Numerical Solutions to Singular Reaction-Diffusion Equation over   Elliptical Domains
  Solid fuel ignition models, for which the dynamics of the temperature is independent of the single-species mass fraction, attempt to follow the dynamics of an explosive event. Such models may take the form of singular, degenerate, reaction-diffusion equations of the quenching type, that is, the temporal derivative blows up in finite time while the solution remains bounded. Theoretical and numerical investigations have proved difficult for even the simplest of geometries and mathematical degeneracies. Quenching domains are known to exist for piecewise smooth boundaries, but often lack theoretical estimates. Rectangular geometries have been primarily studied. Here, this acquired knowledge is utilized to determine new theoretical estimates to quenching domains for arbitrary piecewise, smooth, connected geometries. Elliptical domains are of primary interest and a Peaceman-Rachford splitting algorithm is then developed that employs temporal adaptation and nonuniform grids. Rigorous numerical analysis ensures numerical solution monotonicity, positivity, and linear stability of the proposed algorithm. Simulation experiments are provided to illustrate the accomplishments. 
Matthew Alan Beauregard
math.NA