High-Order Asymptotic-Preserving Projective Integration Schemes for Kinetic Equations
We study a projective integration scheme for a kinetic equation in both the diffusive and hydrodynamic scaling, on which a limiting diffusion or advection equation exists. The scheme first takes a few small steps with a simple, explicit method, such as a spatial centered flux/forward Euler time integration, and subsequently projects the results forward in time over a large, macroscopic time step. With an appropriate choice of the inner step size, the time-step restriction on the outer time step is similar to the stability condition for the limiting equation, whereas the required number of inner steps does not depend on the small-scale parameter. The presented method is asymptotic-preserving, in the sense that the method converges to a standard finite volume scheme for the limiting equation in the limit of vanishing small parameter. We show how to obtain arbitrary-order, general, explicit schemes for kinetic equations as well as for systems of nonlinear hyperbolic conservation laws, and provide numerical results.
KeywordsKinetic Equation Projective Integration Collision Kernel Asymptotic Preserve Semiconductor Equation
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- 8.A. Giuseppe, A. Anile, Moment equations for charged particles: global existence results, in Modeling and Simulation in Science, Engineering and Technology (Birkhäuser, Boston, 2004), pp. 59–80Google Scholar
- 10.P. Lafitte, A. Lejon, G. Samaey, A higher order asymptotic-preserving integration scheme for kinetic equations using projective integration, pp. 1–25. http://arxiv.org/abs/1404.6104
- 11.P. Lafitte, W. Melis, G. Samaey, A relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws. (submitted)Google Scholar