We study a class of monotone numerical schemes for time-dependent HamiltonJacobi equations with weak Dirichlet boundary conditions. We get a convergence rate of 1 2 under some usual assumptions on the data, plus an extra assumption on the Hamiltonian H(Du, x) at the boundary ∂Ω. More specifically the mapping p→ H(p, x) must satisfy a monotonicity condition for all p in a certain subset of Rn gi...

In this paper we discuss the entropy satisfaction of the conservative shock-tracking technique developed in [D. K. Mao, J. Comput. Phys., 92 (1991), pp. 422–455], [D. K. Mao, J. Comput. Phys., 103 (1992), pp. 359–369], and [D. K. Mao, SIAM J. Numer. Anal., 32 (1995), pp. 1677–1703] when it is applied to the Godunov scheme. We consider the scalar case in one-space dimension and assume that both ...

In this paper we describe an adaptive Cartesian grid method for modeling timedependent inviscid compressible ow in irregular regions. In this approach a body is treated as an interface embedded in a regular Cartesian mesh. The single grid algorithm uses an unsplit second-order Godunov algorithm followed by a corrector applied to cells at the boundary. The discretization near the uid-body interf...

Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long-time behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small re...

We study generalizations of the high-resolution wave propagation algorithm for the approximation of hyperbolic conservation laws on irregular grids that have a time step restriction based on a reference grid cell length that can be orders of magnitude larger than the smallest grid cell arising in the discretization. This Godunov-type scheme calculates fluxes at cell interfaces by solving Rieman...

In this paper a sequential coupling of mixed finite element and shock-capturing finite volume techniques is proposed, in order to numerically solve the system of partial differential equations arising from the Black-Oil model. The Brezzi–Douglas–Marini space of degree one (BDM1) is used to approximate the Darcy’s velocity in the parabolic-type pressure equation, while the system of mass conserv...

The water hammer phenomenon is the main problem in long-distance pipeline networks. MOC (Method of characteristics) and finite difference methods lead to severe constraints on mesh Courant number, while volume method second-order Godunov scheme has limited intermittent capture capability. These will produce numerical dissipation, affecting computational efficiency at low numbers. Based lax-Frie...

A posteriori error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants u0 and cellwise p-th order reconstructed functions R 0 pu0. Using standard duality arguments, we construct exact error representation formulas for derived functionals that are tailored to the class of high orde...

A class of gas-kinetic BGK schemes for solving quantum hydrodynamic transport based on the semiclassical Boltzmann equation with the relaxation time approximation is presented. The derivation is a generalization to the development of Xu [K. Xu, A gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and Godunov method, from gas-kinetic theory, J. ...

Up to an arbitrary order of the Chapman–Enskog expansion of the kinetic Bhatnagar– Gross–Krook (BGK) equation, a corresponding analytic solution can be obtained. Based on such a compact exact solution, a new gas-kinetic scheme is constructed for the compressible Navier–Stokes equations. Instead of using a discontinuous initial condition in the gas-kinetic BGK–NS method Xu [K. Xu, A gas-kinetic ...