(untitled) (Communications in Computational Physics, 2012)
In this paper, the second in a series, we improve the discretization of the higher spatial derivative terms in a spectral volume (SV) context. The motivation for the above comes from [J. Sci. Comput., 46(2), 314–328], wherein the authors developed a variant of the LDG (Local Discontinuous Galerkin) flux discretization method. This variant (aptly named LDG2), not only displayed higher accuracy than the LDG approach, but also vastly reduced its unsymmetrical nature. In this paper, we adapt the LDG2 formulation for discretizing third derivative terms. A linear Fourier analysis was performed to compare the dispersion and the dissipation properties of the LDG2 and 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 is however 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 and the LDG2 fluxes was used as the final numerical flux. Even a weight of 1.5% for the LDG (i.e., 98.5% for the LDG2) was sufficient to make the scheme stable. This weighted scheme is still predominantly LDG2 and hence generated smaller dissipation and dispersion errors than the LDG formulation. Numerical experiments are performed to validate the analysis. In general, the numerical results are very promising and indicate that the approach has a great potential for higher dimension Korteweg-de Vries (KdV) type problems.