Any order spectral volume methods for diffusion equations using the local discontinuous Galerkin formulation

In this paper, we present and study two spectral volume (SV) schemes of arbitrary order for diffusion equations by using the local discontinuous Galerkin formulation to discretize the viscous flux. The basic idea of the scheme is to rewrite the diffusion equation into an equivalent first-order system first, and then use the SV method to solve the system. The SV scheme is designed with control volumes constructed by using the Gauss points or Radau points in subintervals of the underlying meshes, which leads to two SV schemes referred to as LSV and RSV schemes, respectively. The stability analysis for the linear diffusion equations based on alternating fluxes are provided, and optimal error estimates are established for both the exact solution and the auxiliary variable. Furthermore, a rigorous mathematical proof are given to demonstrate that the proposed RSV method is identical to the standard LDG method when applied to constant diffusion problems. Numerical experiments are presented to demonstrate the stability, accuracy and performance of the two SV schemes for both linear and nonlinear diffusion equations.