IMPROVEMENT OF NUMERICAL DESCRIPTION OF NON-LINEAR SHOCK PROFILES BY USE OF ANALYTICAL SOLUTIONS OF
DIFFERENTIAL APPROXIMATIONS
Alexey Porubov, Daniel Bouche, Guy Bonnaud
An analysis of dispersive/dissipative features of the difference
schemes is developed based on particular asymptotic and exact
travelling wave solutions of the differential approximation (DA)
of the equation under study. It is shown on an example of the
non-linear Burgers' equation, that its asymptotic travelling wave
solution allows us to describe deviations in the shock wave caused
by a scheme dispersion/dissipation. These analytical predictions
may be used to diminish bad deviations by suitable choice of the
parameters of a scheme. Then it is shown, that exact travelling
wave solution of the DA for the non-linear Burgers' equation helps
us to suggest artificial non-linear additions to the schemes to
suppress an influence of the scheme dispersion and/or dissipation.
Application of the analytical solutions is demonstrated using some
familiar schemes, e.g., the Lax-Wendroff scheme.