Stability of parabolic difference approximations to certain mixed initial boundary value problems

Osher, Stanley

doi:10.1090/S0025-5718-1972-0298990-4

Stability of parabolic difference approximations to certain mixed initial boundary value problems
HTML articles powered by AMS MathViewer

by Stanley Osher PDF

Math. Comp. 26 (1972), 13-39 Request permission

Abstract:

We consider the equation \[ {u_t} - a(x,t){u_{xx}} - b(x,t){u_x} - c(x,t)u = f(x,t)\] in a region $0 \leqq x \leqq 1,t \geqq 0$, with inhomogeneous initial and boundary data. We are concerned with stability and estimates on divided differences in the maximum norm for solutions of consistent implicit, multistep, parabolic difference approximations to this problem. Using a parametrix approach, we give sufficient conditions for certain estimates to be valid.

References

Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
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
S. J. Osher, Maximum norm stability for parabolic difference schemes in half-space, Hyperbolic equations and waves (Rencontres, Battelle Res. Inst., Seattle, Wash., 1968) Springer, Berlin, 1970, pp. 61–75. MR 0657803
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
Stanley Osher, Mesh refinements for the heat equation, SIAM J. Numer. Anal. 7 (1970), 199–205. MR 266451, DOI 10.1137/0707013
Gilbert Strang, Wiener-Hopf difference equations, J. Math. Mech. 13 (1964), 85–96. MR 0160335
Gilbert Strang, Implicit difference methods for initial-boundary value problems, J. Math. Anal. Appl. 16 (1966), 188–198. MR 205496, DOI 10.1016/0022-247X(66)90196-X
Olof B. Widlund, Stability of parabolic difference schemes in the maximum norm, Numer. Math. 8 (1966), 186–202. MR 196965, DOI 10.1007/BF02163187
Olof B. Widlund, On the rate of convergence for parabolic difference schemes. I, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 60–73. MR 0264867
Olof B. Widlund, On the rate of convergence for parabolic difference schemes. I, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 60–73. MR 0264867
J. M. Varah, Maximum norm stability of difference approximations to the mixed initial boundary-value problem for the heat equation, Math. Comp. 24 (1970), 31–44. MR 260215, DOI 10.1090/S0025-5718-1970-0260215-1
J. M. Varah, Stability of difference approximations to the mixed initial boundary value problems for parabolic systems, SIAM J. Numer. Anal. 8 (1971), 598–615. MR 300475, DOI 10.1137/0708057