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

Hermite WENO schemes for Hamilton-Jacobi equations

In this paper, a class of weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving Hamilton-Jacobi equations is presented. The idea of the reconstruction in the HWENO schemes comes from the original WENO schemes, however both the function and its first derivative values are evolved in time and used in the reconstruction, while only the function values are evolved and used in the original WENO schemes. Comparing with the original WENO schemes of Jiang and Peng [Weighted ENO schemes for Hamilton-Jacobi equations, SIAM Journal on Scientific Computing 21 (2000) 2126] for Hamilton-Jacobi equations, one major advantage of HWENO schemes is its compactness in the reconstruction. Extensive numerical experiments are performed to illustrate the capability of the method. (c) 2004 Elsevier Inc. All rights reserved

A comparative study of the efficiency of jet schemes

We present two versions of third order accurate jet schemes, which achieve high order accuracy by tracking derivative information of the solution along characteristic curves. For a benchmark linear advection problem, the efficiency of jet schemes is compared with WENO and Discontinuous Galerkin methods of the same order. Moreover, the performance of various schemes in tracking solution contours is investigated. It is demonstrated that jet schemes possess the simplicity and speed of WENO schemes, while showing several of the advantages as well as the accuracy of DG methods.Comment: 12 pages, 6 figures, presented at the conference Mathematical Modeling and Applications to Industrial Problems 201

Level Set Jet Schemes for Stiff Advection Equations: The SemiJet Method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

High-Order Methods for Wave Phenomena

This dissertation describes high-order numerical methods for wave phenomena and is divided into three main sections. The first and second sections concern extensions of a class of high-order methods known as Hermite methods to simulate wave propagation problems and Hamilton-Jacobi equations. The first section extends the capabilities of Hermite methods for the wave equation. Hermite methods have several attractive features; however, they were only applicable to Cartesian geometry. In this section we generalize the Hermite scheme for the wave equation to curvilinear geometry. We also extend the solver to handle a wave propagation problem on a discontinuous medium while maintaining high-order accuracy. The second section discusses a novel numerical method for Hamilton-Jacobi equations. Time-dependent Hamilton-Jacobi equations appear in many applications, e.g., optimal control, differential games, image processing and the calculus of variations. The Hamiltonian generally depends on the gradient of the solution, which gives rise to discontinuities in the derivative of the solution. Once the derivative fails to be continuous the solution is no longer unique; however, there is a physically relevant weak solution known as the viscosity solution. We develop a solver based on Hermite methods that converges to the viscosity solution as the grid is refined even in the presence of kinks, and maintains high-order accuracy in smooth regions. In the last section we generalize the WaveHoltz iteration to elastic media. We look for time-harmonic solutions in the time-domain, as opposed to solving directly in the frequency domain. The result is a fixed-point iteration for solving the elastic Helmholtz equation by instead solving a sequence of elastic wave equations. This is simple to implement, inherits the memory-leanness and scalability of the underlying wave equation discretization.</p