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Stability of Finite Difference Schemes for Hyperbolic Initial Boundary Value Problems: Numerical Boundary Layers

  • Benjamin Boutin (a1) and Jean-François Coulombel (a2)

In this article, we give a unified theory for constructing boundary layer expansions for discretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels.

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*Corresponding author. Email addresses: (B. Boutin), (J. F. Coulombel)
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[1] Courant, R., Friedrichs, K. and Lewy, H., Über die partiellen differenzengle-ichungen der mathematischen physik, Math. Ann., 100(1) (1928), pp. 3274.
[2] Coulombel, J. F. and Gloria, A., Semigroup stability of finite difference schemes for multidimensional hyperbolic initial boundary value problems, Math. Comp., 80(273) (2011), pp. 165203.
[3] Chainais-Hillairet, C. and Grenier, E., Numerical boundary layers for hyperbolic systems in 1-D, M2AN Math. Model. Numer. Anal., 35(1) (2001), pp. 91106.
[4] Coulombel, J. F., Stability of finite difference schemes for hyperbolic initial boundary value problems, In HCDTE Lecture Notes. Part I. Nonlinear Hyperbolic PDEs, Dispersive and Transport Equations, American Institute of Mathematical Sciences, (2013), pp. 97225.
[5] Coulombel, J. F., The Leray-Gårding method for finite difference schemes, J. Éc. Polytech. Math., 2 (2015), pp. 297331.
[6] Dubois, F. and LeFloch, P., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differ. Equations, 71(1)(1988), pp. 93122.
[7] Gustafsson, B., Kreiss, H. O., and Oliger, J., Time dependent problems and difference methods, John Wiley & Sons, 1995.
[8] Gustafsson, B., Kreiss, H. O. and Sundström, A., Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp., 26(119) (1972), pp. 649686.
[9] Gisclon, M. and Serre, D., Conditions aux limites pour un système strictement hyperbolique fournies par le schéma de Godunov, RAIRO Modél. Math. Anal. Numér., 31(3) (1997), pp. 359380.
[10] Goldberg, M. and Tadmor, E., Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II, Math. Comp., 36(154) (1981), pp. 603626.
[11] Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp., 29(130) (1975), pp. 396406.
[12] Gerard-Varet, D., Formal derivation of boundary layers in fluid mechanics, J. Math. Fluid Mech., 7(2) (2005), pp. 179200.
[13] Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations. I, Spr. S. Comp., second ed., 1993.
[14] Hairer, E. and Wanner, G., Solving Ordinary Differential Equations. II, Spr. S. Comp., (14) 1996.
[15] Kreiss, H. O., Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), pp. 277298.
[16] Lubich, C. and Nevanlinna, O., On resolvent conditions and stability estimates, BIT, 31(2) (1991), pp. 293313.
[17] Métivier, G., On the L2 well-posedness of hyperbolic initial boundary value problems, Hyperbolic, 2014.
[18] Strikwerda, J. C. and Wade, B. A., A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, In Linear operators (Warsaw, 1994), Polish Acad. Sci., (1997), pp.339360.
[19] Trefethen, L. N. and Embree, M., Spectra and Pseudospectra, Springer, 175(98) (2005), xviii.
[20] Trefethen, L. N., Group velocity in finite difference schemes, SIAM Rev., 24(2) (1982), pp. 113136.
[21] Wu, L., The semigroup stability of the difference approximations for initial-boundary value problems, Math. Comp., 64(209) (1995), pp. 7188.
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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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