Grid algebra in finite difference methods

Michael P. Lamoureux, Heather K. Hardeman-Vooys

Large, sparse matrices often appear in numerical methods for solving partial differential equations, particularly in finite difference solutions to the wave equation in two and three dimensions. We demonstrate a technique to represent these operators directly on a grid and develop linear algebraic methods to simplify or factor the operator into a form that is easy to solve using back-substitution. We apply the technique to implement an implicit solver for a finite difference algorithm applied to the wave equation in two dimensions.