Programs in Physics & Physical Chemistry
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|Manuscript Title: FORSIM VI: a program package for the automated solution of arbitrarily defined differential equations.|
|Authors: M.B. Carver|
|Program title: FORSIM VI|
|Catalogue identifier: ACYZ_v1_0|
Distribution format: gz
|Journal reference: Comput. Phys. Commun. 17(1979)283|
|Programming language: Fortran.|
|Computer: CDC 175.|
|Operating system: CDC NOS/BE.|
|Word size: 60|
|Keywords: General purpose, Partial Differential equation, Ordinary differential Equation, Method of lines, Differential quadrature.|
Nature of problem:
Mathematical models comprising partial and ordinary differential equations arise in many scientific disciplines. A large class of these models may be automatically transformed into initial value problems involving only coupled ordinary differential equations, which may then be solved numerically by error-controlled integration. The FORSIM package solves a set of user-defined equations in this manner. These may be ordinary differential equations, and/or partial differential equations in up to three dimensions.
The user describes his equations and related conditions in a single FORTRAN subroutine, according to certain clearly defined rules. This subroutine may call other routines if necessary. Any partial differential equations are then transformed into ordinary differential equations in time, using locally one-dimensional piecewise approximation formulae in the spatial dimensions. Boundary conditions are incorporated in the approximation formulae. The resulting ordinary differential equations are integrated by algorithms which choose the optimal step size to maintain the estimated local truncation error below a specifiable tolerable value. The definition of the equations and the type and frequency of sampling printout are entirely within the user control; the transformation and integration are performed automatically, but the user may exert optional controls governing these operations.
The program is implemented to accept a maximum of 1000 ordinary differential equations, but this limit may be readily changed in a simple user written routine. The main restriction is upon the spatial system, which should have a regular boundary, but may be any orthogonal coordinate system providing the equations are written in a compatible form.
The integration routines in FORSIM have been intensively developed and tested. They include an option to accelerate predictor corrector convergence by an automatic sparse matrix assessment of the system Jacobian, and the application of the error criterion is also unique. The transformation of partial differential equations into ordinary differential equations for two and three dimensions is analogous to the simple one-dimensional case. A user's manual complete with bibliography, background theory, programming philosophy and applications examples is available, and is in fact essential, as the range of possible application is too immense to give anything but an overview here.
Running time is a complicated function of the number of differential equations, the integration option used, the behaviour of the equations and the requested accuracy.
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