A New Discontinuous Galerkin Finite Element Method for Directly Solving the Hamilton-Jacobi Equations

In this paper, we improve upon the discontinuous Galerkin (DG) method for Hamilton-Jacobi (HJ) equation with convex Hamiltonians in (Y. Cheng and C.-W. Shu, J. Comput. Phys. 223:398-415,2007) and develop a new DG method for directly solving the general HJ equations. The new method avoids the reconstruction of the solution across elements by utilizing the Roe speed at the cell interface. Besides, we propose an entropy fix by adding penalty terms proportional to the jump of the normal derivative of the numerical solution. The particular form of the entropy fix was inspired by the Harten and Hyman's entropy fix (A. Harten and J. M. Hyman. J. Comput. Phys. 50(2):235-269, 1983) for Roe scheme for the conservation laws. The resulting scheme is compact, simple to implement even on unstructured meshes, and is demonstrated to work for nonconvex Hamiltonians. Benchmark numerical experiments in one dimension and two dimensions are provided to validate the performance of the method

Arc length based WENO scheme for Hamilton-Jacobi Equations

In this article, novel smoothness indicators are presented for calculating the nonlinear weights of weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and gives a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.Comment: 14 pages, 9 figure

High-order computational scheme for a dynamic continuum model for bi-directional pedestrian flows

In this article, we present a high-order weighted essentially non-oscillatory (WENO) scheme, coupled with a high-order fast sweeping method, for solving a dynamic continuum model for bi-directional pedestrian flows. We first review the dynamic continuum model for bi-directional pedestrian flows. This model is composed of a coupled system of a conservation law and an Eikonal equation. Then we present the first-order Lax-Friedrichs difference scheme with first-order Euler forward time discretization, the third-order WENO scheme with third-order total variation diminishing (TVD) Runge-Kutta time discretization, and the fast sweeping method, and demonstrate how to apply them to the model under study. We present a comparison of the numerical results of the model from the first-order and high-order methods, and conclude that the high-order method is more efficient than the first-order one, and they both converge to the same solution of the physical model. © 2010 Computer-Aided Civil and Infrastructure Engineering.postprin

Anisotropic Fast-Marching on cartesian grids using Lattice Basis Reduction

We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We prove the consistence of the algorithm, and illustrate its efficiency by numerical experiments. The algorithm relies on the computation at each grid point of a special system of coordinates: a reduced basis of the cartesian grid, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.Comment: 28 pages, 12 figure

Some non monotone schemes for Hamilton-Jacobi-Bellman equations

We extend the theory of Barles Jakobsen to develop numerical schemes for Hamilton Jacobi Bellman equations. We show that the monotonicity of the schemes can be relaxed still leading to the convergence to the viscosity solution of the equation. We give some examples of such numerical schemes and show that the bounds obtained by the framework developed are not tight. At last we test some numerical schemes.Comment: 24 page