Highly efficient, oscillation free solution of the transport equation over long times and large spaces

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Abstract
Transport problems with sharp transitions or high convection often stress even the most advanced computational methods beyond feasibility limits. One promising line of attack involves transforming problems to a moving, deforming coordinate system (MDCS). Using MDCS as recently developed, one may concentrate computational attention when and where it is most appropriate, also possibly greatly reducing the effective magnitude of terms which cause trouble. In principle, MDCS may be used with finite difference, finite element, or other numerical methods, according to preference. In the particular finite element system advanced here, mesh movement may be variable in time and space, so that physically fixed boundaries may remain fixed in the transformed coordinate system. A simple finite difference system is used in time. When the method is applied to the standard transport equation test case, great efficiency and accuracy are obtained in early time. Even very steep fronts may be represented well, with no oscillations, using space and time step sizes well beyond conventional (Peclet, Courant number) constraints. More important problems are solved, involving transport over much longer times and larger spaces, when diffusion has some chance to operate and hence a parabolic model is more warranted. The same accuracy and stability are achieved as during early time, using no more nodes and time steps overall than during early time runs. This constitutes orders of magnitude savings in effort over what the application of conventional constraints would require. Relative efficiency from application of MDCS to other problems would depend on the particulars of each case.