Quenching study of two-dimensional fractional reaction–diffusion equation from combustion process

The quenching phenomenon and its physical characters are critical issues in the study of combustion process. In this paper, a two-dimensional temporal fractional combustion model with Caputo derivative is considered. We design an ingenious adaptive finite difference discontinuous Galerkin method to solve this fractional combustion model. To catch quenching moment more accurately, an explicit bisection adaptive strategy is proposed for discretization of fractional derivative. Moreover, discontinuous Galerkin method is employed for spatial discretization to settle large gradient changes and non-smoothness property of solution in adjacent area of quenching moment. The physical properties of combustion process, such as positivity, monotonicity and stability, are theoretically or numerically investigated. Finally, we carry out systematic numerical experiments for such abstract combustor with different shapes and various parameter settings, which illustrate that the dynamics of such model depends on the intensity of initial inputs, the order of fractional derivative and the area of spatial domain.