Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I
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- by Stanley Osher PDF
- Math. Comp. 23 (1969), 335-340 Request permission
Abstract:
H. O. Kreiss, [2], has recently extended the stability theory of difference approximations to include the hyperbolic system \[ u_t = A{u_x},0 \leqq x,t\] , with $A$ a diagonal matrix. Appropriate boundary and initial conditions are given. The amplification matrix $\hat Q(\xi )$ need not be diagonal. However, he required that $|\hat Q(\xi )| \leqq 1$. We use certain results in matrix theory and Wiener-Hopf factorization to replace this restrictive assumption by certain reasonable assumptions on accuracy of $\hat Q(\xi )$ and smoothness of an associated positive-definite symmetric matrix. This technique will be important in half-space problems in many space variables since for such problems the amplification matrix will certainly not be diagonal.References
- Stanley Osher, Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc. 137 (1969), 177–201. MR 237982, DOI 10.1090/S0002-9947-1969-0237982-4
- Heinz-Otto Kreiss, Stability theory for difference approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 241010, DOI 10.1090/S0025-5718-1968-0241010-7
- 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, N.Y., 1966, pp. 141–166. MR 0214305
- I. C. Gohberg and M. G. Kreĭn, Systems of integral equations on a half line with kernels depending on the difference of arguments, Amer. Math. Soc. Transl. (2) 14 (1960), 217–287. MR 0113114, DOI 10.1090/trans2/014/09
- Heinz-Otto Kreiss, On difference approximations of the dissipative type for hyperbolic differential equations, Comm. Pure Appl. Math. 17 (1964), 335–353. MR 166937, DOI 10.1002/cpa.3160170306
Additional Information
- © Copyright 1969 American Mathematical Society
- Journal: Math. Comp. 23 (1969), 335-340
- MSC: Primary 65.67
- DOI: https://doi.org/10.1090/S0025-5718-1969-0246530-8
- MathSciNet review: 0246530