Search

In this paper we show that the use of spectral Galerkin methods for the approximation of the Boltzmann equation in the velocity space permits one to obtain spectrally accurate numerical solutions at a reduced computational cost. We prove that the spectral algorithm preserves the total mass and approximates with infinite-order accuracy momentum and energy. Consistency of the method is also proved, and a stability result for a smoothed positive scheme is given. We demonstrate that the Fourier coefficients associated with the collision kernel of the equation have a very simple structure and in some cases can be computed explicitly. Numerical examples for homogeneous test problems in two and three dimensions confirm the advantages of the method. [ABSTRACT FROM AUTHOR]

Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a small relaxation limit governed by reduced systems of a parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve, and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which are thus able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type. [ABSTRACT FROM AUTHOR]