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

Author:
Alfred Carasso

Journal:
Math. Comp. **24** (1970), 785-792

MSC:
Primary 65.68

MathSciNet review:
0281374

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Abstract | References | Similar Articles | Additional Information

Abstract: We derive a posteriori bounds for and its difference quotient , where and 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 and .

**[1]**Alfred Carasso,*Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators*, Math. Comp.**23**(1969), 717–729. MR**0258291**, 10.1090/S0025-5718-1969-0258291-7**[2]**Alfred Carasso and Seymour V. Parter,*An analysis of “boundary-value techniques” for parabolic problems.*, Math. Comp.**24**(1970), 315–340. MR**0284019**, 10.1090/S0025-5718-1970-0284019-9**[3]**Alfred Carasso,*Long-range numerical solution of mildly non-linear parabolic equations.*, Numer. Math.**16**(1970/1971), 304–321. MR**0286301****[4]**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****[5]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[6]**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**0047885****[7]**H. O. Kreiss & O. B. Widlund,*Difference Approximations for Initial Value Problems for Partial Differential Equations*, Department of Computer Sciences, Report NR 7, Upsala University, 1967.**[8]**Milton Lees,*Approximate solutions of parabolic equations*, J. Soc. Indust. Appl. Math.**7**(1959), 167–183. MR**0110212****[9]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[10]**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**0224312**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0281374-0

Keywords:
Parabolic equations,
Crank-Nicolson method,
limiting steady state,
computations over long times

Article copyright:
© Copyright 1970
American Mathematical Society