On stabilization of an explicit difference scheme for a nonlinear parabolic equation

We developed a technique for the numerical solution of a nonlinear equation describing the diffusion transfer of radiation energy. The method is based on the introduction of the second time derivative with a small parameter into the parabolic equation and an explicit difference scheme. The explicit approximation of the original equation makes it possible to implement an algorithm that is effectively adapted to the architecture of high-performance computing systems. The new scheme provides a second-order resolution of the nonlinearity in time with an acceptable time step. A heuristic algorithm for choosing the parameters of a three-level difference scheme is proposed. Perspective applications of the method are problems in astrophysics, for example, the simulation of a strongly radiating shock wave breakout at the surface of a star at the stage of its evolution known as a supernova explosion.