In this paper we extend the result from [9] (V. Jovanović, C. Rohde, Error estimates for finite volume approximations of classical solutions for nonlinear systems of balance laws, SIAM J. Numer. Anal., 43 (2006)), where, among other things, an h — error estimate in the L — norm for the elastodynamics system has been established. We first derive the general error estimate from [9, Theorem 4.4] i...

We propose a simple, fast sweeping method based on the Lax–Friedrichs monotone numerical Hamiltonian to approximate viscosity solutions of arbitrary static Hamilton–Jacobi equations in any number of spatial dimensions. By using the Lax–Friedrichs numerical Hamiltonian, we can easily obtain the solution at a specific grid point in terms of its neighbors, so that a Gauss–Seidel type nonlinear ite...

Staggered grid finite volume methods (also called central schemes) were introduced in one dimension by Nessyahu and Tadmor in 1990 in order to avoid the necessity of having information on solutions of Riemann problems for the evaluation of numerical fluxes. We consider the general case in multidimensions and on general staggered grids which have to satisfy only an overlap assumption. We interpr...

We study time step restrictions due to linear stability constraints of Runge-Kutta Discontinuous Galerkin methods on triangular grids. The scalar advection equation is discretized in space by the Discontinuous Galerkin method with either the Lax-Friedrichs flux or the upwind flux, and integrated in time with various Runge-Kutta schemes designed for linear wave propagation problems or non-linear...

We introduce two types of finite difference methods to compute the L-solution and the proper viscosity solution recently proposed by the second author for semi-discontinuous solutions to a class of Hamilton-Jacobi equations. By regarding the graph of the solution as the zero level curve of a continuous function in one dimension higher, we can treat the corresponding level set equation using the...

In this paper, we present an intensive investigation of the finite volume method (FVM) compared to difference methods (FDMs). order show main in way approaching solution, take Burgers equation and Buckley–Leverett as examples simulate previously mentioned methods. On one hand, results using schemes Lax–Friedrichs Lax–Wendroff. other apply Godunov’s scheme method. Moreover, how starting with a v...

We introduce microscopic and macroscopic stochastic traffic models including accidents. The model is based on a Follow-the-Leader approach whereas the described by scalar conservation law with space dependent flux function. Accidents are introduced as interruptions of deterministic evolution directly linked to situation. Based Lax-Friedrichs discretization convergence shown. Numerical simulatio...

The usual geometrical-optics expansion of the solution for the Helmholtz equation of a point source in an inhomogeneous medium yields two equations: an eikonal equation for the traveltime function, and a transport equation for the amplitude function. However, two difficulties arise immediately: one is how to initialize the amplitude at the point source as the wavefield is singular there; the ot...