Symbolic Computation of Linear and Nonlinear Modified (Partial Differential) Equations

Jean-Antoine Désidéri, Margarita Spiridonova,

Date: July 18th (Thursday)
Time: 11:35-12:00
Abstract The so-called ``equivalent'' (Lerat, Peyret, 1974) or ``modified'' (Warming, Hyett, 1974) partial differential equation is a very useful, classical and basic tool of analysis of finite-difference schemes for time-dependent problems, mostly but not only, of hyperbolic type. Although limited in application to simple (usually scalar) model equations, it allows in particular a rigorous analysis of the truncation error, in which dispersive and diffusive errors are identified. Thus new schemes are systematically compared with well-known ones by this analysis applied to a test equation (often the wave or heat equation). Furthermore, the approach can be used constructively to build new schemes in which particular types of error are absent (to a certain order of accuracy).

Originally, Warming and Hyett developed codes of symbolic computation of modified equations using, to our knowledge, the language MACSYMA. We have developed a collection of programs of the same nature for symbolic derivation of the modified equation associated with a given finite-difference equation defined in a rather general format using the computer algebra system MAPLE.

This contribution intends to report on the implementation of these programs and their experimentation related to various finite-difference schemes. For the case of linear (hyperbolic or parabolic) partial differential equations, we have examined classical schemes and less classical examples in which equations in one or two space dimensions including a source term are approximated by upwind schemes. For the case of nonlinear equations, the MacCormack scheme applied to Burgers equation was studied as well as Runge-Kutta-type methods applied to general hyperbolic equations.

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