Towards Perfectly Matched Layers for time-dependent space fractional PDEs

Transcription

1 Towards Perfectly Matced Layers for time-dependent space fractional PDEs Xavier Antoine, Emmanuel Lorin To cite tis version: Xavier Antoine, Emmanuel Lorin. Towards Perfectly Matced Layers for time-dependent space fractional PDEs <al > HAL Id: al ttps://al.arcives-ouvertes.fr/al Submitted on 20 Dec 2018 HAL is a multi-disciplinary open access arcive for te deposit and dissemination of scientific researc documents, weter tey are publised or not. Te documents may come from teacing and researc institutions in France or abroad, or from public or private researc centers. L arcive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recerce, publiés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires publics ou privés.

2 Towards Perfectly Matced Layers for time-dependent space fractional PDEs Xavier ANTOINE a, Emmanuel LORIN b,c a Institut Elie Cartan de Lorraine, Université de Lorraine, UMR 7502, Inria Nancy-Grand Est, F Vandoeuvre-lès-Nancy Cedex, France b Scool of Matematics and Statistics, Carleton University, Ottawa, Canada, K1S 5B6 c Centre de Recerces Matématiques, Université de Montréal, Montréal, Canada, H3T 1J4 Abstract Perfectly Matced Layers PML) are proposed for time-dependent space fractional PDEs. Witin tis approac, widely used powerful Fourier solvers based on FFTs can be adapted witout muc effort to compute Initial Boundary Value Problems IBVP) for well-posed fractional equations wit absorbing boundary layers. We analyze matematically te metod and propose some illustrating numerical experiments. Keywords: fractional partial differential equations; unbounded domain; perfectly matced layer; Fourier pseudospectral approximation; time splitting sceme Contents 1 Introduction 2 2 Fractional Perfectly Matced Layers FPMLs) and teir approximations Fractional Perfectly Matced Layers FPMLs) Approximate FPMLs for RL operators Focus on FPMLs for te fractional laplacian FPMLs pseudospectral approximation scemes: implementation and numerical examples IBVP wit space fractional operators of order α less tan IBVP wit space fractional operators of order α larger or equal to Higer dimensional FPDEs Approximate FPMLs for RL operators Focus on FPMLs for FPDE involving te 2D fractional laplacian addresses: xavier.antoine@univ-lorraine.fr Xavier ANTOINE), elorin@mat.carleton.ca Emmanuel LORIN) Preprint submitted to Elsevier December 20, 2018

3 5 A few results in matematical and numerical analysis Some first well-posedness results for te IVP and IBVP Some partial stability results of te discretization scemes Conclusion Introduction Over te last decade, te study of Fractional Partial Differential Equations FPDEs) became a uge domain of investigation in many areas of science and engineering [24, 28, 35, 36]. Terefore, simulating accurately and efficiently a FPDE is a major callenge in computational science. In tis paper, we are interested in suitably truncating te infinite spatial computational domain for one-dimensional and iger-dimensional) space fractional Initial Value Problems IVPs) of te form t ut, x) + v [α] x) α x ut, x) = fut, x)), 1) or t ut, x) + v [α] x) α xut, x) = fut, x)), 2) were x R and t > 0, {v [α] } are smoot real- or purely complex-valued functions, R being a finite set of strictly positive real numbers, and f is square integrable. To tis system is added an initial data ut = 0, x) = u 0 x). Spatially bounding te above IVPs naturally leads to Initial Boundary Value Problems IBVPs) set on te truncated domain. Simple boundary conditions like omogeneous Diriclet, Neumann or Robin boundary conditions as well as periodic boundary conditions can be used in some situations to get a finite computational domain. However, tis coice of boundary condition does not always provide an admissible approximate solution. For integer order Partial Differential Equations PDEs) α N), many approaces can be used to build [1, 5, 19, 43] absorbing, transparent or non-reflecting boundary conditions ABC, TBC, NRBC), according e.g. to te structure and pysical/matematical properties of te PDEs, te kind of expected sceme to be used... Usually, constructing suc boundary conditions requires nontrivial matematical analysis and can often be recast as te fundamental problem of building te exact Diriclet-to-Neumann operator or some of its approximations for te associated PDE. In addition, several complex matematical definition, well-posedness) and computational issues stability, accuracy, computational complexity) are usually related to suc boundary conditions. For te space FPDEs, to te best of autors knowledge, only a few derivations of ABCs/TBCs/NRBCs ave been recently obtained [6, 20, 27, 29]. Furtermore, closely related are newly designed truncation tecniques for nonlocal models see e.g. [22, 23]), generalizing in some sense te notion of FPDEs. Let us remark tat suc boundary conditions were also nontrivially derived for time-fractional PDEs [32, 47, 49]. 2

4 Finally, let us note tat tese boundary conditions cannot be a priori implemented into a Fourier pseudospectral approximation sceme [2, 3, 7, 14, 33, 40] based on FFTs since te enforced boundary conditions are not periodic. An alternative metod to avoid spurious unpysical reflection at te domain boundary consists in using te metod of Perfectly Matced Layers PMLs) wic as been extensively studied for many integer order PDE models see e.g. [8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 25, 26, 43, 44, 46]). PMLs usually enjoy some good matematical and numerical properties: easiness of implementation, flexibility, accuracy, stability,... wic made tem extremely popular since teir introduction in te seminal paper by Bérenger [9] in 1994 for simulating electromagnetic waves. Even if PMLs are very attractive, to te best of our knowledge, te extension to te space FPDEs as never been studied. Te aim of tis paper is to contribute to deriving PMLs for some classes of FPDEs, resulting in Fractional PMLs FPMLs). As for te integer case, tese FPMLs are easy to integrate into te FPDE matematical formulation. Of course, tey also need to be fixed for eac FPDE, as for te standard case, and in particular concerning te absorption profiles and teir tuning parameters. Terefore, in te present paper, instead of focusing on one specific FPDE, we explain te general idea and illustrate te accuracy of FPMLs troug explicit examples. For a better understanding of te full approac, more focused future investigations are required to develop optimal FPMLs for a given FPDE. In addition, we also introduce a specific Fourier based pseudospectral discretization sceme for approximating te FPMLs model wic appears to be very flexible and accurate for te resulting modified problem. However, it is also clear tat many oter discretization approaces could be investigated, wic sould be again furter studied. In particular, one can freely coose te boundary condition imposed at te outer domain boundary. Terefore, tis allows for adapting many already existing discretization scemes. Even if we start wit Riemann-Liouville operators [24, 35, 36], te extension to oter definitions of fractional derivative operators sould be possible, as we notice for Caputo-type derivative operators. Moreover, we also prospect te extension to iger-dimensional problems and FPDEs involving fractional laplacians [31, 34]. Te metod can be adapted but some nontrivial questions remain open for furter improvements. Finally, well-posedness of te truncated problem is partially addressed ere as well as a stability analysis of te sceme. Finally, extending te metod to time-fractional PDE seems possible. Te paper is organized as follows. In Section 2, we present te derivation and possible approximations of PMLs for one-dimensional space FPDEs. Te case of fractional laplacians is also considered. Section 3 is dedicated to te numerical approximation of te FPDE wit FPML by a well-designed pseudospectral metod combined wit time-splitting scemes. Several numerical experiments illustrate te ideas and metods presented in tis paper. Extension to iger dimensional problems is discussed in Section 4. In Section 5, we analyze te well-posedness of te FPDE wit FPML, as well as te stability of an implicit sceme derived in tis paper. We finally conclude in Section 6. 3

