Numerical solution of quenching problems using mesh-dependent variable temporal steps

In this paper, we introduce a new adaptive method for computing the numerical solutions of a class of quenching parabolic equations which exhibit a solution with one singularity. Our method systematically generates an irregular mesh with mesh-dependent temporal increments based on the solution behavior from which an implicit finite difference scheme associated with the irregular mesh is constructed. The convergence and stability of the finite difference scheme is analyzed for the solution before quenching. An equivalent linearized model is used to justify the stability of the method near quenching as well. A numerical example is provided to demonstrate the viability of the proposed method.