Marching Methods for Elliptic Problems: Part 1

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Abstract
The history and a basic algorithm for solving elliptic problems by direct marching methods are reviewed. Accurate operation counts for initialization and repeat solutions are given and are shown to compare well with other direct methods and with iterative methods. The instability of the marching method is described, and a simple estimate of the resulting mesh size limitation is given. Then the following applications are described: various boundary conditions (Dirichlet, Neumann, Robin, and periodic), irregular meshes, irregular boundaries, interior boundaries, variable coefficient diffusion equations, advection terms (including cell Reynolds number effects and the destabilizing effects of upwind differencing), Helmholtz terms, cross derivatives, turbulence terms, and an expanding grid based on the Fibonacci sequence.