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Predictor‐corrector methods for parabolic partial differential equations
Author(s) -
Jacques I. B.
Publication year - 1983
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620190311
Subject(s) - predictor–corrector method , mathematics , convergence (economics) , nonlinear system , numerical stability , ordinary differential equation , stability (learning theory) , mathematical analysis , partial differential equation , numerical analysis , algebraic equation , algebraic number , backward differentiation formula , method of lines , differential equation , differential algebraic equation , physics , computer science , quantum mechanics , machine learning , economics , economic growth
In this paper we extend predictor‐corrector methods, commonly used for the numerical solution of ordinary differential equations (o.d.e.s), to parabolic partial differential equations (p.d.e.s), typically of the form u t = au xx + ƒ( u , u x , x , t ). We describe linear multistep methods for p.d.e.s, the nonlinear algebraic equations arising from implicit formulae being solved using a corrector analogous to those used for o.d.e.s. A sufficient condition for convergence of the iteration is then derived and is found, in most cases, to be far less restrictive than that obtained from the usual method of lines approach. Numerical results are presented to investigate the necessity of this condition. They also indicate that we can accelerate convergence by reducing the time increment. This allows us to achieve convergence within a prescribed number of iterations and so to construct PC m methods corresponding to P ( EC ) m methods for o.d.e.s. Numerical results are also given to test the absolute stability of the Crank‐Nicolson corrector for various predictors P , and iterations, m .