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The original fast sweeping method, which is an efficient iterative method for stationary Hamilton–Jacobi equations, relies on natural ordering provided by a rectangular mesh. We propose novel ordering strategies so that the fast sweeping method can be extended efficiently and easily to any unstructured mesh. To that end we introduce multiple reference(More)
We construct high order fast sweeping numerical methods for computing viscosity solutions of static Hamilton-Jacobi equations on rectangular grids. These methods combine high order weighted essentially non-oscillatory (WENO) approximation to derivatives, monotone numerical Hamiltonians and Gauss Seidel iterations with alternating-direction sweepings. Based(More)
We propose Gaussian-beam based Eulerian methods to compute semi-classical solutions of the Schrödinger equation. Traditional Gaussian beam type methods for the Schrödinger equation are based on the Lagrangian ray tracing. We develop a new Eulerian framework which uses global Cartesian coordinates, level-set based implicit representation and Liouville(More)
The geometric optics approximation to high frequency anisotropic wave propagation reduces the anisotropic wave equation to a static Hamilton–Jacobi equation. This equation is known as the anisotropic eikonal equation and has three different coupled wave modes as solutions. We introduce here a level set-based Eulerian approach that captures all three of(More)
We introduce a new multiscale Gaussian beam method for the numerical solution of the wave equation with smooth variable coefficients. The first computational question addressed in this paper is how to generate a Gaussian beam representation from general initial conditions for the wave equation. We propose fast multiscale Gaussian wavepacket transforms and(More)
In the high frequency regime, the geometrical-optics approximation for the Helmholtz equation with a point source results in an eikonal equation for traveltime and a transport equation for amplitude. Because the point-source traveltime field has an upwind singularity at the source point, all formally high-order finite-difference eikonal solvers exhibit(More)
The viscosity solution of static Hamilton-Jacobi equations with a point-source condition has an upwind singularity at the source, which makes all formally high-order finite-difference scheme exhibit first-order convergence and relatively large errors. To obtain designed high-order accuracy, one needs to treat this source singularity during computation. In(More)
The solution for the eikonal equation with a point-source condition has an upwind singularity at the source point as the eikonal solution behaves like a distance function at and near the source. As such, the eikonal function is not differentiable at the source so that all formally high-order numerical schemes for the eikonal equation yield first-order(More)
We develop a fast sweeping method for static Hamilton-Jacobi equations with convex Hamiltonians. Local solvers and fast sweeping strategies apply to structured and unstructured meshes. With causality correctly enforced during sweepings numerical evidence indicates that the fast sweeping method converges in a finite number of iterations independent of mesh(More)