O(N) unconditionally stable methods through kernel based Successive Convolution

January 14, 2022
Rothe’s method, transvers method of lines and the Method of Lines Transpose are all approaches for solving time dependent PDEs that approach the problem by turning the PDE into a boundary value problem and then addressing the resulting BVP with an efficient spatial discretization. Traditionally, this approach was coupled with an implicit formulation of a time marching method as the very first step in setting up the numerical integrator. In the Method of Lines Transpose approach, the resulting BVP has been addressed using kernel based methods and efficiently evaluated using fast kernel tricks. In this talk, we discuss a refactoring of these kernel based methods into a form where differential operators of any PDE can be express through Successive Convolution independent of the time integration strategy. The method leads to a formulation that makes explicit time stepping methods provably unconditionally stable for linear problems and behaves unconditionally stable for non-linear problems. The method is O(N) and relies on WENO integration to address problems with discontinuities. Boundary conditions pose an interesting challenge for the method, and will be discussed during this talk. We demonstrate the method by a applying it to the Hamilton Jacobi and Degenerate Advection Diffusion equations in 1D and 2D. This is joint work with my student, Mr. Bill Sands, and three former post docs, Drs. Hyoseon Yang, Yan Jiang and Wei Gou.
Supplementary Materials