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

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.

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.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1600
Classification : 65M06,  65M12,  35L02,  35K05,  35Q41
Mots clés : evolution equations, difference approximations, transparent boundary conditions, stability
@article{AFST_2019_6_28_2_259_0,
     author = {Coulombel, Jean-Fran\c{c}ois},
     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},
     mrnumber = {3957682},
     zbl = {07095683},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/afst.1600/}
}
TY  - JOUR
AU  - Coulombel, Jean-François
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  - http://archive.numdam.org/articles/10.5802/afst.1600/
UR  - https://www.ams.org/mathscinet-getitem?mr=3957682
UR  - https://zbmath.org/?q=an%3A07095683
UR  - https://doi.org/10.5802/afst.1600
DO  - 10.5802/afst.1600
LA  - en
ID  - AFST_2019_6_28_2_259_0
ER  - 
Coulombel, Jean-François. Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 2, pp. 259-327. doi : 10.5802/afst.1600. http://archive.numdam.org/articles/10.5802/afst.1600/

[1] Antoine, Xavier; Arnold, Anton; Besse, Christophe; Ehrhardt, Matthias; Schädle, Achim 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] Antoine, Xavier; Besse, Christophe 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] Antoine, Xavier; Besse, Christophe; Szeftel, Jérémie 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] Arnold, Anton; Ehrhardt, Matthias; Sofronov, Ivan 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] Audiard, Corentin 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] Baumgärtel, Hellmut Analytic perturbation theory for matrices and operators, Operator Theory: Advances and Applications, 15, Birkhäuser, 1985, 427 pages | MR 878974 | Zbl 0591.47013

[7] Benzoni-Gavage, Sylvie; Serre, Denis 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] Besse, Christophe; Ehrhardt, Matthias; Lacroix-Violet, Ingrid 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] Besse, Christophe; Mésognon-Gireau, Benoît; Noble, Pascal 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] Coulombel, Jean-François 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] Coulombel, Jean-François 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] Coulombel, Jean-François 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] Coulombel, Jean-François 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] Coulombel, Jean-François The Leray–Gårding method for finite difference schemes, J. Éc. Polytech., Math., Volume 2 (2015), pp. 297-331 | MR 3426750 | Zbl 1328.65175

[15] Coulombel, Jean-François; Gloria, Antoine 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] Ducomet, Bernard; Zlotnik, Alexander 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] Ehrhardt, Matthias 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] Ehrhardt, Matthias; Arnold, Anton Discrete transparent boundary conditions for the Schrödinger equation, Riv. Mat. Univ. Parma, Volume 4* (2001), pp. 57-108 | Zbl 0993.65097

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

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

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

[25] Gustafsson, Bertil; Kreiss, Heinz-Otto; Sundström, Arne 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] Hagstrom, Thomas 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] Hairer, Ernst; Nørsett, Syvert P.; Wanner, Gerhard Solving ordinary differential equations. I. Nonstiff problems, Springer Series in Computational Mathematics, 8, Springer, 1993, xv+528 pages | Zbl 0789.65048

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

[29] Halpern, Laurence 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] Han, Houde; Yin, Dongsheng 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] Kato, Tosio Perturbation theory for linear operators, Classics in Mathematics, Springer, 1995, xxi+619 pages | Zbl 0836.47009

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

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

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

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

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

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

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

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

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

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

[44] Strikwerda, John C.; Wade, Bruce A. 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] Szeftel, Jérémie 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] Szeftel, Jérémie Absorbing boundary conditions for the one-dimensional nonlinear Schrödinger equations, Numer. Math., Volume 103 (2006) no. 1, pp. 103-127 | Zbl 1130.35119

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

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

Cité par Sources :