A positivity preserving variational method for multi-dimensional convection–diffusion–reaction equation

Abstract

A new positivity preserving variational (PPV) procedure is proposed to solve the convection–diffusion–reaction (CDR) equation. Through the generalization of stabilized finite element methods, the present variational procedure offers minimal phase and amplitude errors for different regimes associated with convection, diffusion and reaction effects. By means of Fourier analysis, we first review the shortcomings of the Galerkin/Least-Squares (GLS) and the Subgrid Scale (SGS) methods during the change in sign of the reaction coefficient that motivates us for the present linear stabilization as a combined GLS-SGS methodology. Discrete upwind operator with a solution-dependent nonlinear term is then introduced in high gradient regions, which enables the positivity preserving property in the variational formulation. Direct extension to multi-dimensions is carried out by considering the principle streamline and crosswind directions. The efficacy of the method is demonstrated by the systematic accuracy and stability analyses in one- and two-dimensions. Results show the reduction of oscillations in the solution in one- and two-dimensional cases and a remarkable reduction in the phase error is observed for the cases with negative reaction coefficient. The proposed formulation provides a superior solution in the reaction-dominated as well as the convection-dominated regimes due to the minimization of spurious oscillations and accurate capturing of the high gradient regions.