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 first-order convergence and relatively lar...

Homotopy continuation is an efficient tool for solving polynomial systems. Its efficiency relies on utilizing adaptive stepsize and adaptive precision path tracking, and endgames. In this article, we apply homotopy continuation to solve steady state problems of hyperbolic conservation laws. A third-order accurate finite difference weighted essentially non-oscillatory (WENO) scheme with Lax-Frie...

In this paper, 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 bidirectional flow. We first review the dynamic continuum model for bi-directional flow. This model is composed of a coupled system of a conservation law and an Eikonal equation. Then we present the first-order Lax-F...

The theory of abstract Friedrichs operators, introduced by Ern, Guermond and Caplain (2007), proved to be a successful setting for studying positive symmetric systems first order partial differential equations (Friedrichs, 1958), nowadays better known as systems. Recently, Antoni\'c, Michelangeli Erceg (2017) presented purely operator-theoretic description allowing application the universal ope...

A diffusion regulation parameter, which operates based on the jump in the Mach number, is presented for implementation in Euler solvers. This diffusion regulation parameter adjusts itself automatically in different regimes of the flow and leads to the exact capturing of steady contact discontinuities which are aligned with the grid-lines. This diffusion regulator parameter reduces numerical dis...

We propose a unified discontinuous Petrov–Galerkin (DPG) framework with optimal test functions for Friedrichs-like systems, which embrace a large class of elliptic, parabolic, and hyperbolic partial differential equations (PDEs). The well-posedness, i.e., existence, uniqueness, and stability, of the DPG solution is established on a single abstract DPG formulation, and two abstract DPG methods c...