Advection Solver Performance with Long Time Steps, and Strategies for Fast and Accurate Numerical Implementation

This note addresses the subject of advection around the theme of applying Characteristic Maps as a method of constructing solvers. Typically the accuracy of solvers is characterized as an asymptotic function of a short time step, short usually defined by restricting the value of the CFL parameter. Within the range of the CFL limit the asymptotic function is expressed as a formula in terms of a power of the time step, i.e. O(∆t), for some power p. A key issue is the implementation of solvers that are accurate for “long” time steps. Long time steps might be described as situations that violate CFL restrictions, or at least situations in which it is ambiguous whether the solver can be expected to work based on the CFL restriction. Of course, an advection solver can be used well beyond its CFL limit, with consequent error in the advection. Characterizing the magnitude of that error is of interest. A common approach for computing error is to advect a field over a period of time with some number of steps, then reverse the advection so that, in principle, the field should return to its original form. The error measure is related to the disparity between the original field and the advected one. Defining advection solvers in terms of characteristic maps offers another measure of accuracy. Characteristic maps depend only on the velocity field, and exist independent of the material subject to advection. Given a “reliable” characteristic map, the error of a particular solver could be assessed without invoking any particular material model. This approach is employed in section 7, along with “traditional” evaluation methods. These issues are addressed here in three stages: