A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms II: Improving the Third Derivative Spatial Discretization Using the LDG2 Method (Communications in Computational Physics, 2012)
In this paper, the second in a series, we improve the discretization of thehigher spatial derivative terms in a spectral volume (SV) context. The motivation forthe above comes from [J. Sci. Comput., 46(2), 314–328],  wherein the authors developeda variant of the LDG (Local Discontinuous Galerkin) flux discretization method. Thisvariant (aptly named LDG2), not only displayed higher accuracy than the LDG approach, but also vastly reduced its unsymmetrical nature. In this paper, we adapt theLDG2 formulation for discretizing third derivative terms. A linear Fourier analysiswas performed to compare the dispersion and the dissipation properties of the LDG2and the LDG formulations. The results of the analysis showed that the LDG2 scheme(i) is stable for 2nd and 3rd orders and (ii) generates smaller dissipation and dispersion errors than the LDG formulation for all the orders. The 4th order LDG2 scheme ishowever mildly unstable: as the real component of the principal eigen value briefly becomes positive. In order to circumvent the above, a weighted average of the LDG andthe LDG2 fluxes was used as the final numerical flux. Even a weight of 1.5% for theLDG (i.e., 98.5% for the LDG2) was sufficient to make the scheme stable. This weightedscheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation. Numerical experiments are performed tovalidate the analysis. In general, the numerical results are very promising and indicatethat the approach has a great potential for higher dimension Korteweg-de Vries (KdV)type problems.