A posteriori bounds in the numerical solution of mildly nonlinear parabolic equations

Carasso, Alfred

doi:10.1090/S0025-5718-1970-0281374-0

A posteriori bounds in the numerical solution of mildly nonlinear parabolic equations
HTML articles powered by AMS MathViewer

by Alfred Carasso PDF

Math. Comp. 24 (1970), 785-792 Request permission

Abstract:

We derive a posteriori bounds for $(V - \hat V)$ and its difference quotient ${(V - \hat V)_x}$, where $V$ and $\hat V$ are, respectively, the exact and computed solution of a difference approximation to a mildly nonlinear parabolic initial boundary problem, with a known steadystate solution. It is assumed that the computation is over a long interval of time. The estimates are valid for a class of difference approximations, which includes the CrankNicolson method, and are of the same magnitude for both $(V - \hat V)$ and $(V - \hat V)x$.

References

Alfred Carasso, Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators, Math. Comp. 23 (1969), 717–729. MR 258291, DOI 10.1090/S0025-5718-1969-0258291-7
Alfred Carasso and Seymour V. Parter, An analysis of “boundary-value techniques” for parabolic problems, Math. Comp. 24 (1970), 315–340. MR 284019, DOI 10.1090/S0025-5718-1970-0284019-9
Alfred Carasso, Long-range numerical solution of mildly non-linear parabolic equations, Numer. Math. 16 (1970/71), 304–321. MR 286301, DOI 10.1007/BF02165002
Jim Douglas Jr., A survey of numerical methods for parabolic differential equations, Advances in Computers, Vol. 2, Academic Press, New York, 1961, pp. 1–54. MR 0142211
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Fritz John, On integration of parabolic equations by difference methods. I. Linear and quasi-linear equations for the infinite interval, Comm. Pure Appl. Math. 5 (1952), 155–211. MR 47885, DOI 10.1002/cpa.3160050203

Difference Approximations for Initial Value Problems for Partial Differential Equations

Milton Lees, Approximate solutions of parabolic equations, J. Soc. Indust. Appl. Math. 7 (1959), 167–183. MR 110212
Robert D. Richtmyer and K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR 0220455
O. B. Widlund, On difference methods for parabolic equations and alternating direction implicit methods for elliptic equations, IBM J. Res. Develop. 11 (1967), 239–243. MR 224312, DOI 10.1147/rd.112.0239