Asymptotic-Preserving Discretization Schemes
Saturday, May 15, 2004 - 9:30am - 10:20am
Jim Morel (Los Alamos National Laboratory)
Asymptotic limits associated with partial differential equations are limits in which certain nondimensional parameters in an equation are made small relative to other nondimensional parameters. The asymptotic solution is generally found to satisfy an equation that is much simpler than the original full equation. When the scale lengths associated with the asymptotic solution are much larger than the smallest scalelengths associated with the full equation, it becomes essential from a numerical point of view to use a discretization scheme for the full equation that preserves the asymptotic limit. An asymptoticpreserving scheme is one that yields accurate asymptotic solutions whenever the scale lengths associated with the asymptotic solution are resolved by the mesh. If a scheme is not asymptotic-preserving, accurate asymptotic solutions will be obtained only if the smallest scale lengths associated with the full equation are resolved by the mesh. Because asymptotic scale lengths can be arbitrarily larger than the smallest scale lengths of the full equation, this requirement can make asymptotic calculations prohibitively expensive for discretization schemes that are not asymptotic-preserving. We discuss spatial discretization schemes for the radiation transport equation in the asymptotic diffusion limit. The smallest spatial scale lengths associated with the transport equation are on the order of a mean-free-path (the mean-distance between particle interactions). A truncation error analysis for any consistent transport spatial discretization scheme will indicate that convergence to a smooth solution is guaranteed whenever the spatial cell widths measured in mean-free-paths go to zero. However, the scale length associated with the diffusion limit can be arbitrarily large with respect to a mean-free-path. Thus it can be essential to use asymptotic-preserving discretization schemes in highly diffusive calculations. We show that the simple upwind scheme does not preserve the asymptotic diffusion limit, while a discontinuous Galerkin scheme with a linear trial space does preserve this limit. Both theory and computational examples are presented.