Analysis of any order Runge-Kutta Spectral Volume Schemes for 1D Hyperbolic Equations

TL;DR

This paper provides a comprehensive analysis of arbitrary-order Runge-Kutta spectral volume schemes for 1D hyperbolic equations, establishing stability and convergence properties through a matrix-based framework.

Contribution

It introduces a general matrix transfer process framework for analyzing stability and convergence of RKSV schemes, including error estimates and key indicators.

Findings

Stability of RKSV(s,k) schemes is proven under CFL condition.

Error estimate of order O(h^{k+1} + τ^s) is established.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we analyze any-order Runge-Kutta spectral volume schemes (RKSV(s,k)) for solving the one-dimensional scalar hyperbolic equation. The RKSV(s,k) was constructed by using the $s$-th explicit Runge-Kutta method in time-discretization which has {\it strong-stability-preserving} (SSP) property, and by letting a piecewise $k-$th degree($k\geq 1 $ is an arbitrary integer) polynomial satisfy the local conservation law in each control volume designed by subdividing the underlying mesh with $k$ Gauss-Legendre points (LSV) or right-Radau points (RRSV).For the RKSV(s,k), we would like to establish a general framework which use the matrix transferring process technique for analyzing the stability and the convergence property. The framework for stability is evolved based on the energy equation, while the framework for error estimate is evolved based on the error equation. And the evolution process is represented by matrices.After the evolution is completed, three key indicative pieces of information are obtained: the termination factor $\zeta$, the indicator factor $\rho$, and the final evolved matrix. We prove that for the RKSV(s,k), the {\it stability } holds and the $L_2$ norm error estimate is $\mathcal{O}(h^{k+1}+\tau^s)$, provided that the CFL condition is satisfied. Our theoretical findings have been justified by several numerical experiments.