On accuracy of finite-difference schemes in calculations of centered rarefaction waves

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