Analysis of two fully discrete spectral volume schemes for hyperbolic equations

Abstract

Abstract In this paper, we analyze two classes of fully discrete spectral volume schemes (SV) for solving the one‐dimensional scalar hyperbolic equation. These two schemes are constructed by using the forward Euler (EU) method or the second‐order Runge–Kutta (RK2) method in time‐discretization, and by letting a piecewise k th degree( is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with Gauss–Legendre points (LSV) or right‐Radau points (RRSV). We prove that for the EU‐SV schemes, the weak (2) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. While for the RK2‐SV schemes, the weak (4) stability holds and the norm errors converge with optimal orders , provided that the CFL condition is satisfied. Here and are, respectively, the spacial and temporal mesh sizes and the constant is independent of and . Our theoretical findings have been justified by several numerical experiments.