solving the IBVP. The first, as used in [@Rose-1983; @Zhang-2009], is to solve PME in a large domain containing the free boundary for the whole time period of the simulation. With this approach, there is no need to explicitly treat the free boundary, which makes the approach more amenable to problems with complex solution supports. The main disadvantage of this approach is that the solution has a corner shape between the regions of zero and nonzero solution values and thus its regularity on the whole domain is at most $H^1$. An $H^1$ regularity often means at best a first-order convergence in the numerical solution as the mesh is refined.

The second approach is to solve the problem only in the region of compact support; e.g., see [@BHJ05a; @Duque2014; @Duque2015]. One of the advantages of this approach is that a smaller spatial domain is used and thus fewer mesh points can be used to achieve the same computational accuracy. Moreover, the regularity of the solution is better on the support than on a larger domain since it does not have a corner shape. As a result, the numerical solution can have a higher convergence order than that with the first approach. 