Higher-order accurate finite volume discretization of Helmholtz equations with Pollution Effects Reductions

Abstract

Higher-order accurate finite volume schemes are developed for Helmholtz equations in two dimensions. Through minimizations of local equation error expansions for the flux integral formulation of the equation, we determine quadrature weights for the discretization of the equation. Collocations of local expansions of the solution and the source terms are utilized to formulate weighted quadratures of all local compact fluxes to describe the equation error expansion within the computational domain. In using the source term distribution to account for fluxes along all compact directions about each grid point as the centroid of a local control volume, the right minimizing quadrature weights are determined and optimized for stability and uniform higher-order convergence. As a result, the resulting local residuals form more complete descriptions of the wave number k and the complexities of the associated pollution effects. The leading terms of the residual errors are optimized for pollution effects reductions to ensure stability and robust convergence of the resulting schemes. Numerical results and analysis of the schemes demonstrate the effectiveness of the methodology.