The stability of Runge–Kutta spectral volume methods for 1-D linear hyperbolic equations with constant coefficients

Abstract

This paper focuses on proving the stability of the Runge–Kutta spectral volume (RKSV) scheme for solving one-dimensional equations, with a specific analysis of the Radau spectral volume (RRSV) and Gauss–Legendre spectral volume (LSV) schemes. By comparing the similarities and discrepancies between the Runge–Kutta spectral volume (RKSV) and Runge–Kutta discontinuous Galerkin (RKDG) schemes, we transform the stability analysis of the RKSV scheme into that of the RKDG scheme-an approach that already possesses a well-established theoretical analysis basis. Our key findings reveal two critical results: first, the Runge–Kutta Radau spectral volume (RKRRSV) scheme is entirely equivalent to the RKDG scheme; second, under the framework of a newly defined norm, the Runge–Kutta Gauss–Legendre spectral volume (RKLSV) scheme yields stability results identical to those of the RKDG scheme. Furthermore, numerical experiments are conducted to validate both the stability and optimal convergence properties of the RKSV scheme, providing empirical support for its theoretical conclusions.