The velocity field in coastal and oceanic currents is mostly non-uniform, which will result irregular particle distribution when the fluid represented by an amount of moving discrete particles as smoothed hydrodynamics (SPH). When non-uniformity flow big, i.e., with large deformation, conventional SPH method can hardly solve associated advection-diffusion process (e.g., substance transport). To...

In this work, we present a class of fast first order finite volume solvers, named as PVM (Polynomial Viscosity Matrix), for balance laws or, more generally, for nonconservative hyperbolic systems. They are defined in terms of viscosity matrices computed by a suitable polynomial evaluation of a Roe matrix. These methods have the advantage that they only need some information about the eigenvalue...

We propose a novel Babich-like ansatz consisting of an infinite series of dyadic coefficients (three-by-three matrices) and spherical Hankel functions for solving point-source Maxwell’s equations in an inhomogeneous medium so as to produce the so-called dyadic Green’s function. Using properties of spherical Hankel functions, we derive governing equations for the unknown asymptotics of the ansat...

A Godunov scheme is derived for two-dimensional scalar conservation laws without or with source terms following ideas originally proposed by Boukadida and LeRoux [9] in the context of a staggered Lax-Friedrichs scheme. In both situations, the numerical fluxes are obtained at each interface of a uniform Cartesian computational grid just by means of the “external waves” involved in the entropy so...

In problems where shock waves move slowly relative to the grid, numerical errors in density and momentum may occur at the shock and propagate downstream, thereby contaminating the solution. These errors are analyzed in the present work and a fix is proposed. The errors are divided into two classes: start-up errors and post-shock oscillations. It is shown that a traveling wave analysis of the is...

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This paper continues the convergence analysis in [M. Giles and S. Ulbrich, SIAM J. Numer. Anal., 48 (2010), pp. 882–904] of discrete approximations to the linearized and adjoint equations arising from an unsteady one-dimensional hyperbolic equation with a convex flux function. We consider a simple modified Lax–Friedrichs discretization on a uniform grid, and a key point is that the numerical sm...

An efficient and scalable global discontinuous Galerkin atmospheric model (DGAM) on the sphere is developed. The continuous flux form of the nonlinear shallow water equations on the cubed-sphere (in curvilinear coordinates) are developed. Spatial discretization is a nodal basis set of Legendre polynomials. Fluxes along internal element interfaces are approximated by a Lax-Friedrichs scheme. A t...