logo AFST
Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 2, pp. 259-327.

The aim of this article is to propose a systematic study of transparent boundary conditions for finite difference approximations of evolution equations. We try to keep the discussion at the highest level of generality in order to apply the theory to the broadest class of problems.

We deal with two main issues. We first derive transparent numerical boundary conditions, that is, we exhibit the relations satisfied by the solution to the pure Cauchy problem when the initial condition vanishes outside of some domain. Our derivation encompasses discretized transport, diffusion and dispersive equations with arbitrarily wide stencils. The second issue is to prove sharp stability estimates for the initial boundary value problem obtained by enforcing the boundary conditions derived in the first step. We focus here on discretized transport equations. Under the assumption that the numerical boundary is non-characteristic, our main result characterizes the class of numerical schemes for which the corresponding transparent boundary conditions satisfy the so-called Uniform Kreiss–Lopatinskii Condition. Adapting some previous works to the non-local boundary conditions considered here, our analysis culminates in the derivation of trace and semigroup estimates for such transparent numerical boundary conditions. Several examples and possible extensions are given.

Le but de cet article est de proposer une étude systématique des conditions aux limites transparentes pour les approximations par différences finies d’équations d’évolution. On essaie de maintenir la discussion au plus haut niveau de généralité possible afin d’appliquer la théorie à la plus large classe de problèmes.

On aborde deux problèmes principaux. On construit en premier lieu des conditions aux limites numériques transparentes, c’est-à-dire qu’on exhibe les relations satisfaites par la solution du problème de Cauchy quand les données initiales sont nulles hors d’un certain domaine. Notre construction englobe les discrétisations d’équations de type transport, diffusion ou dispersif avec un « stencil » arbitrairement grand. Le second problème que nous abordons est celui de la stabilité du problème mixte obtenu en imposant les conditions aux limites numériques construites à la première étape. On étudie ici le cas des équations de transport discrétisées. Sous une hypothèse de bord non-caractéristique, notre résultat principal classifie les schémas numériques pour lesquels les conditions aux limites transparentes vérifient la condition dite de Kreiss–Lopatinskii uniforme. En adaptant des travaux antérieurs au cadre non-local considéré ici, notre analyse aboutit finalement à des estimations de trace et de semi-groupe pour les conditions aux limites numériques transparentes. L’article se conclut avec des exemples et de futures extensions possibles.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1600
Classification: 65M06,  65M12,  35L02,  35K05,  35Q41
Keywords: evolution equations, difference approximations, transparent boundary conditions, stability
Jean-François Coulombel 1

1 CNRS and Université de Nantes, Laboratoire de Mathématiques Jean Leray (UMR CNRS 6629), 2 rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
@article{AFST_2019_6_28_2_259_0,
     author = {Jean-Fran\c{c}ois Coulombel},
     title = {Transparent numerical boundary conditions for evolution equations: {Derivation} and stability analysis},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {259--327},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 28},
     number = {2},
     year = {2019},
     doi = {10.5802/afst.1600},
     zbl = {07095683},
     mrnumber = {3957682},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1600/}
}
TY  - JOUR
TI  - Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2019
DA  - 2019///
SP  - 259
EP  - 327
VL  - Ser. 6, 28
IS  - 2
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1600/
UR  - https://zbmath.org/?q=an%3A07095683
UR  - https://www.ams.org/mathscinet-getitem?mr=3957682
UR  - https://doi.org/10.5802/afst.1600
DO  - 10.5802/afst.1600
LA  - en
ID  - AFST_2019_6_28_2_259_0
ER  - 
%0 Journal Article
%T Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2019
%P 259-327
%V Ser. 6, 28
%N 2
%I Université Paul Sabatier, Toulouse
%U https://doi.org/10.5802/afst.1600
%R 10.5802/afst.1600
%G en
%F AFST_2019_6_28_2_259_0
Jean-François Coulombel. Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 28 (2019) no. 2, pp. 259-327. doi : 10.5802/afst.1600. https://afst.centre-mersenne.org/articles/10.5802/afst.1600/

