A pseudospectral method for the solution of the Helmholtz equation

Yung-Ta Li

Department of Mathematics

Fu-Jen Catholic University

ytli@math.fju.edu.tw

    In this talk, we present a pseudospectral method for the Helmholtz equation. The key of the numerical algorithm is to choose a suitable basis associated with the Legendre polynomials that has the following two features: (1) boundary conditions are met and (2) the linear system arising from discretizing the Helmholtz equation under the basis is easily solved. To interpretate the procedure of constructing such a basis, we first introduce two matrix decompositions which are the discrete analogues of the recursion formula and the orthogonal property of the Legendre polynomials, respectively. Subsequently the basis can be constructed through performing row/column operations on the matrix decompositions. Numerical experiments are presented to validate the proposed method.

Keyword: Helmholtz equation; pseudospectral method; Legendre polynomials; tridiagonal matrix.