5 2. Fractional Perfectly Matced Layers FPMLs) and teir approximations Some general definitions and basic properties about fractional derivatives are recalled below [21, 24, 28, 35, 36]. All along te paper, in particular in Section 5, some key results about fractional Sobolev spaces will be needed and recalled. Here, we simply provide te minimal requirement about fractional derivatives for a good understanding of te material presented in te paper Fractional Perfectly Matced Layers FPMLs) For α R + and p = α + 1 α being te integer part of α), we define te special Gamma function Γ, for Rez) > 0, as Γz) := 0 x z 1 e x dx, wit Γp) = p 1)!, for p N. Te fractional derivative tat we consider in tis paper is te so-called Riemann-Liouville RL) fractional derivative of order α over te interval ; x], wic is defined by RL D α x ux) = 1 d p Γp α) dx p x uy) x y) α p+1dy, 3) wile te rigt RL fractional derivative is given by RL xd α ux) = d p 1)p Γp α) dx p + x uy) y x) α p+1dy. Similarly, we introduce te left fractional Riemann-Liouville integral operator of order α as Troug tese notations, we ave te relation RL Ix α ux) = 1 x uy) Γα) x y) 1 αdy. 4) RL D α xux) = dp dx p RL I p α x ux). 5) Let us remark tat many oter fractional derivatives/integrals can also be defined [24, 35, 36], like e.g. te well-known Caputo C) derivative of order α C D α x ux) = 1 x u p) y) Γp α) x y) α p+1dy. 6) In te sequel of te paper, we restrict our study to te Riemann-Liouville fractional partial derivatives wit respect to x, but it may be possible tat some of te ideas can be extended to oter definitions of te fractional operators as suggested below. From now on, te RL 4

6 fractional derivative will be denoted by x α or x α, respectively corresponding to RL Dx α and RL x D α. We again refer to [24, 35, 36] for te matematical properties of fractional operators. Let us now consider te basic approac related to PMLs. Te equations under consideration are defined on an open one-dimensional bounded pysical domain denoted by D Py, but te extension to iger-dimensional problems can be derived see also Section 4). As usual, we add a layer wic is called D PML, surrounding D Pys, stretcing te x-coordinate. Te overall computational domain is ten defined by: D = D Py D PML. For te onedimensional case, D = [ L, L] and D Pys = [ L, L ], wit L < L, te layer tickness being δ x. Usual PMLs require a complex stretcing of te real spatial coordinate x suc as xx) = x + e iθ x were te absorbing function σ : D R is defined [4] as α N ) L σs)ds, 7) σx) = { σ x L), L x < L, 0, x < L. 8) Te rotation angle θ is usually fixed by te problem under study. For example, θ = π/2 is often considered for integer order) time armonic Helmoltz-type problems [8, 12, 13, 17, 43, 44] wile θ = π/4 is more adapted to Scrödinger problems [1, 5, 46, 48]. Here, we derive te PML equation only for te rigt layer, te extension to te left side being straigtforward by a symmetry argument. In addition, te boundary condition at point {L} can be fixed for example as a omogeneous Diriclet or Neumann boundary condition or a periodic boundary condition, since te solution is supposed to be small enoug after being damped. Tis provides a great flexibility for te coice of te discretization metod. We will later coose a periodic boundary condition so tat Fourier pseudospectral approximation metods based on FFTs can be used. Te starting point of PMLs for PDEs wit integer order partial derivatives considers te following damped operator x x := e iθ σx)) x, 9) modifying ence te initial PDE. Following tis idea, if e.g. 0 < α < 1, we can introduce te new complex-valued fractional derivative operators as α x ux) = = x 1 1 d uy) Γ1 α) 1 + e iθ σx)) dx x y) α1 + eiθ σy)) 1 α dy e iθ σ) α x 1 + e iθ σ) 1 α u). 10) 5

7 For 1 < α < 2 p = 2), ten one gets from 3) and by a similar argument 1 1 d α x ux) = Γ2 α) 1 + e iθ σx)) dx 1 d uy) 1 + e iθ σx)) dx x y) α 11 + eiθ σy)) 2 α dy) 1 d = 1 + e iθ σ) dx 1 d 1 + e iθ σ) RL Ix 2 α 1 + e iθ σ) 2 α u))), dx 11) wic degenerates, by evaluating from te inner to te outer parenteses, into te standard formula 1 xux) = 1 d 2 x 1 du uy)dy = Γ1) d x e iθ σ dx, 12) tat could be also obtained from 10) taking te lower limit on α. Similarly, one gets x 2 xu = 1 d 2 Γ2) d x 2u = 1 d 1 + e iθ σ) dx 1 du 1 + e iθ σ) ). 13) dx A generalization of te absorbing RL fractional derivative operators of order α R could be derived following te formula setting α x ux) = dp d x p RL I p α x 1 + e iθ σ) p α u), 14) d p d x = 1 d p 1 + e iθ σ) dx 1 d 1 + e iθ σ) dx...)) p times). In te case of te Caputo derivative 6), ten a natural extension of a PML Caputo fractional derivative operator of order α R is C D α x ux) = 1 x u p) y) Γp α) x y) α p+11 + eiθ σ) α dy = RL Ix p α 1 + e iθ σ) α u p) ). 15) Coosing an absorption function is not trivial and is based on te underlying PDE model. Te first profiles proposed in te literature are based on polynomial functions. Here, we consider te quadratic and cubic functions Type I: σ 0 x + δ x ) 2, Type II: σ 0 x + δ x ) 3, 16) were σ 0 is a real-valued positive parameter to adjust. Oter possibilities derived for example for te Helmoltz equation [12, 13] and later used for te Scrödinger equation [4] include unbounded functions σ suc tat L L σx)dx = +. 17) 6

