ENOC-2008, Saint Petersburg, Russia, June, 30–July, 4 2008 IMPROVEMENT OF NUMERICAL DESCRIPTION OF NON-LINEAR SHOCK PROFILES BY USE OF ANALYTICAL SOLUTIONS OF DIFFERENTIAL APPROXIMATIONS Alexey Porubov Institute of Problems in Mechanical Engineering, Russia porubov.math@mail.ioffe.ru Daniel Bouche CMLA, ENS de Cachan, Cachan, France daniel.bouche@cea.fr Guy Bonnaud CEA, INSTN, Centre de Saclay France guy.bonnaud@cea.fr Abstract 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 the 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 de- viations by suitable choice of the parameters of the 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 the influence of the scheme dis- persion and/or dissipation. Application of the analyti- cal solutions is demonstrated using the Lax-Wendroff scheme. Key words Scheme dispersion, non-linear wave, analytical and numerical solutions 1 Introduction A discrete model described by a scheme often pos- sesses internal dispersive and/or dissipative properties caused by a method of discretization. It gives rise to non-physical deviations in the numerical solution. One can decrease the influence of these bad factors by vary- ing time and space steps and modifying the method of approximation. A possibility to know how to do it is the application of the method of differential approximation (DA) [Lerat and Peyret, 1975; Shokin, 1983; Mukhin et al, 1983; Fletcher, 1991]. This method allows us to study dispersive and dissipative features of a scheme by an analysis of the differential equation called a differen- tial approximation (DA). It is obtained using a substitu- tion of the Taylor expansions of the discrete functions into a difference scheme. An analysis of the resulting partial differential equation (PDE) is possible if the ex- pansion is truncated at some order. However, the DA for a dicretization of a non-linear equation is also non- linear and nonitegrable equation. Then, only particular analytical travelling wave solutions existing at specific initial conditions may be obtained. A natural question arises: may the particular asymptotic and exact solu- tions be used to analyze the features of the DA, thus the features of difference schemes? In this paper, we demonstrate the efficiency of the use of analytical solutions for understanding the deviations in the shock caused by the scheme features on an ex- ample of the non-linear Burgers’ equation, v t +(v 2 ) x − bv xx =0. (1) In general, its DA may be written as [Lerat and Peyret, 1975; Shokin, 1983; Engelberg, 1999] u t +(u 2 ) x −bu xx = −s(u) u xxx +α(u, u x )−q(u) u xxxx , (2) In particular, we obtain for the Lax-Wendroff (LW) scheme [Lerat and Peyret, 1975; Shokin, 1983; Fletcher, 1991] u t +(u 2 ) x − bu xx = △t 2 24 (u 4 ) xxx − △x 2 12 (u 2 ) xxx , (3) One can try to find an exact travelling wave solution of Eq.(3). Another way is to consider the r.h.s. of it as a small perturbation and find an asymptotic solution. In this last case it was suggested in [Mukhin et al, 1983] to linearize the r.h.s. around a constant, say, the value of u -∞ = u(x → −∞). Then Eq.(3) is simplified towards the linearly perturbed Burgers equation, u t + uu x − bu xx = − ( △x 2 −△t 2 u 2 -∞ ) u -∞ 6 u xxx , (4)