Abstract
We propose a second order, fully semi-Lagrangian method for the numerical solution of systems of advection-diffusion-reaction equations, which is based on a semi-Lagrangian approach to approximate in time both the advective and the diffusive terms. The proposed method allows to use large time steps, while avoiding the solution of large linear systems, which would be required by implicit time discretization techniques. Standard interpolation procedures are used for the space discretization on structured and unstructured meshes. A novel extrapolation technique is proposed to enforce second-order accurate Dirichlet boundary conditions. We include a theoretical analysis of the scheme, along with numerical experiments which demonstrate the effectiveness of the proposed approach and its superior efficiency with respect to more conventional explicit and implicit time discretizations.
Highlights
Systems of advection--diffusion--reaction (ADR) equations model the chemical or biochemical processes involving several species transported by a fluid
In evaluating the accuracy of this technique, we should split the error in two components— one associated to internal nodes, which has already been analysed in the previous section, and one related the the treatment of boundary conditions (BCs), which comes into play only in the interval [x0, x0 + ex ]
We start with a simple heat equation, and we level of complexity considering an advection–diffusion equation, a reaction– diffusion equation, an advection-diffusion-reaction system and an advection–diffusion equation on an non-convex domain
Summary
Systems of advection--diffusion--reaction (ADR) equations model the chemical or biochemical processes involving several species transported by a fluid. Even minor efficiency gains in the solution of this very classical problem are of paramount practical importance This explains why numerical methods that allow the use of large time steps are favoured for these applications, see e.g. the discussion in [44]. The standard ways to enhance efficiency for the solution of advection step are either the use of implicit schemes or the application of semi-Lagrangian (SL) techniques, [18,39]. These are coupled to implicit methods for the diffusion and reaction step.
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