Second Order Fully Semi-Lagrangian Discretizations of Advection-Diffusion-Reaction Systems

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.

A high-order Lagrange-Galerkin scheme for a class of Fokker-Planck equations and applications to mean field games

“…Using this formulation, a second-order accurate semi-Lagragian scheme can be derived to approximate m * (see e.g. [6]). However, such a scheme is not conservative, i.e.…”

“…We now describe a variation of the scheme in [6] to deal with the nonlinearity of the Hamiltonian in (HJB) with respect to ∇v (see also [38,45] for related constructions). For a given µ ∈ C([0, T ];…”

“…We choose [0, T ] × O ∆ = [0, 0.25]×(−2, 2) d , with Dirichlet boundary conditions on ∂O ∆ , the latter being equal to the exact solution of (4.7) for the HJB equation and homogeneous for the FP equation. The numerical approximation of the boundary conditions for the HJB equation is based on the technique proposed in [6], while for the FP equation we proceed as in the previous test. In this and in the following test, to compute (3.7) we have used a fourth-order finite difference approximation of the gradient of v ∆ [m], and we have not introduced the mollifier φ ε .…”

“…More precisely, the characteristic curves are discretized with a Crank-Nicolson method (see e.g. [28,39]), as in high-order SL schemes for parabolic equations (see [6]), and we discretize in the space variable by using a LG scheme with a symmetric Lagrangian basis of odd order.…”

“…Let us mention [46] and [35], where the authors propose finite difference based second-order accurate methods to approximate MFG systems. In our approach, we first introduce a high-order accurate SL method for the HJB equation, which can be viewed as a nonlinear extension of the scheme considered in [6], and we couple it with our LG scheme for the FP equation. Because of this coupling, the resulting approximation is not explicit and it is solved by fixed point iterations.…”