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

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 howevermildly 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 tomake the scheme stable.Thisweighted 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.

Improving the High Order Spectral Volume Formulation Using a Diffusion Regulator

The concept of diffusion regulation(DR)was originally proposed by Jaisankar for traditional second order finite volume Euler solvers.This was used to decrease the inherent dissipation associated with using approximate Riemann solvers.In this paper,the above concept is extended to the high order spectral volume(SV)method.The DR formulation was used in conjunction with the Rusanov flux to handle the inviscid flux terms.Numerical experiments were conducted to compare and contrast the original and the DR formulations.These experiments demonstrated(i)retention of high order accuracy for the new formulation,(ii)higher fidelity of the DR formulation,when compared to the original scheme for all orders and(iii)straightforward extension to Navier Stokes equations,since the DR does not interfere with the discretization of the viscous fluxes.In general,the 2D numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.

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

In this paper,we develop a formulation for solving equations containing higher spatial derivative terms in a spectral volume(SV)context;more specifically the emphasis is on handling equations containing third derivative terms.This formulation is based on the LDG(Local Discontinuous Galerkin)flux discretization method,originally employed for viscous equations containing second derivatives.A linear Fourier analysis was performed to study the dispersion and the dissipation properties of the new formulation.The Fourier analysis was utilized for two purposes:firstly to eliminate all the unstable SV partitions,secondly to obtain the optimal SV partition.Numerical experiments are performed to illustrate the capability of this formulation.Since this formulation is extremely local,it can be easily parallelized and a h-p adaptation is relatively straightforward to implement.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.