12 research outputs found
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
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
A discontinuous Galerkin solver for front propagation
International audienceWe propose a new discontinuous Galerkin (DG) method based on [Cheng and Shu, JCP, 2007] to solve a class of Hamilton-Jacobi equations that arises from optimal control problems. These equations are connected to front propagation problems or minimal time problems with non isotropic dynamics. Several numerical experiments show the relevance of our method, in particular for front propagation
High-resolution alternating evolution schemes for hyperbolic conservation laws and Hamilton-Jacobi equations
The novel approximation system introduced by Liu is an accurate approximation to systems of hyperbolic conservation laws. We develop a class of global and local alternating evolution (AE) schemes for one- and two-dimensional hyperbolic conservation law and one-dimensional Hamilton-Jacobi equations, where we take advantage of the high accuracy of the AE approximation. The nature of solutions having singularities, which is generic to these equations in handled using the AE methodology. The numerical scheme is constructed from the AE system by sampling over alternating computational grid points. Higher order accuracy is achieved by a combination of high-order polynomial reconstruction and a stable Runge-Kutta discretization in time. Local AE schemes are made possible by letting the scale parameter [epsilon] reflect the local distribution of nonlinear waves. The AE schemes have the advantage of easier formulation and implementation, and efficient computation of the solution. Theoretical numerical stability is proved mainly for the first and second order schemes of hyperbolic conservation law and Hamilton-Jacobi equations. In the case of hyperbolic conservation law, we have also shown that the numerical solutions converge to the weak solution. The designed methods have the advantage of being Riemann solver free, and the performs comparably to the finite volume/difference methods currently used. A series of numerical tests illustrates the capacity and accuracy of our method in describing the solutions