Solving Two- Dimensional Partial Differential Equations On Vector and Scalar Machines

Abstract

TTGR is a Fortran program that solves time-varying par tial differential equations on a rectangle, using a Galerkin method with fixed tensor product B-splines, nonuniform meshes, and higher order approximations. The algo rithm reduces the partial differential equation to a linear system. Problems requiring enormous resources with a finite element or finite difference method are tractable with the Galerkin approach. However, the construction of Galerkin equations often is more costly than their so lution. The straightforward construction has inner loop lengths on the order of the approximation k, which is typically 4. We rearrange the construction so that the inner loops are over the number of mesh points n on a side. On vector machines this produces a speedup of nearly nlk for constructing the matrix equations for the interior points of the domain. We also discuss the per formance of the linear system solvers in TTGR on vector and scalar machines on problems with different orders of B-spline approximations.