4 research outputs found
A high-order scheme for mean field games
In this paper we propose a high-order numerical scheme for time-dependent
mean field games systems. The scheme, which is built by combining
Lagrange-Galerkin and semi-Lagrangian techniques, is consistent and stable for
large time steps compared with the space steps. We provide a convergence
analysis for the exactly integrated Lagrange-Galerkin scheme applied to the
Fokker-Planck equation, and we propose an implementable version with inexact
integration. Finally, we validate the convergence rate of the proposed scheme
through the numerical approximation of two mean field games systems
Error Estimates for First- and Second-Order Lagrange-Galerkin Moving Mesh Schemes for the One-Dimensional Convection-Diffusion Equation
A new moving mesh scheme based on the Lagrange-Galerkin method for the
approximation of the one-dimensional convection-diffusion equation is studied.
The mesh movement, which is prescribed by a discretized dynamical system for
the nodal points, follows the direction of convection. It is shown that under a
restriction of the time increment the mesh movement cannot lead to an overlap
of the elements and therefore an invalid mesh. For the linear element, optimal
error estimates in the norm are proved in
case of both, a first-order backward Euler method and a second-order two-step
method in time. These results are based on new estimates of the time dependent
interpolation operator derived in this work. Preservation of the total mass is
verified for both choices of the time discretization. Numerical experiments are
presented that confirm the error estimates and demonstrate that the proposed
moving mesh scheme can circumvent limitations that the Lagrange-Galerkin method
on a fixed mesh exhibits.Comment: 27 pages, 3 figures, 4 table
A Semi-Lagrangian Particle Level Set Finite Element Method for Interface Problems
We present a quasi-monotone semi-Lagrangian particle level set (QMSL-PLS) method for moving interfaces. The QMSL method is a blend of first order monotone and second order semi-Lagrangian methods. The QMSL-PLS method is easy to implement, efficient, and well adapted for unstructured, either simplicial or hexahedral, meshes. We prove that it is unconditionally stable in the maximum discrete norm, � · �h,∞, and the error analysis shows that when the level set solution u(t) is in the Sobolev space Wr+1,∞(D), r ≥ 0, the convergence in the maximum norm is of the form (KT/Δt)min(1,Δt � v �h,∞ /h)((1 − α)hp + hq), p = min(2, r + 1), and q = min(3, r + 1),where v is a velocity. This means that at high CFL numbers, that is, when Δt > h, the error is
O( (1−α)hp+hq) Δt ), whereas at CFL numbers less than 1, the error is O((1 − α)hp−1 + hq−1)). We have tested our method with satisfactory results in benchmark problems such as the Zalesak’s slotted disk, the single vortex flow, and the rising bubble