On the Accuracy of Finite-Difference Schemes in Calculations of Centered Rarefaction Waves

Abstract

A comparative accuracy analysis of three finite-difference schemes (the first order UpWind (UW), the second order TVD, and the third order in time WENO5 schemes) is carried out when calculating the nonlinear transport equation of the Cauchy problem with piecewise linear discontinuous periodic initial data. We show that in the case of stable initial discontinuities from which a sequence of shocks is formed, the convergence order of all three schemes between the shocks coincides with their formal accuracy. In the case of unstable initial discontinuities, when a sequence of centered rarefaction waves is formed, all three schemes have the first order of convergence within these waves. We obtain an explicit formula for the disbalances of the difference solutions in a centered rarefaction wave. This formula agrees closely with the numerical calculations in the case of high-order accurate schemes, does not depend on the scheme type, and is determined by the error in approximating the initial data in the vicinity of an unstable strong discontinuity.