This paper is concerned with a posteriori error bounds for wide class of numerical schemes, $$n\times n$$ hyperbolic conservation laws in one space dimension. These estimates are achieved by “post-processing algorithm”, checking that the solution retains small total variation, and computing its oscillation on suitable subdomains. The results apply, particular, to solutions obtained Godunov or L...

This paper presents a Godunov-type large time step (LTS) solver of the non-homogeneous shallow water equations (SWEs). Source terms are decomposed into simple characteristic waves in approximate Riemann solvers (ARS) and exact (ERS), information is transferred over multiple cells per using LTS method. Benchmark simulations presented different solution algorithms (ARS ERS with without entropy fi...

The Godunov method for conservation laws produces numerical solutions that are total-variation diminishing (TVD) and converge to weak solutions which satisfy the entropy condition (Entropy Consistency), but the method is only first order accurate. Many second and higher order accurate Godunov–type methods have been developed by various researchers. Although these high order methods perform very...

A family of Godunov-type central-upwind schemes for the Saint-Venant system of shallow water equations has been first introduced in [A. Kurganov and D. Levy, M2AN Math. Model. Numer. Anal., 36, 397-425, 2002]. Depending on the reconstruction step, the second-order versions of the schemes there could be made either well-balanced or positivity preserving, but fail to satisfy both properties simul...

Discretizations based on unstructured grids have been widely used in other scientific disciplines for modeling problems bounded by complex geometries. Solution-adaptive techniques are relatively easy to implement on unstructured grids and in addition they provide a convenient way to model the multi-scale spanning nature of atmospheric flows. This paper describes the implementation of a higher-o...

We derive a Godunov-type relaxation scheme for turbulent flows with heat transfer. The building block of this approach is a kinetic Boltzmann-type formulation for a model of turbulent thermal flows based on large-eddy simulation modelling. Discrete-velocity moments are taken in order to derive a relaxation system which is in turn solved by a relaxation scheme. High-order schemes are obtained by...

In this paper, we consider first order Hamilton-Jacobi (HJ) equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. For this continuous HJ problem, we propose a finite difference scheme and prove two main results. As a first result, we show bounds on the discrete gradient and time derivative of the numerical solution. Our second res...

A nite-volume method is presented for the computation of compressible ows of two immiscible uids at very di erent densities. The novel ingredient in the method is a twouid linearized Godunov scheme, allowing for ux computations in case of di erent uids (e.g., water and air) left and right of a cell face. A level-set technique is employed to distinguish between the two uids. The level-set equati...