Solution of parabolic partial differential equations in complex geome- tries by a modified Fourier collocation method

The heat equation with Dirichlet boundary conditions is solved in various geometries by a modified Fourier collocation method. The computational domain is embedded in a larger, regular domain with a uniform, Cartesian grid, and the solution is defined to be identically zero outside the original domain. The discontinuities thus introduced across the boundary are handled by the modified Fourier collocation method, such that highly accurate approximations to the spatial derivatives along each grid line can be calculated. One-dimensional applications are presented to demonstrate the accuracy and the robustness of the method. A detected robustness problem with respect to the location of boundary points relative to grid points is discussed. and modifications that stabilize the method are presented. Two-dimensional problems are then solved with high accuracy, and the flexibility with respect to complex geometries is demonstrated.