@Article{CiCP-12-3,
author = {},
title = {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},
journal = {Communications in Computational Physics},
year = {2012},
volume = {12},
number = {3},
pages = {767--788},
abstract = {<p style="text-align: justify;">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],&nbsp; 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 2<sup>nd</sup> and 3<sup>rd</sup> orders and (ii) generates smaller dissipation and dispersion errors than the LDG formulation for all the orders. The 4<sup>th</sup> 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.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.030211.040811a},
url = {https://global-sci.com/article/80797/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}
}