Description of numerical shock profiles of non-linear Burgers’ equation by asymptotic solution of its differential approximations

Full Title:

Description of numerical shock profiles of non-linear Burgers’ equation by asymptotic solution of its differential approximations

Abstract:

An analysis of dispersive/dissipative features of the difference schemes used for simulations of the non-linear Burgers’ equation is developed based on the travelling wave asymptotic solutions of its differential approximation. It is shown that these particular solutions describe well deviations in the shock profile even outside the formal applicability of the asymptotic expansions, namely for shocks of moderate amplitudes. Analytical predictions may be used to improve calculations by suitable choice of the parameters of some familiar schemes, i.e., the Lax-Wendroff, Mac-Cormack etc. Moreover, an improvement of the scheme may be developed by adding artificial terms according to the asymptotic solution.