A High Order Spectral Volume Formulation for Solving Equations Containing Higher Spatial Derivative Terms: Formulation and Analysis for Third Derivative Spatial Terms Using the LDG Discretization Procedure (Communications in Computational Physics, 2011)
In this paper, we develop a formulation for solving equations containinghigher spatial derivative terms in a spectral volume (SV) context; more specifically theemphasis is on handling equations containing third derivative terms. This formulationis based on the LDG (Local Discontinuous Galerkin) flux discretization method, originallyemployed for viscous equations containing second derivatives. A linear Fourieranalysis was performed to study the dispersion and the dissipation properties of thenew formulation. The Fourier analysis was utilized for two purposes: firstly to eliminateall the unstable SV partitions, secondly to obtain the optimal SV partition. Numericalexperiments are performed to illustrate the capability of this formulation. Sincethis formulation is extremely local, it can be easily parallelized and a h-p adaptationis relatively straightforward to implement. In general, the numerical results are verypromising and indicate that the approach has a great potential for higher dimensionKorteweg-de Vries (KdV) type problems.