Stability of finite-difference models containing two boundaries or interfaces

Author:
Lloyd N. Trefethen

Journal:
Math. Comp. **45** (1985), 279-300

MSC:
Primary 65M10

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804924-2

MathSciNet review:
804924

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Abstract: It is known that the stability of finite-difference models of hyperbolic initial-boundary value problems is connected with the propagation and reflection of parasitic waves. Here the waves point of view is applied to models containing two boundaries or interfaces, where repeated reflection of trapped wave packets is a potential new source of instability. Our analysis accounts for various known instability phenomena in a unified way and leads to several new results, three of which are as follows. (1) Dissipativity does not ensure stability when three or more formulas are concatenated at a boundary or internal interface. (2) Algebraic "GKS instabilities" can be converted by a second boundary to exponential instabilities only when an infinite numerical reflection coefficient is present. (3) "GKS-stability" and "*P*-stability" can be established in certain problems by showing that the numerical reflection coefficient matrices have norm less than one.

**[1]**R. M. Beam, R. F. Warming, and H. C. Yee,*Stability analysis of numerical boundary conditions and implicit difference approximations of hyperbolic equations*, J. Comput. Phys.**48**(1982), no. 2, 200–222. MR**683521**, https://doi.org/10.1016/0021-9991(82)90047-X**[2]**M. Berger,*Adaptive Mesh Refinement for Hyperbolic Partial Differential Equations*, Ph. D. Thesis, Department of Computer Science, Stanford University, 1982.**[3]**M. Berger,*Stability of Interfaces with Mesh Refinement*, ICASE Report 83-42.**[4]**David L. Brown,*A note on the numerical solution of the wave equation with piecewise smooth coefficients*, Math. Comp.**42**(1984), no. 166, 369–391. MR**736442**, https://doi.org/10.1090/S0025-5718-1984-0736442-3**[5]**Gerald Browning, Heinz-Otto Kreiss, and Joseph Oliger,*Mesh refinement*, Math. Comp.**27**(1973), 29–39. MR**0334542**, https://doi.org/10.1090/S0025-5718-1973-0334542-6**[6]**Melvyn Ciment,*Stable matching of difference schemes*, SIAM J. Numer. Anal.**9**(1972), 695–701. MR**0319383**, https://doi.org/10.1137/0709058**[7]**M. Giles,*Eigenmode Analysis of Unsteady One-Dimensional Euler Equations*, NASA Contract Report 172217, ICASE 1983, submitted to*AIAA J.***[8]**M. Giles & W. Thompkins, Jr.,*Asymptotic Analysis of Numerical Wave Propagation in Finite Difference Equations*, Gas Turbine and Plasma Phys. Lab. Rep. 171, Massachusetts Institute of Technology, 1983.**[9]**Moshe Goldberg and Eitan Tadmor,*Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II*, Math. Comp.**36**(1981), no. 154, 603–626. MR**606519**, https://doi.org/10.1090/S0025-5718-1981-0606519-9**[10]**Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström,*Stability theory of difference approximations for mixed initial boundary value problems. II*, Math. Comp.**26**(1972), 649–686. MR**0341888**, https://doi.org/10.1090/S0025-5718-1972-0341888-3**[11]**Bertil Gustafsson,*The choice of numerical boundary conditions for hyperbolic systems*, J. Comput. Phys.**48**(1982), no. 2, 270–283. MR**683522**, https://doi.org/10.1016/0021-9991(82)90050-X**[12]**Heinz-Otto Kreiss,*Difference approximations for the initial-boundary value problem for hyperbolic differential equations*, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, 1966, pp. 141–166. MR**0214305****[13]**Heinz-Otto Kreiss,*Initial boundary value problems for hyperbolic systems*, Comm. Pure Appl. Math.**23**(1970), 277–298. MR**0437941**, https://doi.org/10.1002/cpa.3160230304**[14]**H.