A pseudo-spectral method for solving the Helmholtz equation in curvilinear nonorthogonal coordinates

A pseudo-spectral method is presented which is preconditioned to the solution of the two-dimensional Helmholtz equation in nonorthogonal curvilinear coordinates. A transformation is defined for the coordinates of the Helmholtz system of equations with Dirichlet, Neumann or mixed boundary conditions. The spectral decomposition of the Helmholtz equations is performed on the basis of Tchebychev polynomials, thereby obtaining the nonorthogonal coordinate system. A finite difference scheme is used to generate a preconditioning matrix for an approximate factorization of the Helmholtz operator to obtain an operator that is applicable for iteratively solving the resulting system of equations. The Richardson method is used for the iterative calculations. Use of the method is illustrated by describing the flow inside a two-dimensional channel with a variable section. The technique is also useful for solving the Poisson equation.