8 Among tese, we use te following discontinuous and continuous functions Type III: σ 0 x, Type IV: σ 0 x 2, Type V: σ 0 x σ 0, Type VI: σ 0 δ x x σ 0. 2 δ 2 x 18) Let us remark tat oter coices of absorbing functions may be more suited to fractional equations, but tis point is not analyzed in te current paper. It is well-known tat tuning te parameter σ 0 in PMLs is relatively problem dependent. As noticed e.g. in [4, 12, 13], Types III-VI usually exibit a larger stability region for a numerical discretization for Helmoltz and Scrödinger problems Approximate FPMLs for RL operators Even if te above Fractional PML FPML) operators could be used, we propose ere an approximate and modified version of a PML fractional derivative operator wic can be related to te previous RL operators. Let us first recall tat, for te RL operators, we ave te following Fourier symbols F α x u)ξ) = iξ) α Fu)ξ), F RL I α x u)ξ) = iξ) α Fu)ξ), 19) were F is te Fourier transform according to x and ξ is te corresponding dual co-variable. Terefore, if one considers te principal symbol σ p in te sense of pseudodifferential operators [42] of te operator α x defined by 14), one gets, for α R, σ p α x ) = 1 + e iθ σ) α iξ) α. 20) Consequently, anoter fractional PML operator tat could be used to bound te computational domain is 1 α x := 1 + e iθ σ) α α x, 21) ten generalizing te well-know formula 9) for integer order derivatives to fractional orders of differentiation. Different strategies can be developed for constructing te absorbing operators. Even if 21) is a reasonable approximation for α < 1, tis is not te unique coice. For α > 1, we would like to mimic tis for te FPML 14) corresponding to α = p, wenever p 1 < α < p. Witin tis aim, we set, for any α R +, wit α = n α + q α R +, β α := α n α + 1 R +, 22) n α := { α 1, if α N, α, if α R N, 23) 7

9 were α is te integer part of α, and q α := α n α R + is strictly less tan 1. Te main idea consists in decomposing te derivatives α x, wit α R +, as σ x α ) = iξ) α = iξ) βα ) nα+1 = iξ) qα iξ) nα, x α = x βα ) nα+1 = x qα x nα. We ten consider te following various FPML operators wic are approximations up to some lower order operators of te original FPML operators), for α > 1, were β α 0, 1), for α x Approac 0. For n α defined by 23), te first approac corresponds to 11) α x u = 1 d 1 + e iθ σ) dx e iθ σ) d dx RL Ix nα α 1 + e iθ σ) nα+1 α u)). Approac 1. Tis approac directly mimics 14), ten leading to σ α x ) 1 + e iθ σ) βα iξ) βα) n α+1. For u smoot enoug, tis leads to te corresponding approximation 1 α x u 1 + e iθ σ ) β α x βα ) nα+1 u. 24) Approac 2a. A closely related approac consists in approximating te symbol σ α x ) as follows yielding σ α x ) 1 + e iθ σ) 1 iξ) ) q α 1 + e iθ σ) 1 iξ) ) n α, α x u e iθ σ ) q α qα x e iθ σ ) nαu. x 25) Approac 2b. Symmetrically, one can consider α x u 1 nα nα 1 + eiθ σ) x e iθ σ ) q α x qα )u. 26) Approac 3. As previously explained, te last approac proposes to select te principal symbol to α x, i.e. 1 α x u 1 + e iθ σ ) α x α u. 27) 8

10 In Approaces 1 and 2, if α = p N {0, 1}, te RHS of 24) and 25) becomes 14). Example 1. As an illustration and to compare te various strategies, we report u in logscale) at T = 10 on Fig. 1 for te different FPML approaces, wen solving te FPDE { t ut, x) vx) 3/2 x ut, x) = 0, t > 0, x R, ut, 0) = u 0 x), x R, 28) were u 0 x) = N 1 e 2x2 +ik 0 x N being a L 2 D)-normalization constant), for k 0 = 1, te multiplicative function being vx) = e x 2. Te pysical domain of computation is D Pys := L ; L ) wile te extended domain wit FPMLs is D := L; L), setting L = 8 and L = 0.95L. Te approximate FPMLs is cosen using te Type I absorbing function 16), wit σ 0 = 10, θ = π/8, and periodic boundary conditions at te endpoints of te layers. More specifically, we are solving te IBVP t ut, x) vx) 3/2 x ut, x) = 0, t > 0, x D, ut, L) = ut, L), t > 0, 29) ut, 0) = u 0 x), x D, were 3/2 x is one of te FPML operators. Since α = 3/2, ten we ave: n α = 1, q α = 1/2 and β α = 3/4. Te reference solution is computed on a larger domain so tat no reflection arises at te rigt boundary. In addition, we also report te solution computed on D but wit periodic boundary conditions and no FPML. Te discretization parameters are: N x = 1001 and t = 10 2 N t = 2500). Te numerical metod is described later in te paper Section 3). Let us remark tat any oter discretization could be used for te FPML metod, te sceme depending on te boundary conditions set at te endpoints. Tis first example sows tat te different FPML approaces ave relatively similar absorption features, wic are all very effective. In particular, and to fix te ideas in te paper, we consider now te FPML based on te operator proposed in Approac 1, except if specified oterwise, wic is well-suited in te numerical simulation wile being simple. Since numerous parameters need to be tuned for optimizing te FPML e.g. Type of PML, σ 0, θ, δ x ) and because tis is problem dependent, we will address in details te question of selecting te best absorbing operator approximation for specific FPDE in a future work Focus on FPMLs for te fractional laplacian Let us now specifically look at te problem of building FPMLs related to FPDEs involving te 1D fractional laplacian α x, 0 < α 1, wic appears in many fractional models [6, 22, 23, 24, 28, 34]. Let us remark ere tat we do not use te operator x ) α and rater include te normalization constant e iπα in te multiplicative function v if necessary. First, we notice tat σ α x) = e iαπ ξ 2α = e iβαπ ξ 2βα ) nα+1 = e iqαπ ξ 2qα e inαπ ξ 2nα, α x = βα x ) nα+1 = qα x nα x. 9

11 Figure 1: Example 1. Comparison of te amplitude u logscale) for te Approaces 0, 1, 2a, 2b, 3, wit te FPML of Type I, pure Periodic Boundary Conditions i.e. witout any PML), and a reference solution computed on a large domain. For u smoot enoug, tis leads to te following possible coices of operators to define te FPMLs Approac 0. α xu 1 1 = 1/2 x 1/2 x 1 + eiθ σ) 1 + eiθ σ) Approac 1. α xu 1 ) 2nα+1) 1 + e iθ σ ) β α βα/2 x u, Approac 2a. Approac 2b. Approac 3. α xu e α xu 1 α xu iθ σ) qα qα/2 x 1 + e iθ σ ) 1/2 1/2 x e iθ σ ) 2α α xu e iθ σ ) 1/2 1/2 x ) 2nα+1) 1 Notice tat, for te Approac 1 and for x 2α, one considers RL Ix nα α 1 + e iθ σ) nα+1 α u), ) 2nα+1) u, 1 + eiθ σ) qα qα/2 σ 2α x ) 1 + e iθ σ) β 2α iξ) β 2α ) n 2α +1. x u, For α in R N, we ave β 2α = 2α/1 + 2α ) and n 2α = 2α. Ten, one gets te approximation σ x 2α ) 2α 1 + e iθ σ) 2α ) 1+ 2α iξ) 1+ 2α 2α +1, leading to te approximate FPML operator x 2α 2α 1 ) 1+ 2α 2α +1u 1 ) n2α +1 u 1 + e iθ σ) 2α x = 1+ 2α 1 + e iθ σ) β β 2α 2α x u. 31) 10 30)

