A stable and convergent finite-difference model which conserves the positivity and the dissipativity of Gibbs’ free energy for a nonlinear combustion equation

Abstract

This manuscript establishes the theoretical numerical features of a finite-difference scheme that conserves the dissipation of Gibbs’ free energy for a nonlinear multidimensional combustion equation. Our system considers fractional derivatives of the Caputo class in the temporal variable, along with Riesz fractional operators in the spatial coordinates. This system dissipates the Gibbs’ free energy, and some discrete paradigms have been reported already in the literature to preserve this feature of the solutions in the energy domain. In the present work, a discrete model for solving such system is considered. We demonstrate mathematically the solubility of the discrete system using suitable constraints on the computational parameters and establish the consistency. Additionally, we theoretically demonstrate the convergence and stability of the computational method. Computer simulations confirm the veracity of the theoretical properties proved mathematically in this manuscript.