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A Method of Projection of Delta Waves in a Godunov Scheme and Application to Pressureless Fluid Dynamics

Some systems of PDEs, such as the one of pressureless fluid dynamics, show delta waves in the solution of the Riemann problem. A method of projection of these delta waves in a Godunov scheme is proposed. It provides a modification of the original Le Roux et al. scheme in case of changes in sign of velocity [R. Baraille, G. Bourdin, F. Dubois, and A. Y. Le Roux, C. R. Acad. Sci. Paris, 314 (1992), pp. 147-152]. Stability and convergence of the scheme are proved in one space dimension for the system of pressureless fluids. As an application, this method has been extended to classical systems of fluid dynamics used for the numerical simulation of large structure formation in cosmology, in presence of expansion of the background. This method of projection of delta waves can also be applied to systems of conservation laws that have delta waves in the solution to the Riemann problem.