SIMULATION OF DISCONTINUOUS SOLUTIONS IN EVOLUTIONARY EQUATIONS. NEW APPROACHES FOR REDUCING NON-PHYSICAL OSCILLATIONS AND SMOOTHING IN NUMERICAL SOLUTIONS

摘要

In this paper some theoretical considerations for understanding the errors in numerical computations are proposed. It is strictly considered for some cases as extra smoothing of fronts as the origin of artificial oscillations in the solutions. It is confirmed that the smoothing is originated by dissipation in schemes and oscillations by dispersion of schemes. Some new methods of improving numerical solutions of evolution equations are proposed on the base of theoretical considerations. In the case of linear equations proposed tools can increase the order of the accuracy. The artificial viscosity and artificial dispersion for difference schemes of gas dynamics are proposed as the first examples. A new class of tools for improving numerical solutions is proposed - 'Langoliers'. 'Langoliers' are special difference operators which should be applied at each time steps after the running of original difference schemes. The design of 'Langoliers' allows to reduce the dissipative and dispersive errors of schemes. The examples are anti-diffusion. anti-dispersion and specially constructed difference schemes. Different illustrative examples of such tools are considered for gas dynamics equations and for wave equation.