J. G. Blom
Abstract
This article deals with an adaptive-grid finite-difference solver for time-dependent two-dimensional systems of partial differential equations. It describes the ANSI Fortran 77 code, VLUGR2, autovectorizable on the Cray Y-MP, that is based on this method. The robustness and the efficiency of the solver, both for vector and scalar processors, are illustrated by the application of the code to two example problems arising from a groundwater-flow model.
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Software for "VLUGR2: a vectorizable adaptive-grid solver for PDEs in 2D"
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Index Terms
- Algorithm 758: VLUGR2: a vectorizable adaptive-grid solver for PDEs in 2D
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Algorithm 759: VLUGR3: a vectorizable adaptive-grid solver for PDEs in 3D—Part II. code description
This article describes an ANSI Fortran 77 code, VLUGR3, autovectorizable on the Cray Y-MP, that is based on an adaptive-grid finite-difference method to solve time-dependent three-dimensional systems of partial differential equations.
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Reviews
Warren E. Ferguson
VLUGR2 and VLUGR3, for partial differential equations (PDEs) in two and three spatial dimensions, respectively, are vectorizable Fortran 77 codes that solve initial boundary-value problems involving a system of PDEs. Both codes use an adaptive-grid finite-difference method and are tuned for a system of time-dependent parabolic PDEs. The system of PDEs is solved using a “method of lines” approach. In this approach, the PDEs are first discretized in space, and the resulting ordinary differential equations (ODEs) or differential-algebraic equations (DAEs) in time are solved using a second-order two-step implicit backward differentiation formula (BDF) method. The resulting system of equations created by the BDF method is solved by the user's choice of one of three solvers: (1) bi-conjugate gradient method stabilized (BiCGStab) with ILU preconditioning, (2) generalized conjugate residual orthonormalization (GCRO) with a simple block diagonal scaling, or (3) a matrix-free version of (2). The spatial discretization is carried out by central finite-differences in the interior and one-sided finite-differences on the boundary. As time evolves, the spatial grid is adapted via the local uniform grid r efinement method. This method starts with a coarse grid that covers the entire spatial domain and recursively introduces finer grids on spatial subdomains with high spatial activity. High spatial activity is detected by a function that monitors the spatial curvature of the solution. The fine-grid nodal solution values are injected into the coinciding coarser-grid nodes. When a grid cell must be refined, the cell is divided into four equal parts. When interpolation is needed to determine solution values, linear interpolation is used. For at least VLUGR3, there is a restriction on the shape of the spatial domain. The use of the codes is demonstrated by using them to solve a system of parabolic PDEs where each PDE is similar to Burger's equation.
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