Circulant matrix methods for the numerical solution of partial differential equations by FFT convolutions

Abstract

A method is described for obtaining finite difference approximation solutions of multidimensional partial differential equations satisfying boundary conditions specified on irregularly shaped bondaries by using circulant matrices and fast Fourier transform (FFT) convolutions. Circulant meshes are used to embed the region of interest within a cyclic region. Even with irregular boundaries, solutions can sometimes be found in closed form, although often iterations will be required. There is a degree of commonality between different problems, without regard to the number of dimensions involved, because the matrices characterize the linear homogeneous differential operators (or inverses) involved rather than the operator and boundary conditions combined. The inverse circulant can be found easily if it exists, and both the system circulant and its inverse can be stored as one-dimensional entities (cyclic strings). Although the circulant matrices are associated with linear operators, this does not prevent the solution of some nonlinear equations. Most of the method's computation effort occurs in FFT operations. Sets of equations can be handled in a similar way using block circulant matrices. It is expected that the method will find uses in parallel computing algorithms.