-O. Kreiss & J. Oliger,*Methods for the Approximate Solution of Time Dependent Problems*, Global Atmospheric Research Programme Publ. Series No. 10, Geneva, 1973.**[15]**Joseph Oliger,*Fourth order difference methods for the initial boundary-value problem for hyperbolic equations*, Math. Comp.**28**(1974), 15–25. MR**0359344**, https://doi.org/10.1090/S0025-5718-1974-0359344-7**[16]**Joseph Oliger,*Hybrid difference methods for the initial boundary-value problem for hyperbolic equations*, Math. Comp.**30**(1976), no. 136, 724–738. MR**0428727**, https://doi.org/10.1090/S0025-5718-1976-0428727-0**[17]**Joseph Oliger,*Constructing stable difference methods for hyperbolic equations*, Numerical methods for partial differential equations (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 42, Academic Press, New York-London, 1979, pp. 255–271. MR**558221****[18]**Stanley Osher,*Systems of difference equations with general homogeneous boundary conditions*, Trans. Amer. Math. Soc.**137**(1969), 177–201. MR**0237982**, https://doi.org/10.1090/S0002-9947-1969-0237982-4**[19]**Stanley Osher,*Hyperbolic equations in regions with characteristic boundaries or with corners*, Numerical solution of partial differential equations, III (Proc. Third Sympos. (SYNSPADE), Univ. Maryland, College Park, Md., 1975) Academic Press, New York, 1976, pp. 413–441. MR**0474850****[20]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[21]**L. Sarason and J. A. Smoller,*Geometrical optics and the corner problem*, Arch. Rational Mech. Anal.**56**(1974/75), 34–69. MR**0346334**, https://doi.org/10.1007/BF00279820**[22]**J. C. South & M. M. Hafez,*Stability Analysis of Intermediate Boundary Conditions in Approximate Factorization Schemes*, Proc. 6th AIAA Computational Fluid Dynamics Conference, Danvers, MA, 1983.**[23]**J. A. Trapp & J. D. Ramshaw, "A simple heuristic method for analyzing the effects of boundary conditions on numerical stability,"*J. Comput. Phys.*, v. 20, 1976, pp. 238-242.**[24]**L. N. Trefethen,*Wave Propagation and Stability for Finite Difference Schemes*, Ph. D. Thesis, Department of Computer Science, Stanford University, 1982.**[25]**Lloyd N. Trefethen,*Group velocity in finite difference schemes*, SIAM Rev.**24**(1982), no. 2, 113–136. MR**652463**, https://doi.org/10.1137/1024038**[26]**Lloyd N. Trefethen,*Group velocity interpretation of the stability theory of Gustafsson, Kreiss, and Sundström*, J. Comput. Phys.**49**(1983), no. 2, 199–217. MR**699214**, https://doi.org/10.1016/0021-9991(83)90123-7**[27]**Lloyd N. Trefethen,*Instability of difference models for hyperbolic initial-boundary value problems*, Comm. Pure Appl. Math.**37**(1984), no. 3, 329–367. MR**739924**, https://doi.org/10.1002/cpa.3160370305**[28]**Lloyd N. Trefethen,*Stability of hyperbolic finite-difference models with one or two boundaries*, Large-scale computations in fluid mechanics, Part 2 (La Jolla, Calif., 1983) Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 311–326. MR**818794****[29]**Eli Turkel,*Progress in computational physics*, Comput. & Fluids**11**(1983), no. 2, 121–144. MR**701941**, https://doi.org/10.1016/0045-7930(83)90006-3**[30]**Robert Vichnevetsky and John B. Bowles,*Fourier analysis of numerical approximations of hyperbolic equations*, SIAM Studies in Applied Mathematics, vol. 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1982. With a foreword by Garrett Birkhoff. MR**675265****[31]**H. C. Yee, R. M. Beam & R. F. Warming,*Stable Boundary Approximations for a Class of Implicit Schemes for the One-Dimensional Inviscid Equations of Gas Dynamics*, AIAA 5th Computational Fluid Dynamics Conference, 1981.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804924-2

Article copyright:
© Copyright 1985
American Mathematical Society