[1] Xavier Antoine; Anton Arnold; Christophe Besse; Matthias Ehrhardt; Achim Schädle A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations, Commun. Comput. Phys., Volume 4 (2008) no. 4, pp. 729-796 | Zbl: 1364.65178

[2] Xavier Antoine; Christophe Besse Construction, structure and asymptotic approximations of a microdifferential transparent boundary condition for the linear Schrödinger equation, J. Math. Pures Appl., Volume 80 (2001) no. 7, pp. 701-738 | Article | Zbl: 1129.35324

[3] Xavier Antoine; Christophe Besse; Jérémie Szeftel Towards accurate artificial boundary conditions for nonlinear PDEs through examples, Cubo, Volume 11 (2009) no. 4, pp. 29-48 | MR: 2571793 | Zbl: 1184.35014

[4] Anton Arnold; Matthias Ehrhardt; Ivan Sofronov Discrete transparent boundary conditions for the Schrödinger equation: fast calculation, approximation, and stability, Commun. Math. Sci., Volume 1 (2003) no. 3, pp. 501-556 | Article | Zbl: 1085.65513

[5] Corentin Audiard Non-homogeneous boundary value problems for linear dispersive equations, Commun. Partial Differ. Equations, Volume 37 (2012) no. 1, pp. 1-37 | MR: 2864804 | Zbl: 1246.35048

[6] Hellmut Baumgärtel Analytic perturbation theory for matrices and operators, Operator Theory: Advances and Applications, Volume 15, Birkhäuser, 1985, 427 pages | MR: 878974 | Zbl: 0591.47013

[7] Sylvie Benzoni-Gavage; Denis Serre Multi-dimensional hyperbolic partial differential equations. First-order systems and applications, Oxford Mathematical Monographs, Oxford University Press, 2007, xxv+508 pages | Zbl: 1113.35001

[8] Christophe Besse; Matthias Ehrhardt; Ingrid Lacroix-Violet Discrete artificial boundary conditions for the linearized Korteweg–de Vries equation, Numer. Methods Partial Differ. Equations, Volume 32 (2016) no. 5, pp. 1455-1484 | Article | MR: 3535627 | Zbl: 1348.65124

[9] Christophe Besse; Benoît Mésognon-Gireau; Pascal Noble Artificial boundary conditions for the linearized Benjamin–Bona–Mahony equation, Numer. Math., Volume 139 (2018) no. 2, pp. 281-314 | Article | MR: 3802673 | Zbl: 1397.65130

[10] Jean-François Coulombel Stability of finite difference schemes for hyperbolic initial boundary value problems, SIAM J. Numer. Anal., Volume 47 (2009) no. 4, pp. 2844-2871 | MR: 2551149 | Zbl: 1205.65245

[11] Jean-François Coulombel Stability of finite difference schemes for hyperbolic initial boundary value problems II, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 10 (2011) no. 1, pp. 37-98 | MR: 2829318 | Zbl: 1225.65089

[12] Jean-François Coulombel Stability of finite difference schemes for hyperbolic initial boundary value problems, HCDTE lecture notes. Part I. Nonlinear hyperbolic PDEs, dispersive and transport equations (AIMS Series on Applied Mathematics) Volume 6, American Institute of Mathematical Sciences, 2013, pp. 97-225 | MR: 3340992 | Zbl: 1284.65116

[13] Jean-François Coulombel Fully discrete hyperbolic initial boundary value problems with nonzero initial data, Confluentes Math., Volume 7 (2015) no. 2, pp. 17-47 | MR: 3466438 | Zbl: 1355.65116

[14] Jean-François Coulombel The Leray–Gårding method for finite difference schemes, J. Éc. Polytech., Math., Volume 2 (2015), pp. 297-331 | MR: 3426750 | Zbl: 1328.65175

