bulent, quasi-elastic beam theory, long waves on heterogeneous lattices, and14attern evolution. Acknowledgements
This research was partly supported by the AustralianResearch Council with grant DP180100050.
A Computer algebra derives macroscale PDE
The following computer algebra derives the effective advection-diffusion pde (6b),or any higher-order generalisation, for the microscale nonlocal Zappa sys-tem (3). This code uses the free computer algebra package Reduce. Analo-gous code will work for other computer algebra packages, and/or for cognateproblems (Roberts 2015 b , e.g.). % a d v e c t i o n − d i f f u s i o n PDE o f Zappa t r a n s p o r t i n a c h a n n e l % AJR, 20 Jan 2017 −−
20 Jan 2020 > % t r u n c a t e t o t h i s o r d e r o f e r r o r % uu ( n ):= d f ( uu , x , n ) { d f ( uu ( ˜ n ) , x)= > uu ( n+1) , d f ( uu ( ˜ n ) , t)= > d f ( g , x , n ) } ;8 o p e r a t o r mean ; l i n e a r mean ; % a v e r a g e a c r o s s c h a n n e l { mean ( 1 , y)= > mean ( y ˆ˜˜p , y)= > (1+( − } ;1011 % P r e p r o c e s s n o n l o c a l x − jumping : i n e s s e n c e f i n d s t h e % k e r n e l i n t e g r a l s are ( − v )ˆ n
13 depend w, x ; % dummy f u n c t i o n f o r u ( x ) % Taylor expand w( x i )=w( x+z ) where z=x i − x
15 jmp:= for n : = 0 : deg ((1+d ) ˆ 9 9 , d ) sum dˆn ∗ d f (w, x , n ) ∗ z ˆn/ f a c t o r i a l ( n ) $16 jmp:= i n t ( exp ( z /v ) ∗ jmp , z ) $ % i n t e g r a t e ex p ( ( x i − x )/ v )w( x ) % e v a l from z= − i n f t o 0 f o r t h e c o n v o l u t i o n
18 jmp:= sub ( z =0 ,jmp/v) − w$1920 % i t e r a t e from q u a s i − e q u i l i b r i u m s t a r t
21 u:=uu ( 0 ) $ g :=0 $22 for i t : = 1 : 9 9 do b e g i n23 r e s := − d f ( u , t )+sub ( { w=u , v=1 − y ˆ 2 } , jmp)+( − u+ mean ( u , y ) ) ;24 w r i t e l e n g t h r e s := length ( r e s ) ;25 g := g+(gd:= mean ( r e s , y ) ) ;26 u:=u+r e s − gd ;27 i f r e s =0 then w r i t e ” S u c c e s s : ” , i t := i t +10000;28 end ;29 w r i t e ”The r e s u l t i n g slo w m a n i f o l d and e v o l u t i o n i s ” ;30 u:=u ; duudt := g ; end ; References
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