12 As an example, from 30) and 31) and taking α = 3/4, n α = 0, β α = 3/4, n 2α = 1, β 2α = 3/4, one gets 3/4 x u e iθ σ) 3/4 3/8 x ) 2u, 3/2 x u 1 ) 2u. 1 + eiθ σ) 3/4 3/4 x Example 2. To compare te different FPMLs, we plot u in logscale) at T = 10 on Fig. 2 for several coices of FPMLs wen solving te FPDE linear fractional Scrödinger-type equation) { i t ut, x) + vx) 3/4 x ut, x) = 0, t > 0, x R, 32) ut, 0) = u 0 x), x R, wit u 0 x) = N 1 e 2x2 +ik 0 x were N is te L 2 D)-normalization constant), te function v is vx) = e x 2 +3iπ/4 and te wave number is k 0 = 5. Te pysical domain of computation is D Pys := [ L ; L ] wile te entire domain wit FPMLs is D := L; L), coosing L = 8 and L = 0.95L. Regarding te FPMLs, we fix te Type I absorbing function 16), for σ 0 = 10, θ = π/8, and we consider periodic boundary conditions, i.e., we solve for example te IBVP 1 i t ut, x) + vx) 1 + e ut, L) = ut, L), t > 0, ut, 0) = u 0 x), x D. iθ σx)) 3/4 3/8 x ) 2ut, x) = 0, t > 0, x D, Te discretization parameters are again N x = 1001 and t = 10 2 N t = 1000). We conclude tat all te FPMLs are igly absorbing and relatively similar in terms of quality up to te optimization of te tuning parameters). From now on, we fix te FPML based on Approac FPMLs pseudospectral approximation scemes: implementation and numerical examples In tis section, we propose a general metodology for solving IBVPs using Fourier-based pseudospectral metods [2, 3, 7, 14, 33, 40] wit FPML. Naturally, oter approximation scemes could also be used. We start wit fractional equations involving spatial orders of derivation r < 1 in Subsection 3.1, and next consider te more general case r R + in Subsection 3.2. Initially, te problem of interest is an IVP set in R. For obvious computational reasons, te problem must be solved as an IBVP on a truncated domain, as it is commonly te case. Considering te well-posedness, suitable boundary conditions need to be imposed at te boundary of te truncated domain. Wen Fourier-based metods are used, periodic boundary conditions are naturally fixed, wic are ten potentially inappropriate for computing delocalized solutions. Te proposed computational metodology developed ere allows for fixing tis issue in te case of FPDEs, based on te previous FPMLs combined wit a pseudospectral discretization and time splitting scemes )

13 Figure 2: Example 2. Comparison of te amplitude u logscale) for te Approaces 0, 1, 2a, 2b, 3, wit te FPML of Type I, pure Periodic Boundary Conditions i.e. witout any PML), and reference solution computed on a large domain IBVP wit space fractional operators of order α less tan 1 We consider te following IBVP, wic is assumed to be well-posed in [0; T ] D Py see Section 5), were te equation contains fractional operators of order α strictly less tan 1. For a given initial data ut = 0, ) = u 0 tat we assume to be compactly supported in D Py, we want to solve te IVP t ut, x) + v [α] x) α x ut, x) = fut, x)), were {v [α] } are smoot functions, f is square integrable, and R is a finite set of strictly positive real numbers suc tat te IBVP is well-posed, for 0 < r := max R < 1. Generalizing te idea presented in Subsection 2.2 for te Approac 1, we set { 1, if x < L Sx) =, 1 + e iθ σx), if L 34) x < L, and consider te modified IBVP on [0; T ] D wit FPMLs t ut, x) + v [α] x) S α x) α x ut, x) = fut, x)), 35) wit periodic boundary conditions ut, L) = ut, L) and initial data ut = 0, ) = u 0. We will also study te case of FPDEs of te form t ut, x) + v [α] x) α xut, x) = fut, x)), 12

14 wic leads to te new system t ut, x) + v [α] x) S α x) α xut, x) = fut, x)), wit initial data ut = 0, ) = u 0, and periodic boundary conditions at te domain interface { L; L}. Let us denote te set of grid-points by D Nx = { x k1 }k 1 O Nx, O Nx = { k 1 N/ k 1 = 0,, N x 1 }, and te uniform mes size by = x := x k1 +1 x k1 = 2L/N x for te entire domain D). Te corresponding discrete wave numbers are defined by ξ p = pπ/l, for p { N x /2,, N x /2 1}. Concerning te approximations, we use te following notation û p t) = N x 1 k 1 =0 u k1 t)e iξpx k 1 +L), ũ k1 t) = 1 N x/2 1 N x p= N x/2 û p t)e iξpx k 1 +L). 36) Finally, we define te Fourier-based approximations of te fractional derivative operators of order α > 0 as α x ut n, x k1 ) { [[ x α ]]u k1 t n ) } := 1 N x/2 1 N x p= N x/2 iξp )α ũp t n )e iξpx k 1 +a). 37) Similarly, for te fractional laplacian, we introduce α xut n, x k1 ) { [[ α x]]u k1 t n ) } := 1 N x/2 1 N x p= N x/2 ξp 2α e iαπ ũp t n )e iξpx k 1 +a). 38) We denote by u n te approximation of u at time t n on D Nx, and by S and v [α] values of S and v [α] evaluated on te grid set D Nx. operator splitting in time of 35) reads u n+1/2 = u n + t 2 f u n+1/2 ), v [α] u n = u n+1/2 t u n+1 = u n + t 2 f ) u n+1. S α te respective A simple second-order semi-explicit [[ α x ]]u n+1/2, In principle, te above sceme is constrained by a CFL condition wic limits its interest and application, in particular wen te equation involves ig-order fractional) derivatives. 13

15 We conjecture tat te above one-dimensional sceme is stable under a CFL condition of te form t/ x r C, for some C > 0 [41]. Te larger r < 1, te more restrictive te above CFL condition. To avoid tis problem, we rater consider an implicit version of te splitting sceme u n+1/2 = u n + t 2 f u n+1/2 ), u n = u n+1/2 t S α [[ x α ]]u n, 39) u n+1 = u n + t 2 f ) u n+1. Te sceme 39) requires te use of a linear system solver for te second fractional step, but ensures in principle better L 2 -stability of te sceme. Regarding te spatial discretization of te second equation in 39), from t n to t n = t n + t wit initial data u n+1/2 := { n+1/2} u j, it is equivalent to te linear system j O Nx were A n u n v [α] = u n+1/2, 40) A n = I + t S α [[ x α ]], 41) I being te identity matrix. From an implementation and computational point of view, tis approac seems to be more complex, since te equation involves space variable coefficients. However, te system solution can be obtained tanks to te combination of a Krylov subspace iterative solver [38] like te GMRES wit matrix-free evaluations based on FFTs. In addition, analytical preconditioners could be added to improve te convergence rate. We refer to [2, 3] for te introduction of tis idea for te Gross-Pitaevskii equation in te framework of Bose- Einstein condensates wic can be directly extended ere to FPDEs. In te present paper, we use te GMRES witout preconditioner [38, 39]. Example 3. To analyze ow te proposed sceme is working, we consider te following linear FPDE { t u v [9/10] 9/10 x u = 0, t > 0, x R, u 0 x) = N 1 e x 5)2 +ik 0 x, x R, wit N a normalizing coefficient, v [9/10] x) = e x2 /200 /2 and k 0 = 1. For te FPML, we use te Type I absorbing function given by 16), were σ 0 = and θ = π/8, and wit a layer tickness given by δ x := 0.05 for te domain D = [ 10, 10]. Te modified FPDE problem wit FPML is ten t ut, x) v[9/10] x) S 9/10 x) 9/10 x ut, x) = 0, t > 0, x D, 42) ut, L) = ut, L), t > 0, u 0 x) = N 1 e x 5)2 +ik 0 x, x D, 14 v [α]

