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

摘要

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 highorder 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 RBC_(2p-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. E-mail address: alain.lerat@ensam.eu.