The semigroup stability of the difference approximations for initial-boundary value problems

Author:
Lixin Wu

Journal:
Math. Comp. **64** (1995), 71-88

MSC:
Primary 65N06; Secondary 34G10, 65N12

MathSciNet review:
1257582

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For semidiscrete approximations and one-step fully discretized approximations of the initial-boundary value problem for linear hyperbolic equations with diagonalizable coefficient matrices, we prove that the Kreiss condition is a sufficient condition for the semigroup stability (or stability). Also, we show that the stability of a fully discretized approximation generated by a locally stable Runge-Kutta method is determined by the stability of the semidiscrete approximation.

**[1]**M. S. Agronovich,*Theorem of matrices depending on parameters and its application to hyperbolic systems*, Functional Anal. Appl.**6**(1972), 85-93.**[2]**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**, 10.1090/S0025-5718-1981-0606519-9**[3]**Bertil Gustafsson,*Numerical boundary conditions*, Large-scale computations in fluid mechanics, Part 1 (La Jolla, Calif., 1983), Lectures in Appl. Math., vol. 22, Amer. Math. Soc., Providence, RI, 1985, pp. 279–308. MR**818773****[4]**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**, 10.1090/S0025-5718-1972-0341888-3**[5]**Reuben Hersh,*Mixed problems in several variables*, J. Math. Mech.**12**(1963), 317–334. MR**0147790****[6]**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****[7]**Heinz-Otto Kreiss,*Stability theory for difference approximations of mixed initial boundary value problems. I*, Math. Comp.**22**(1968), 703–714. MR**0241010**, 10.1090/S0025-5718-1968-0241010-7**[8]**Heinz-Otto Kreiss,*Initial boundary value problems for hyperbolic systems*, Comm. Pure Appl. Math.**23**(1970), 277–298. MR**0437941****[9]**Heinz-Otto Kreiss,*Problems with different time scales for partial differential equations*, Comm. Pure Appl. Math.**33**(1980), no. 3, 399–439. MR**562742**, 10.1002/cpa.3160330310**[10]**H.-O. Kreiss,*Difference approximation for initial boundary value problems*. I, preprint.**[11]**Heinz-O. Kreiss and Lixin Wu,*On the stability definition of difference approximations for the initial-boundary value problem*, Appl. Numer. Math.**12**(1993), no. 1-3, 213–227. Special issue to honor Professor Saul Abarbanel on his sixtieth birthday (Neveh, 1992). MR**1227187**, 10.1016/0168-9274(93)90119-C**[12]**Daniel Michelson,*Stability theory of difference approximations for multidimensional initial-boundary value problems*, Math. Comp.**40**(1983), no. 161, 1–45. MR**679433**, 10.1090/S0025-5718-1983-0679433-2**[13]**Stanley Osher,*Systems of difference equations with general homogeneous boundary conditions*, Trans. Amer. Math. Soc.**137**(1969), 177–201. MR**0237982**, 10.1090/S0002-9947-1969-0237982-4**[14]**Jeffrey Rauch,*\cal𝐿₂ is a continuable initial condition for Kreiss’ mixed problems*, Comm. Pure Appl. Math.**25**(1972), 265–285. MR**0298232****[15]**John C. Strikwerda,*Initial-boundary value problems for the method of lines*, J. Comput. Phys.**34**(1980), no. 1, 94–107. MR**558146**, 10.1016/0021-9991(80)90114-X

Retrieve articles in *Mathematics of Computation*
with MSC:
65N06,
34G10,
65N12

Retrieve articles in all journals with MSC: 65N06, 34G10, 65N12

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1995-1257582-6

Keywords:
Hyperbolic,
semigroup stability,
Runge-Kutta methods

Article copyright:
© Copyright 1995
American Mathematical Society