Steady discrete shocks of 5th and 7th-order RBC schemes and shock profiles of their equivalent differential equations

Abstract

An exact expression of steady discrete shocks was recently obtained by the author in [9] for a class of residual-based compact schemes (RBC) applied to the inviscid Bürgers equation in a finite domain. Following the same lines, the analysis is extended to an infinite domain for a scalar conservation law with a general convex flux. For the dissipative high-order schemes considered, discrete shocks in infinite domain or with boundary conditions at short distance (Rankine–Hugoniot relations) are found to be very close. Besides, the present analytical description of shock capturing in infinite domain is explicit and so simple that it could lead to a new approach for correcting parasitic oscillations of high order RBC schemes. In a second part of the paper, exact solutions are also derived for equivalent differential equations (EDE) approximating RBC2p−1 schemes (subscript denotes the accuracy order) at orders 2p and 2p+1. Although EDE involves Taylor expansions around steep structures, agreement between the exact EDE shock-profiles and the discrete shocks is remarkably good for RBC5 and RBC7 schemes. In addition, a strong similarity is demonstrated between the analytical expressions of discrete shocks and EDE shock profiles.