A Stable and Conservative Hybrid Scheme for the Frank-Kamenetskii Equation

We consider the Frank-Kamenetskii partial differential equation as a model for combustion in Cartesian, cylindrical and spherical geometries. Due to the presence of a singularity in the equation stemming from the Laplacian operator, we consider a specific conservative continuous formulation thereof, which allows for a discrete energy estimate. Furthermore, we consider multiple methodologies across multiple domains. On the left domain, close to the singularity, we employ the Galerkin method which allows us to integrate over time appropriately, and on the right domain we implement the finite difference method. We also derive a condition at the singularity that removes a potentially artificial boundary layer. The summation-by-parts (SBP) methodology assists us in coupling these two numerical schemes at the interface, so that we end up with a provably stable and conservative hybrid numerical scheme. We provide numerical support for the theoretical derivations and apply the procedure to a realistic case.