A pseudospectral method for solving the Helmholtz equation in curvilinear nonorthogonal coordinates
Abstract
A pseudospectral method is presented which is preconditioned to the solution of the twodimensional 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 twodimensional channel with a variable section. The technique is also useful for solving the Poisson equation.
 Publication:

Academie des Sciences Paris Comptes Rendus Serie Sciences Mathematiques
 Pub Date:
 December 1985
 Bibcode:
 1985CRASM.301.1327F
 Keywords:

 Computational Fluid Dynamics;
 Helmholtz Equations;
 NavierStokes Equation;
 Spectral Methods;
 Two Dimensional Flow;
 Boundary Conditions;
 Boundary Value Problems;
 Channel Flow;
 Coordinate Transformations;
 Finite Difference Theory;
 Iterative Solution;
 Fluid Mechanics and Heat Transfer