Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions

Abstract

In this paper, we apply positivity-preserving local discontinuous Galerkin (LDG) methods to solve parabolic equations with blow-up solutions. This model is commonly used in combustion problems. However, previous numerical methods are mainly based on a second order finite difference method. This is because the positivity-preserving property can hardly be satisfied for high-order ones, leading to incorrect blow-up time and blow-up sets. Recently, we have applied discontinuous Galerkin methods to linear hyperbolic equations involving δ-singularities and obtained good approximations. For nonlinear problems, some special limiters are constructed to capture the singularities precisely. We will continue this approach and study parabolic equations with blow-up solutions. We will construct special limiters to keep the positivity of the numerical approximations. Due to the Dirichlet boundary conditions, we have to modify the numerical fluxes and the limiters used in the schemes. Numerical experiments demonstrate that our schemes can capture the blow-up sets, and high-order approximations yield better numerical blow-up time.