Abstract
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.
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Lafitte, P., Lejon, A., Melis, W., Roose, D., Samaey, G. (2015). High-Order Asymptotic-Preserving Projective Integration Schemes for Kinetic Equations. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_38
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DOI: https://doi.org/10.1007/978-3-319-10705-9_38
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