Stabilization of coupled convection–diffusion-reaction equations for continuum dislocation transport

The plasticity of crystalline materials can be described at the meso-scale by dislocations transport models, typically formulated in terms of dislocation densities. This leads to sets of coupled nonlinear partial differential equations involving diffusive and convective transport mechanisms. Since exact solutions for these systems are not available, numerical approximations are needed to efficiently solve them. The properties of these systems of equations cause most traditional numerical methods to fail, even for the case of a single equation. For systems of equations the problem is even more challenging due to the lack of fundamental principles guiding numerical discretization strategies. Special strategies must be developed and carefully applied to obtain physically meaningful and numerically stable approximations. The objective of this paper is to construct a coefficient perturbation-based stabilization technique for general systems of equations and to apply it to the modelling of one-dimensional dislocation transport. A detailed numerical study is carried out in order to demonstrate its ability to render well-behaved and physically admissible numerical approximations. (paper)