# FIDE

FIDE: a REDUCE package for automation of FInite difference method for solving pDE. The article deals with automation of the process of numerical solving of partial differential equations systems (PDES) by means of computer algebra. For solving PDES the finite difference method is applied. The computer algebra system REDUCE and the numerical programming language FORTRAN are used in the methodology presented, its main aim being to speed up the process of preparing numerical programs for solving PDES. Quite often, especially for complicated systems, this process is a tedious and time consuming task. In the process several stages can be found in which computer algebra can be used for performing routine analytical calculations, namely: transformation of differential equations into different coordinate systems, discretization of differential equations, analysis of difference schemes, and generation of numerical programs. The FIDE package consists of the following modules that have been built, tested, and documented: EXPRESS for transforming PDES into any orthogonal coordinate system. IIMET for discretization of PDES by integro-interpolation method. APPROX for determining the order of approximation of difference schemes. CHARPOL for calculating the amplification matrix and characteristic polynomial of difference schemes, which are needed in the Fourier stability analysis. HURWP for locating polynomial roots necessary in verifying the von Neumann stability condition. LINBAND for generating the block of FORTRAN code, which solves a system of linear algebraic equations with band matrix appearing frequently in difference schemes. The FIDE package is applied to two physical problems. The first one is the nonlinear Schrödinger equation, which describes several physical phenomena, e.g. the Langmuir waves in plasma. The second one is the Fokker-Planck equation in diffusive approximation, which is used to describe the kinetics of electrons in plasma, including interactions of electrons with a laser beam through inverse bremsstrahlung. The numerical programs have been tested and compared with similar calculations published so far

This software is also peer reviewed by journal TOMS.

## Keywords for this software

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## References in zbMATH (referenced in 5 articles )

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