Theoretical results on accuracy of numerical schemes for differential equations are built on specific assumptions about the level of solution smoothness/regularity. But if a physically relevant solution is singular, this can severely degrade the convergence rates of “standard” numerical methods. When the exact location & type of singularity are known in advance, we can use the “factoring” techniques to circumvent this difficulty. The idea is to rewrite the original solution as a product (or a sum) of two functions: the first is chosen to have the exact right type of singularity at that location; the second is (at least locally) smooth but unknown and we recover it by solving a modified equation.

We will illustrate this idea for ODE initial value problems and Eikonal PDEs with a point source. In the latter case, the “rarefaction fan” of characteristics yields a localized blow-up in second derivatives of the solution and decreases the rate of convergence even for simple (first-order upwind) discretizations.

However, rarefaction fans can also result from general (inhomogeneous) boundary conditions or discontinuities in coefficients of the equation. This talk will present a method for “dynamic factoring” in 2-dimensional Eikonal problems. The goal is to treat rarefaction fans as they are discovered in the process of solving the PDE on the grid.

Joint work with Dongping Qi (SJTU-Cornell).