16 wic is approximated by I t v 9/10 S 9/10 [[ 9/10 x ]] ) u n+1 = u n. 43) Te discretization parameters are N x = 1001 and t = We plot in Fig. 3 te solutions {x, t, log ux, t) ), x, t) D [0; T ]} for te FPML of Type I, for te solution wit pure periodic boundary conditions no FPML) and for a reference solution computed on te larger domain [ 20, 20]). We observe tat te FPML properly works, and tat tere is almost no reflection at te boundary. A small amount of te solution passes troug te boundary and comes back on te left boundary because of te periodic boundary condition. Figure 3: Example 3. Solutions {x, t, log ux, t) ), x, t) D [0; T ]} wit periodic boundary conditions Left), FPML of Type I Center) and reference solution Rigt). Example 4. Very similar ideas can be adapted to te case of equations involving fractional laplacians wit FPMLs. Rater tan presenting te general teory, we illustrate tis troug an example considering te following nonlinear FPDE { i t u + v [1/2] 1/4 x u + v [3/4] 3/8 x u = fu), t > 0, x R, u 0 x) = N 1 e x+2)2 +ik 0 x, x R, were v [1/2] = e x2 /20+iπ/4, v [3/4] x) = e x2 /10+3iπ/8, f : u iκ u 2 u for κ = 10, and k 0 = 10. To include te FPML on D := [ L; L], te following modified problem is solved t u + v[1/2] S 1/4 1/2 x u + v[3/4] S 3/4 3/8 x u = fu), 44) wit initial data and periodic boundary conditions. It is approximated as follows u n+1/2 = u n + iκ t u n 2 u n+1/2, 2 v [1/2] I + t [[ 1/4 x ]] + t v3/4 [[ 3/8 x ]] ) u n = u n+1/2 S 1/2 S 3/4 u n+1 = u n 15, + iκ t u n 2 u n+1 2.

17 Alternatively, te ODEs involving te source term can be treated semi-analytically, i.e. u n+1/2 = u n exp i tκ t u n 1 2), 2 v [1/2] I + t [[ 1/4 x ]] + t v3/4 [[ x 3/8 ]] ) u n = u n+1/2, S 1/2 S 3/4 u n+1 = u n exp i tκ t u n 2). 2 For te simulation, we fix t = and N x = 1001 for te discretization. Regarding te FPML parameters, we coose L = 4, δ x = 0.075L and σ 0 = 10 for te Type I absorbing function. We compare {x, t, log ux, t) ), x, t) D [0; T ]} in Fig. 4 te solution wit periodic boundary conditions no FPML), wit FPML and for a reference solution computed on te domain, [ 16, 16]). Tis clearly illustrates te accuracy of te FPML combined wit te splitting sceme and pseudospectral approximation. Figure 4: Example 4. Solution {x, t, log ux, t) ), x, t) D [0; T ]} wit periodic boundary conditions Left), FPML of Type I Center) and reference solution Rigt). In te second part of te test, we compare te solution wit tree different types of FPML: Type I wit σ 0 = 10, Type II wit σ 0 = 10 2, Type VI wit σ 0 = see Fig. 5). Te FPML of Types I & II ave an accurate absorbing effect, wile Type VI-PML is less efficient for tis example. We also compare on Fig. 6 te absorbing effect of te FPML of Type I wit different values of σ 0, i.e. σ 0 := 10 1, 1, 10, 10 2, Te best absorptions are obtained for te values σ 0 = 10 and IBVP wit space fractional operators of order α larger or equal to 1 We assume now tat te IBVP contains fractional operators of order larger or equal to 1. We impose ut = 0, ) = u 0, and consider t ut, x) + v [α] x) α x ut, x) = fut, x)), 45) were {v [α] } and f are smoot functions, and R is a finite set of positive real numbers suc tat te IBVP is well-posed. In te following, we propose to analyze te Approaces 16

18 Figure 5: Example 4. Comparison te FPML wit Type I, II and VI absorption profiles. 1 and 2 from Subsection 2.2. Basically, te main idea consists in decomposing te derivatives α x, wit α R +. We can rewrite te equation 45) as t ut, x) + α v [α] nα+1 x) x ) nα+1 ut, x) = fut, x)). 46) We ten consider te modified IBVP on D including te FPML, following Approac 1 from Subsection 2.2, t ut, x) + 1 ) nα+1 v [α] x) S βα x) βα x ut, x) = fut, x)), 47) wit periodic boundary conditions at D and compactly supported initial data u 0. By using te same notations as above, we ave te following approximation βα x ut n, x k1 ) { [[ x βα ]]u k1 t n ) } := 1 N x N x/2 1 p= N x/2 iξp ) βα ũp t n )e iξpx k 1 +L), 48) 17

19 Figure 6: Example 4. Comparison of te FPML of Type I wit different values of σ 0 = 10 1, 1, 10, 10 2, 10 3 ). were β α is defined in 22). Te implicit sceme reads u n+1/2 = u n + t 2 f u n+1/2 ), u n = u n+1/2 t u n+1 = u n + t 2 f ) u n+1. 1 v [α] S βα ) nα+1 [[ x βα ]] u n, Approac 2a consists in solving te modified IBVP wit FPML 49) t ut, x) ) nαut, v [α] x) S qα x) qα x Sx) x x) = fut, x)), 50) 18

20 were q α is defined in 22). Te corresponding second-order implicit splitting sceme is given by u n+1/2 = u n + t 2 f u n+1/2 ), u n = u n+1/2 t u n+1 = u n + t 2 f u n+1 v [α] S qα ). 1 ) nαu [[ x qα n ]] [[ x ]], S Practically, te second equation in 49) and 51) are linear systems solved by using te GMRES or BiCGStab iterative solvers [39] applied e.g. to [ I + t v [α] S qα 1 ) nα ] [[ x qα ]] [[ x ]] u n = u n+1/2. S We use ere a GMRES algoritm to solve tese linear systems. Similarly, for te fractional laplacian and for te initial data ut = 0, ) = u 0, we consider 51) t ut, x) + v [α] x) α xut, x) = fut, x)), 52) were {v [α] } and f are regular functions. Te set R is a finite sequence of positive real numbers so tat te IBVP is well-posed. Applying te Approac 1 from Subsection 2.2 yields t ut, x) + 1 ) nα+1 v [α] x) S βα x) βα/2 x ut, x) = fut, x)), wile Approac 2 leads to t ut, x) + v [α] x) Sx) qα qα/2 x 1 Sx) 1/2 1/2 x ) 2nα+1) ut, x) = fut, x)), for t > 0 and x D. For well-posedness, periodic boundary conditions are added. Example 5. In tis example, te linear fractional system under consideration is { t ut, x) vx) 3/2 x ut, x) = fut, x)), t > 0, x R, ut, 0) = u 0 x), x R, 53) were te initial data is u 0 x) = N 1 e x+20/3) 2 +ik 0 x, wit k 0 = 1. Te multiplicative function is te gaussian vx) = e x 2 and te nonlinear cubic term is fu) = 10i u 2 u. A FPML is next introduced into te formulation, wit periodic boundary conditions. Te 19