[15] Jean-François Coulombel; Antoine Gloria Semigroup stability of finite difference schemes for multidimensional hyperbolic initial-boundary value problems, Math. Comput., Volume 80 (2011) no. 273, pp. 165-203 | MR: 2728976 | Zbl: 1308.65142

[16] Bernard Ducomet; Alexander Zlotnik On stability of the Crank–Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I, Commun. Math. Sci., Volume 4 (2006) no. 4, pp. 741-766 | Zbl: 1119.65085

[17] Matthias Ehrhardt Absorbing boundary conditions for hyperbolic systems, Numer. Math., Theory Methods Appl., Volume 3 (2010) no. 3, pp. 295-337 | MR: 2798552 | Zbl: 1240.65239

[18] Matthias Ehrhardt; Anton Arnold Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma, Volume 4* (2001), pp. 57-108 | Zbl: 0993.65097

[19] Etienne Emmrich Convergence of the variable two-step BDF time discretisation of nonlinear evolution problems governed by a monotone potential operator, BIT, Volume 49 (2009) no. 2, pp. 297-323 | MR: 2507603 | Zbl: 1172.65026

[20] Etienne Emmrich Two-step BDF time discretisation of nonlinear evolution problems governed by monotone operators with strongly continuous perturbations, Comput. Methods Appl. Math., Volume 9 (2009) no. 1, pp. 37-62 | MR: 2641310 | Zbl: 1169.65046

[21] Israel C. Gohberg; I. A. Felʼdman Convolution equations and projection methods for their solution, Translations of Mathematical Monographs, Volume 41, American Mathematical Society, 1974 (Translated from the Russian) | MR: 355675 | Zbl: 0278.45008

[22] Moshe Goldberg On a boundary extrapolation theorem by Kreiss, Math. Comput., Volume 31 (1977) no. 138, pp. 469-477 | MR: 443363 | Zbl: 0359.65080

[23] Moshe Goldberg; Eitan Tadmor Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II, Math. Comput., Volume 36 (1981) no. 154, pp. 603-626 | MR: 606519 | Zbl: 0466.65054

[24] Bertil Gustafsson; Heinz-Otto Kreiss; Joseph Oliger Time dependent problems and difference methods, Pure and Applied Mathematics, John Wiley & Sons, 1995, xi+642 pages | Zbl: 0843.65061

[25] Bertil Gustafsson; Heinz-Otto Kreiss; Arne Sundström Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comput., Volume 26 (1972) no. 119, pp. 649-686 | MR: 341888 | Zbl: 0293.65076

[26] Thomas Hagstrom Radiation boundary conditions for the numerical simulation of waves (Acta Numerica) Volume 8, Cambridge University Press, 1999, pp. 47-106 | MR: 1819643 | Zbl: 0940.65108

[27] Ernst Hairer; Syvert P. Nørsett; Gerhard Wanner Solving ordinary differential equations. I. Nonstiff problems, Springer Series in Computational Mathematics, Volume 8, Springer, 1993, xv+528 pages | Zbl: 0789.65048

[28] Ernst Hairer; Gerhard Wanner Solving ordinary differential equations. II. Stiff and differential-algebraic problems, Springer Series in Computational Mathematics, Volume 14, Springer, 1996, xvi+614 pages | Zbl: 0859.65067

[29] Laurence Halpern Absorbing boundary conditions for the discretization schemes of the one-dimensional wave equation, Math. Comput., Volume 38 (1982) no. 158, pp. 415-429 | MR: 645659 | Zbl: 0482.65053

[30] Houde Han; Dongsheng Yin Absorbing boundary conditions for the multidimensional Klein–Gordon equation, Commun. Math. Sci., Volume 5 (2007) no. 3, pp. 743-764 | MR: 2352500 | Zbl: 1143.35306

[31] Tosio Kato Perturbation theory for linear operators, Classics in Mathematics, Springer, 1995, xxi+619 pages | Zbl: 0836.47009

[32] Heinz-Otto Kreiss Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comput., Volume 22 (1968), pp. 703-714 | MR: 241010

