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Usefulness of computer algebra methods in numerical simulations

Michel Fournié
Laboratoire M.I.P., UMR-CNRS 5640
118 route de Narbonne
31062 Toulouse Cedex 04, France
fournie@mip.ups-tlse.fr

Abstract:

Different kind of Symbolic-Numeric interfaces used to - define, analyse, implement - new schemes approximating Partial Differential Equations are presented. A symbolic approach to define stable (Fourier method) and accurate (direct computation of the truncation error) schemes for bidimensional unsteady diffusion-convection problems is presented. For steady state problem, we describe symbolic and automatic procedures for deriving finite difference approximations. We present new robust schemes used for the simulation of semiconductor devices and detail the treatment of the boundary conditions (elimination of the ghosts points), the linearization of the problem using Newton iteration (computation of the Jacobian matrix) and the generation of FORTRAN code. The elimination of some numerical instabilities and the reduction of the time required for numerical tests are considered and limited by symbolic manipulations (using the computer algebra system AXIOM).





IMACS ACA'99 Symbolic-Numeric Interface session