21 computational domain is D = [ L; L], for L = 10 and L = 0.95L. Setting α = 3/2, we ave: n α = 1, q α = 1/2 and β α = 3/4. If one cooses te Approac 1 see Eq. 47), we get te FPML equation t ut, x) wile for Approac 2a see Eq. 50) we ave t ut, x) vx) 1 ) S βα x) βα x S βα x) βα x ut, x) = fut, x)), vx) 1 ) S qα x) qα x Sx) x ut, x) = fut, x)). We fix t = and N x = 2001 for te discretization. In Fig. 7, we report ux, t) logscale) on D [0; T ] wit T = 6), for respectively te periodic solution witout FPML, te solution wit FPML Approac 2a) of Type I wit σ 0 = 10, θ = π/8), and a reference solution computed on [ 20, 20]). We clearly see tat te FPMLs wit pseudospectral approximation performs very well. For completeness, we also provide in Fig. 8 a zoom of te solution on D [T/2; T ], were we see tat te FPML solution disperses a little bit. We next compare in Fig. 9 Left) te absorption effects of te FPML Approac 2a) for tree different profiles: Type I-, Type II wit σ 0 = 10) and Type VI wit σ 0 = 10 1 ), setting δ x = 0.05L and θ = π/8. In addition, we also report on Fig. 9 Rigt) te L 2 D)-norm of te solution in logscale) vs. te time variable t, i.e. {t, log ut, ) L 2 D)) : t [0; T ]}. Figure 7: Example 5. Solution {x, t, log ux, t) ), x, t) D [0; T ]} wit periodic boundary conditions no FPMLs) Left), wit FPML of Type I Center) and reference solution Rigt). Example 6. Let us now consider te following nonlinear cubic Scrödinger equation i t u = 1/2 x ) 2 u + κ u 2 u, t > 0, x R. 54) tat we write as a FPDE since x := 1/2 x ) 2 ) to sow tat te proposed formulation can also work well even in tis situation. Te modified FPML equation [4] reads i t u = 1 S 1/2 x 20 1 S 1/2 x ) u, 55)

22 Figure 8: Example 5. Solution {x, t, log ux, t) ), x, t) D [T/2, T ]} wit periodic boundary conditions Left), FPML wit absorption function of Type I Center), and reference solution Rigt) Figure 9: Example 5. Left) Amplitude u of te solution in logscale) at final time T = 7.5, on te domain D, for te FPMLs Approac 2a) wit absorption function of Type I, II σ 0 = 10) and VI σ 0 = 10 1 ), for θ = π/8. Rigt) Time evolution of te L 2 D)-norm of te solution for te FPMLs wit absorption function of Types I, II, VI, wit periodic boundary conditions and for te reference solution. wit S given by Eqs. 34). For stability reasons, we still use te following implicit splitting sceme u n+1/2 = u n i κ t u n 2 n+1/2 u, 2 u n = u n+1/2 + i t 1 1 [[ x 1/2 ]] [[ 1/2 S S u n+1 = u n u n 2 u n+1 i κ t 2. ) x ]] u n, Te initial data is te centered normalized Gaussian u 0 x) = N 1 e x2 /2+ik 0 x, wit wave number k 0 = 5. We fix κ = 10. Te FPML of Type I, wit σ 0 = , θ = π/8 and δ x = L is used. Te discretization parameters are fixed to t = 10 3 and N x = 1001 for te computational domain D = [ 8, 8]. In Fig. 10, we report {x, t, log ux, t) ), x, t) D [0; T ]} for te periodic boundary conditions case no FPML), for te FPML solution and te reference solution computed on [ 16, 16]). Tis illustrates te property tat te FPML 21

23 and its implementation is very efficient and accurate. We draw in Fig. 11 te amplitude u logscale) of te tree solutions at time T = Figure 10: Example 6. Solution {x, t, log ux, t) ), x, t) D [0; T ]} wit periodic boundary conditions Left), FPML wit absorption function of Type I Center) and reference solution Rigt) Figure 11: Example 6. log ut, x) at time T = 1.25 wit periodic boundary conditions, FPML wit Type I absorption function for σ 0 = , δ x = L and θ = π/8. Let us remark ere tat te FPML tat we use for te integer order PDE does not matc wit te standard way of writing a PML for Scrödinger-like PDEs and Helmoltztype equations). Indeed, usually, te PML modification of te laplacian [5] is x 1 1 ) x S S x, 56) wile ere, we ave x 1 S 1/2 x 1 S 1/2 x ). 57) 22

24 For te transformation 56), te involved operators are local and based on x wile 1/2 x appearing in 57) corresponds to a nonlocal operator. Tis operator is nontrivial to numerically approximate if we are not using a Fourier pseudospectral approximation sceme. Te possibility of writing two kinds of PMLs for te integer case is related to te fact tat te symbol of x is ξ 2, wic can be written as ξ ξ nonlocal) but also ξ ξ local). Example 7. Te equation under consideration is now te fractional nonlinear cubic Scrödinger equation i t u + v 3/4 x u = fu), t > 0, x R, wit initial data u 0 x) = N 1 e x+4) 2 +ik 0 x for k 0 = 5). Te function v is given by vx) = e x 2 +3iπ/4 and fu) = 10i u 2 u. For te FPML, we fix te computational domain to D = [ L; L], wit L = 8 and L = 0.85L. We set α = 3/2, leading to n α = 1, q α = 1/2 and β α = 3/4. According to te metods developed above, we can eiter solve: Approac 1 or Approac 2a i t u + i t u + v S 3/4 3/8 x v S 1/2 1/4 x 1 S 3/4 3/8 x ) u = fu), t > 0, x D, 1 ) S 1/2 u = fu), t > 0, x D, x wit initial data u 0 and periodic boundary conditions. We fix t = 10 2, N x = 501, and T = 10. In Fig. 12, we plot {x, t, log ux, t) ), x, t) D [0; T ]} for te periodic boundary conditions no PML), wit te FPML of Type II wit σ 0 = 10, θ = π/8) and finally te reference solution computed on [ 16, 16]). Tis again sows tat te FPMLs are very accurate wen implemented in te Fourier pseudospectral metod wit time splitting. Figure 12: Example 7. Solution {x, t, log ux, t) ), x, t) D [0; T ]} wit periodic boundary conditions Left), FPML wit absorption function of Type I Center) and reference solution Rigt). We also compare on Fig. 13 Left) te absorption effects of te FPMLs wit tree different types of absorption function: Type I, Type II wit σ 0 = 10), and Type VI wit 23