[33] Heinz-Otto Kreiss Initial boundary value problems for hyperbolic systems, Commun. Pure Appl. Math., Volume 23 (1970), pp. 277-298 | MR: 437941 | Zbl: 0327.65070

[34] Peter D. Lax Functional analysis, Pure and Applied Mathematics, John Wiley & Sons, 2002, xx+580 pages | Zbl: 1009.47001

[35] Nikolai K. Nikolski Operators, functions, and systems: an easy reading. Vol. 1. Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, Volume 92, American Mathematical Society, 2002 (franslated from the French by Andreas Hartmann) | MR: 1864396 | Zbl: 1007.47001

[36] Stanley Osher Systems of difference equations with general homogeneous boundary conditions, Trans. Am. Math. Soc., Volume 137 (1969), pp. 177-201 | MR: 237982 | Zbl: 0174.41701

[37] Stanley Osher Stability of parabolic difference approximations to certain mixed initial boundary value problems, Math. Comput., Volume 26 (1972), pp. 13-39 | MR: 298990 | Zbl: 0254.65065

[38] Meng Zhao Qin Difference schemes for the dispersive equation, Computing, Volume 31 (1983) no. 3, pp. 261-267 | Article | MR: 722326

[39] Robert D. Richtmyer; Keith W. Morton Difference methods for initial-value problems, Interscience Tracts in Pure and Applied Mathematics, Volume 4, John Wiley & Sons, 1967, xiv+405 pages | MR: 220455 | Zbl: 0155.47502

[40] Walter Rudin Real and complex analysis, McGraw-Hill Book Co., 1987 | Zbl: 0925.00005

[41] Leonard Sarason On hyperbolic mixed problems, Arch. Ration. Mech. Anal., Volume 18 (1965), pp. 310-334 | MR: 172002 | Zbl: 0137.06506

[42] Gilbert Strang Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys., Volume 41 (1962), pp. 147-154 | Zbl: 0111.31601

[43] Gilbert Strang Wiener-Hopf difference equations, J. Math. Mech., Volume 13 (1964), pp. 85-96 | MR: 160335 | Zbl: 0197.07104

[44] John C. Strikwerda; Bruce A. Wade A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, Linear operators (Warsaw, 1994) (Banach Center Publications) Volume 38, Polish Academy of Sciences, 1997, pp. 339-360 | MR: 1457017 | Zbl: 0877.15029

[45] Jérémie Szeftel Design of absorbing boundary conditions for Schrödinger equations in d , SIAM J. Numer. Anal., Volume 42 (2004) no. 4, pp. 1527-1551 | Zbl: 1094.35037

[46] Jérémie Szeftel Absorbing boundary conditions for the one-dimensional nonlinear Schrödinger equations, Numer. Math., Volume 103 (2006) no. 1, pp. 103-127 | Zbl: 1130.35119

[47] Lloyd N. Trefethen Instability of difference models for hyperbolic initial-boundary value problems, Commun. Pure Appl. Math., Volume 37 (1984) no. 3, pp. 329-367 | MR: 739924 | Zbl: 0575.65095

[48] Robert Vichnevetsky; John B. Bowles Fourier analysis of numerical approximations of hyperbolic equations, SIAM Studies in Applied Mathematics, Volume 5, Society for Industrial and Applied Mathematics, 1982, xii+140 pages (With a foreword by Garrett Birkhoff) | MR: 675265 | Zbl: 0495.65041

[49] Chunxiong Zheng; Xin Wen; Houde Han Numerical solution to a linearized KdV equation on unbounded domain, Numer. Methods Partial Differ. Equations, Volume 24 (2008) no. 2, pp. 383-399 | MR: 2382787 | Zbl: 1140.65070

[50] Andrea Zisowsky; Matthias Ehrhardt Discrete transparent boundary conditions for parabolic systems, Math. Comput. Modelling, Volume 43 (2006) no. 3-4, pp. 294-309 | MR: 2214640 | Zbl: 1135.35313

Cited by Sources: