A High Order Spectral Volume Method for Equations Containing Third Spatial Derivatives Using an Interior Penalty Formulation

Abstract

In this paper, we present an interior penalty formulation for solving equations containing third spatial derivative terms, in the context of a high order spectral volume method. The motivation for this approach comes from the “penalization method” formulated by Kannan and Wang (Kannan R, Wang Z.J., “A Study of Viscous Flux Formulations for a p-Multigrid Spectral Volume Navier Stokes Solver, “Journal of Scientific Computing, 41(2), 2009, 165 - 199) developed for discretizing the second order viscous fluxes. A linear Fourier analysis was performed to compare and contrast the dispersion and the dissipation properties of the new formulation and the existing LDG (Local Discontinuous Galerkin) formulation. The analyses performed on the linear and the cubic partitions showed improvement over the traditional LDG formulation. The analysis performed on the quadratic partition showed significant dispersion, when compared to the LDG scheme. The new formulation is easy to implement and is highly symmetrical. Numerical experiments were conducted and the results were in accord with the analysis results. In general, the formulation is general and can be used even for higher dimension problems.