25 σ 0 = ), for δ x = 0.05L and θ = π/8. A similar test was performed were we compare te Approaces 1 and 2 on Fig. 13 Rigt) wit FPML of Types I & II σ 0 = 10). Te comparisons sow a similar beavior for bot approaces for te FPML wit Type II absorption function, but te second approac seems more appropriate for te FPML wit Type I function. Similar tests were also performed wit an explicit sceme. However, te latter requires very small time steps for stability reasons and terefore implicit scemes are strongly recommended see Section 5) Figure 13: Example 7. Left) Amplitude u logscale) of te solution at final time T = 10 and on D = [ 8, 8]. Te FPMLs Approac 2a) are based on te absorption functions of Type I, II σ 0 = 10) and VI σ 0 = 10 1, wit θ = π/8. Rigt) Amplitude u logscale) of te solution at final time T = 10 on D = [ 10, 10]. We use te FPML wit absorption profiles of Types I, II, θ = π/8, σ 0 = 10 2 and for te Approaces 1 and 2. Remark 3.1. Te metod developed in tis paper can straigtforwardly be extended to wellposed quasilinear equations t u + v [α] u) α x u = fu), t > 0, x R. For te sake of simplicity, let us assume tat f = 0. For example, for Approac 1 in Subsection 2, we first rewrite te equation on D, including te FPML, as t u + 1 ) nα+1 v [α] u) S βα x) βα x u = 0, were S is defined by 34) and periodic boundary conditions are added. sceme, we ave For te implicit u n+1 = u n t 1 v [α] u n ) 24 S βα ) nα+1 [[ x βα ]] u n+1.

26 4. Higer dimensional FPDEs 4.1. Approximate FPMLs for RL operators We extend now te ideas and metods to two-dimensional problems te 3D case can be treated by a direct extension). Te numerical solution of te equation under consideration is approximated on an open two-dimensional bounded rectangular pysical domain denoted by D Py. As usual, we add a layer, denoted by D PML, surrounding D Pys, stretcing te ν-coordinates, wit ν = x, y. Te overall computational domain is ten: D = D Py D PML. In two dimensions, one gets D = [ L x, L x ] [ L y, L y ] and D Pys = [ L x, L x] [L y, L y] see e.g. Fig. 14 Left)). Standard PMLs metods [5] require a stretcing of te real spatial coordinates following te cange of variables ν ν = ν + e iθ σ ν s)ds, 58) wit ν = x, y, and were te absorbing functions are defined by { σν ν L σ ν ν) = ν ), L ν ν < L ν, 0, ν < L ν. Finally, we set L ν S ν ν) := 1 + e iθν σν), and define te operators along te ν-direction γ ν 1 S γ ν) γ ν = e iθ σν) ) γ γ ν, 59) were γ is a given derivation order. Following te 1D case, tis leads to various coices of FPMLs for te 2D case by extension along eac direction, by adapting γ. Example 8. Rater tan developing te general case tat will be studied in a fortcoming paper dedicated to multidimensional FPDEs), we consider te following FPDE i t ut, x, y) + x 3/2 ut, x, y) /2 y ut, x, y) = 0, t > 0, x, y) R 2, 60) ut = 0, x, y) = u 0 x, y), x, y) R 2. We propose to compute te solution to tis system in te bounded domain [0; T ] D, wit D = [ 2.5; 2.5] 2 and T = 1. Te initial data is cosen wit a support close to te boundary since we are mainly interested in te absorbing features) see Fig. 14) u 0 x, y) = e 5x+5/3)2 +y 2 ) 2ix+iy. 25

27 y L y +δ y L y D: Domain of Pysical Interest x L y L y δ y L x δ x L x L x L x +δ x Figure 14: Example 8. Left) Domain wit FPML. Rigt) Amplitude of te initial data. Following Approac 1, te modified equation wit FPML and periodic boundary conditions reads i t u ) 3/4 Sx 3/4 x 3/4 Sx 3/4 x u periodic BC on [0; T ] D, u0, x, y) = u 0 x, y), x, y) D. 1 S 3/4 y y 3/4 1 S 3/4 y y 3/4 ) u = 0, in [0; T ] D, We apply a directional splitting wit semi-implicit discretization for solving te IBVP. Te real space grid as N x N y = points and te time step is t = In Fig. 15, we compare snapsots of te amplitude of te solution at times t = 0.1, 0.4, 1 and teir logaritm for te last line at t = 1) for i) a solution of reference computed on a larger domain), ii) a solution wit periodic boundary conditions witout FPML) and iii) a solution wit FPML by considering te Type VI profile σ 0 /ν 2 σ 0 /δ 2 ν, wit σ 0 = 10 3, θ = π/64, δ ν = L ν for L ν = 2.5). As we can observe, te FPML solution reproduces correctly te reference solution. Wen using te logscale representation, we can see tat small residuals of te wave field pass troug te bottom boundary and come back at te top interface since we are using periodic boundary conditions. In comparison, te full wave is transmitted from te bottom to te top boundary because of te periodic boundary condition. Te absorbing layer used in te above computations is based on 59) leading to 61) i.e. using one of te previous 1D strategies). For completeness of te study, we also consider te solution based on te absorption operator built on a direct approac, i.e. replacing te operator in 60) by α ν 1 S α ν) α ν = 61) e iθ σν) ) α α ν, 62) 26

28 Figure 15: Example 8. Tree first lines for te times t = 0.1, 0.4, 1): amplitude of te solution: Left) reference, Center) FPML wit Type VI profile, Rigt) periodic BC. Fourt line: logaritm of te amplitude of te solution at time t = 1: Left) reference, Center) FPML wit Type VI profile, Rigt) periodic BC. were α= 3/2) is te complete derivation order for eac direction ν = x, y. To tis aim, we report in Fig. 16, te logaritm of te amplitude of te solution at t = 1, by using 62) Left) and 59) Rigt), wic confirms tat 59) sould be rater selected, as discussed in Section 2. 27

