We develop a theoretical tool to examine the properties of numerical schemes for advection equations. To magnify the defects of a scheme we consider a convection-reaction equation 0021-9 doi:10. q Th * Co E-m ut þ ðjujq=qÞx 1⁄4 u; u; x 2 R; t 2 Rþ; q > 1: It is shown that, if a numerical scheme for the advection part is performed with a splitting method, the intrinsic properties of the scheme a...

In this paper we propose to revisit the notion of simple Riemann solver both in Lagrangian and Eulerian coordinates following seminal work Gallice ”Positive entropy stable Godunov-type schemes for gas dynamics MHD equations or coordinates” Numer. Math., 94, 2003. We provide relation between forms systems conservation laws 1D. Then an approximate (simple) is derived based on notions positivity p...

We consider hyperbolic scalar conservation laws with discontinuous flux function of the type ∂tu+ ∂xf(x, u) = 0 with f(x, u) = fL(u)1 R−(x) + fR(u)1 R+(x). Here fL,R are compatible bell-shaped flux functions as appear in numerous applications. In [1] and [2], it was shown that several notions of solution make sense, according to a choice of the so-called (A,B)-connection. In this note, we remar...

We propose a low Mach number, Godunov-type finite volume scheme for the numerical solution of the compressible Euler equations of gas dynamics. The scheme combines Klein’s non-stiff/stiff decomposition of the fluxes (J. Comput. Phys. 121:213-237, 1995) with an explicit/implicit time discretization (Cordier et al., J. Comput. Phys. 231:56855704, 2012) for the split fluxes. This results in a scal...

Abstract. Scalar conservation laws with a flux function discontinuous in space are approximated using a Godunov-type method for which a convergence theorem is proved. The case where the flux functions at the interface intersect is emphasized. A very simple formula is given for the interface flux. A numerical comparison between the Godunov numerical flux and the upstream mobility flux is present...

The focus of the research activity is on the Smoothed Particle Hydrodynamics (SPH) method and its applications to environmental flow problems. The purpose is to model fully three-dimensional non-hydrostatic free surface flows and impact problems with the SPH method applying it to inviscid and viscous fluids. After a thorough literature review of the SPH method, the research work is divided up i...

The Russian landmark paper ‘‘Application of the principle of minimizing the derivative to the construction of finite-difference schemes for computing discontinuous solutions of gas dynamics’’ by Vladimir P. Kolgan is discussed in a historical–technical perspective. The 1972 paper already featured a Godunov-type scheme for the Euler equations with second-order spatial accuracy, and a limiter to ...

We describe a second-order accurate sequential algorithm for solving two-phase multicomponent flow in porous media. The algorithm incorporates an unsplit second-order Godunov scheme that provides accurate resolution of sharp fronts. The method is implemented within a block structured mesh refinement framework that allows grids to dynamically adapt to features of the flow and enables efficient p...

The adjoint method provides a computationally efficient means of calculating the gradient for applications in constrained optimization. In this article, we consider a network of scalar conservation laws with general topology, whose behavior is modified by a set of control parameters in order to minimize a given objective function. After discretizing the corresponding partial differential equati...

This paper presents two finite-volume (FV) schemes for solving linear transport problems on the cubedsphere grid system. The schemes are based on the central-upwind finite-volume (CUFV) method, which is a class of Godunov-type method for solving hyperbolic conservation laws, and combines the attractive features of the classical upwind and central FV methods. One of the CUFV schemes is based on ...