Unified Analysis of Any Order Spectral Volume Methods for Diffusion Equations

In this paper, a new class of spectral volume (SV) methods are proposed, analyzed, and implemented for diffusion equations, with the viscous flux taken as an interior penalty or direct discontinuous Galerkin formulation. The control volumes are constructed by using four kinds of special points (including Legendre–Gauss, Legendre–Gauss–Lobatto, right Legendre–Gauss–Radau and left Legendre–Gauss–Radau points) in subintervals of the underlying meshes, which leads to four different SV schemes. A framework for the stability analysis and error estimates of the four SV schemes is established. In particular, the influence of the choice of the parameters in the numerical fluxes on the convergence rate and the optimal choices of coefficients for each SV scheme are discussed and provided. Numerical experiments are presented to demonstrate the stability and accuracy of the four SV schemes for both linear and nonlinear diffusion equations.