29 Figure 16: Example 8. Logaritm of te amplitude of te solution at time t = 1: Left) by using te direct formulation 62), Rigt) by using 59). Example 9. Let us now consider te 2D FPDE i t ut, x, y) + 9/10 x ut, x, y) /10 y ut, x, y) = 0, t > 0, x, y) R 2, u0, x, y) = u 0 x, y), x, y) R 2. 63) Te initial data is cosen as te L 2 -normalized gaussian u 0 x, y) = N 1 e 5x+10/3)2 +y 2 ) 10ix 2iy. 64) We consider te bounded spatial domain of computation D = [ 5, 5] 2 wile te time interval is [0; T ], wit T = 1. Te truncated FPDE wit FPML is ten based on Approac 1 following i t u + 1 S 9/10 x 1 9/20 x S 9/10 x ) x 9/20 u S 9/10 y 1 9/20 y S 9/10 y ) y 9/20 u = 0, 65) in [0; T ] D, adding periodic boundary conditions at D and considering te initial data 64). Te discretization sceme uses a directional time splitting wit semi-implicit discretization of te IBVP. Te spatial grid involves N x N y = points wile t = We report in Fig. 17 a snapsot of te amplitude of te te wave field at t = 0.7 wit periodic boundary conditions witout any FPML), te reference solution computed on a large domain, and te FPML solution. Here, we are using te Approac 1 see Eq. 65)) for te Type II profile, setting σ 0 = , θ = π/4, and δ ν = 10 1 L ν. A very good absorption is observed wit te proposed metodology. Let us remark tat some tuning is still necessary to properly coose te FPML parameters, wic is problem-dependent as usual. 28

30 Figure 17: Example 9. Top-left) amplitude of te initial data, Top-rigt) periodic solution, Bottom-left) reference solution and Bottom-rigt) of te FPML solution wit Type II profile. Example 9bis. We now sligtly modify system 63) as i t ut, x, y) + 9/10 x ut, x, y) /4 y ut, x, y) = 0, t > 0, x, y) R 2, ut = 0, x, y) = u 0 x, y), x, y) R 2, 66) were te initial data is given by 64). Te FPML system related to Approac 1 is i t u + 1 S 9/10 x 1 9/20 x S 9/10 x ) 9/20 x u S 3/4 y 1 3/8 y S 3/4 y 3/8 y ) u = 0, 67) in [0; T ] D, wit D = [ 5, 5] 2 and for T = 1. Te sceme as well as te discretization parameters are te same as for Example 9. We report in Fig. 18 te amplitude of te initial data, and te solution at time t = 0.63) wit periodic boundary conditions no FPML), a 29

31 reference solution, and te FPML solution based on te Type II profile, wit σ 0 = , θ = π/4, and δ ν = 10 1 L ν. We again clearly observe an effective accuracy of te FPML implemented witin te pseudospectral sceme. Figure 18: Example 9bis. Top-left) amplitude of te initial data, Top-rigt) periodic solution, Bottomleft) reference solution and Bottom-rigt) of te FPML solution wit Type II profile Focus on FPMLs for FPDE involving te 2D fractional laplacian As seen in subsection 2.3, accurate FPMLs can be derived for 1D FPDEs involving fractional laplacians. Since FPMLs can be obtained for RL fractional operators, tis is also relatively expected for te 1D fractional laplacian since we ave some relations between bot operators, like e.g. considering te 1D formula, < x < +, ) α ux) = 2α ux). 68) x 2α For te 2D case, tis is muc less clear ow to proceed and a relation as te above one does not exist see e.g. [31, 34] ). We discuss te question of te extension of FPMLs to 2D 30

32 FPDEs and prospect a first possible approac. Neverteless, tis example also sows tat furter developments are still needed. Let us consider te time-dependent two-dimensional FPDE i t ut, x, y) + v [α] x, y) ) α ut, x, y) = 0, for t, x, y) R R 2, 69) ut = 0, x, y) = u 0 x, y), wit x, y) R 2. Te Fourier spectral definition of te fractional laplacian is given by ) α u = F 1 ξ x 2 + ξ y 2 ) α Fu)ξ x, ξ y )), were ξ x respectively ξ y ) is te Fourier dual variable in direction x respectively y), Fu) is te two-dimensional Fourier transform of u and F 1 is te associated inverse Fourier transform. For α = 1, let us recall tat te modified PML laplacian operator as a total symbol given by σ PML ) = σ 1 S x x 1 S x x ) + 1 S y y 1 S y y )) = σ 1 2 Sx 2 x + 1 x 1 ) x + 1 S x S x Sy 2 2 y + 1 y 1 ) y ) S y S y 70) = 1 ξ Sx 2 x 2 + i 1 x 1 )ξ x + 1 S x S x Sy ξ 2 y 2 + i 1 y 1 )ξ y. S y S y By analogy, we propose te 2D FPML laplacian wit variable coefficients based on te pseudodifferential operator definition wit te symbol ) α PMLu := F 1 a α x, y, ξ x, ξ y )Fu)ξ x, ξ y )), a α x, y, ξ x, ξ y ) = σ PML )) α. From a practical point of view, wen α is not an integer, it is no longer possible to split te real space and Fourier variables. In te latter case, a direct implementation is very inefficient, and ten requires some approximations. A possible approac is owever based on te decomposition of te fractional laplacian : α = α 1. As a consequence, te following approximation is proposed were σ PML,cst ) = α PML α PML,a := PML,cst ) α 1 PML, 71) 1 S 2 x,cst ξ x 2 i S x,cst ξ Sx,cst 3 x + 1 ξ Sy,cst 2 y 2 i S y,cst ξ Sy,cst 3 y ) 72) 31

33 and S x,cst, S y,cst, S x,cst, S y,cst are te respective approximations to S x, S y, S x, S y based on a constant cst) profile. In practice, we solve on [0; T ] D te approximation of system 69) by i t ut, x, y) + v [α] x, y) PML,a ) α ut, x, y) = 0, for t, x, y) [0; T ] D, 73) ut = 0, x, y) = u 0 x, y), for x, y) D, wit periodic boundary conditions on [0; T ] D. Example 10. To illustrate te approac, we consider te following system wit α = 9/10) i t ut, x, y) + e9πi/ /10 ut, x, y) = 0, for t, x, y) R R 2, 74) ut = 0, x, y) = u 0 x, y), for x, y) R 2, wit u 0 x, y) = 10e 15x 2)2 +y 16/3) 2 )+5ix 5iy. 75) Te computational domain is D = [ 8, 8] 2 and te final time is T = 350. Following our strategy, te modified FPDE wit FPML wic is solved consists in replacing 9/10 in 74) by 9/10 PML,a = 1/10 PML,cst PML, were PML,cst is defined by expression 72). In tis example, we simply take S x,cst = S y,cst = e iθ and terefore S x,cst = S y,cst = 0. We apply a directional splitting wit semiimplicit discretization to solve te corresponding IBVP 73). Te real space grid involves N x N y = points and we fix t = We report in Fig. 19 te amplitude of te initial data, of te solution wit periodic boundary conditions witout FPML), te solution of reference computed on a larger domain) and te FPML-based solution. For tis last solution, we use a profile of Type I, wit σ 0 = , θ = π/4, and δ ν = 0.15L ν. 5. A few results in matematical and numerical analysis In tis section, we develop some analysis of fractional PDEs, and teir approximation wit FPML using te pseudospectral metod Some first well-posedness results for te IVP and IBVP To analyze te stability of te overall metod, let us first focus on te well-posedness of 1D linear equations wit constant coefficients, i.e. we consider t ut, x) + v [α] x α ut, x) = 0, t, x) R R, 76) u0, x) = u 0 x), x R, 32