Stability of difference approximations of dissipative type for mixed initialboundary value problems. I
Author:
Stanley Osher
Journal:
Math. Comp. 23 (1969), 335340
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 , with a diagonal matrix. Appropriate boundary and initial conditions are given. The amplification matrix need not be diagonal. However, he required that . We use certain results in matrix theory and WienerHopf factorization to replace this restrictive assumption by certain reasonable assumptions on accuracy of and smoothness of an associated positivedefinite symmetric matrix. This technique will be important in halfspace problems in many space variables since for such problems the amplification matrix will certainly not be diagonal.
 [1]
Stanley
Osher, Systems of difference equations with
general homogeneous boundary conditions, Trans.
Amer. Math. Soc. 137 (1969), 177–201. MR 0237982
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 [2]
HeinzOtto
Kreiss, Stability theory for difference
approximations of mixed initial boundary value problems. I, Math. Comp. 22 (1968), 703–714. MR 0241010
(39 #2355), http://dx.doi.org/10.1090/S00255718196802410107
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HeinzOtto
Kreiss, Difference approximations for the initialboundary value
problem for hyperbolic differential equations, Numerical Solutions of
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John Wiley & Sons, Inc., New York, 1966, pp. 141–166. MR 0214305
(35 #5156)
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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
(22 #3954)
 [5]
HeinzOtto
Kreiss, On difference approximations of the dissipative type for
hyperbolic differential equations, Comm. Pure Appl. Math.
17 (1964), 335–353. MR 0166937
(29 #4210)
 [1]
 S. Osher, ``Systems of difference equations with general homogeneous boundary conditions,'' Trans. Amer. Math. Soc., v. 137, 1969, pp. 177201. 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. 703714. MR 0241010 (39:2355)
 [3]
 H. O. Kreiss, ``Difference approximations for the initialboundary value problem for hyperbolic differential equations,'' in Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966), Wiley, New York, 1966, pp. 141166. 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. 372; English transl., Amer. Math. Soc. Transl., v. (2) 14, 1960, pp. 217288. 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. 335353. MR 29 #4210. MR 0166937 (29:4210)
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DOI:
http://dx.doi.org/10.1090/S00255718196902465308
PII:
S 00255718(1969)02465308
Article copyright:
© Copyright 1969
American Mathematical Society
