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Stability of difference approximations of dissipative type for mixed initial-boundary value problems. I

Author: Stanley Osher
Journal: Math. Comp. 23 (1969), 335-340
MSC: Primary 65.67
MathSciNet review: 0246530
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Abstract: H. O. Kreiss, [2], has recently extended the stability theory of difference approximations to include the hyperbolic system

$\displaystyle u_t = A{u_x},0 \leqq x,t$

, 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 $ \vert\hat Q(\xi )\vert \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 [Enhancements On Off] (What's this?)

  • [1] S. Osher, ``Systems of difference equations with general homogeneous boundary conditions,'' Trans. Amer. Math. Soc., v. 137, 1969, pp. 177-201. MR 0237982 (38:6259)
  • [2] H. O. Kreiss, ``Stability theory for difference approximations for mixed initial boundary value problems. I,'' Math. Comp., v. 22, 1968, pp. 703-714. MR 0241010 (39:2355)
  • [3] H. O. Kreiss, ``Difference approximations for the initial-boundary value problem for hyperbolic differential equations,'' in Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966), Wiley, New York, 1966, pp. 141-166. MR 35 #5156. MR 0214305 (35:5156)
  • [4] I. C. Gohberg & M. G. Krein, ``Systems of integral equations on a half line with kernels depending on the difference of arguments,'' Uspehi Mat. Nauk, v. 13, 1958, no. 2 (80), pp. 3-72; English transl., Amer. Math. Soc. Transl., v. (2) 14, 1960, pp. 217-288. MR 21 #1506; MR 22 #3954. MR 0113114 (22:3954)
  • [5] H. O. Kreiss, ``On difference approximations of the dissipative type for hyperbolic differential equations,'' Comm. Pure Appl. Math., v. 17, 1964, pp. 335-353. MR 29 #4210. MR 0166937 (29:4210)

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Article copyright: © Copyright 1969 American Mathematical Society

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