This work concerns the solution of generalized Riemann problems. To this end, we consider the ADER scheme of Titarev & Toro (2002), which relies on a generalization of the classical Godunov scheme. Another solution method is the power series expansion of LeFloch & Raviart (1988). We analyze the two resulting approximation schemes, where we show that for scalar 1d problems the Toro-Titarev solve...

Godunov type schemes, based on exact or approximate solutions to the Riemann problem, have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws. However, there are many instances in which a particular scheme produces inappropriate results. In this paper we consider several situations in which Roe's scheme gives incorrect results (or blows up...

We present a splitting method for the one-dimensional Saint-Venant-Exner equations used describing bed evolution in shallow water systems. adapt flux vector approach of Toro and Vazquez-Cend\`on identify one subsystem conservative (advection system) non-conservative (pressure system), both having very simple eigenstructure compared to full system. The final numerical scheme is constructed using...

We prove the convergence of Godunov-type scheme for a scalar conservation law in one space dimension with possibly infinitely many spatial discontinuities, which may have accumulation points. In contrast to study appearing Ghoshal, Jana and Towers [Convergence Godunov an Audusse–Perthame adapted entropy solution laws BV flux, Numer. Math. 146(3) (2020) 629–659], we do not restrict flux be unimo...

In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the...

An efficient formulation for the robust shape optimization of aerodynamic objects is introduced in this paper. The formulation has three essential features. First, an Euler solver based on a second-order Godunov scheme is used for the flow calculations. Second, a genetic algorithm with binary number encoding is implemented for the optimization procedure. The third ingredient of the procedure is...

We present a multilane traffic model based on balance laws, where the nonlocal source term is used to describe lane changing rate. The modelling framework includes consideration of local and flux functions. Based Godunov type numerical scheme, we provide BV estimates discrete entropy inequality. Together with $L^1$-contractivity property, prove existence uniqueness weak solutions. Numerical exa...