A simulation expressivity of the quenching phenomenon in a two-sided space-fractional diffusion equation

Abstract

The aims of this paper are to investigate and propose a numerical approximation for a quenching type diffusion problem associated with a two-sided Riemann-Liouville space-fractional derivative. The approach adopts weighted Grünwald formulas for suitable spatial discretization. An implicit Crank-Nicolson scheme combined with adaptive technology is then implemented for a temporal integration. Monotonicity, positivity preservation and linearized stability are proved under suitable constraints on spatial and temporal discretization parameters. Two specially designed simulation experiments are presented for illustrating and outreaching properties of the numerical method constructed. Connections between the two-sided fractional differential operator and critical values including quenching time, critical length and quenching location are investigated.