Numerical stability of discretization schemes for nonlinear hyperbolic systems is obtained most currently via upwinding. In this paper we aim to report on the interaction of upwinding and forcing terms. It turns out that, for a given scheme, one must discretize properly in space the forcing terms in order to maintain the accuracy. As a matter of fact, when one uses an upwind method to discretiz...

The point-source traveltime field has an upwind singularity at the source point. Consequently, all formally high-order, finite-difference eikonal solvers exhibit firstorder convergence and relatively large errors. Adaptive upwind finite-difference methods based on high-order Weighted Essentially NonOscillatory (WENO) RungeKutta difference schemes for the paraxial eikonal equation overcome this ...

Balance laws arise from many areas of engineering practice specifically from the fluid mechanics. Many numerical methods for the solution of these balanced laws were developed in recent decades. The numerical methods are based on two views: solving hyperbolic PDE with a nonzero source term (the obvious description of the central and central-upwind schemes; (Kurganov & Levy, 2002; LeVeque, 2004)...

The point source traveltime eld has an upwind singularity at the source point. Consequently, all formally high-order nite-diierence eikonal solvers exhibit rst-order convergence and relatively large errors. Adaptive upwind nite-diierence methods based on high-order Weighted Essentially NonOscillatory (WENO) Runge-Kutta diierence schemes for the paraxial eikonal equation overcome this diiculty. ...

A computationally efficient high-order solver is developed to compute the wall distances by solving relevant partial differential equations, namely: Eikonal, Hamilton–Jacobi (HJ) and Poisson equations. In contrast upwind schemes widely used in literature, we explore suitability of central difference (explicit/compact) for wall-distance computation. While equation, performed approximately 1.4–2....

We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d’Alembert’s exact solution. We construct conservative finite differe...

We present a systematic and constructive methodology to devise various hybridized discontinuous Galerkin (HDG) methods for linearized shallow water equations. At the heart of our development is an upwind HDG framework obtained by hybridizing the upwind flux in the standard discontinuous Galerkin (DG) approach. The chief idea is to first break the uniqueness of the upwind flux across element bou...