In this paper, we present total variation diminishing (TVD) multigrid methods for computing the steady state solutions for systems of hyperbolic conservation laws. An efficient multigrid method should avoid spurious numerical oscillations. This can be achieved by designing methods which preserve monotonicity and TVD properties through the use of upwind interpolation and restriction techniques. ...

It is well known that finite difference or finite volume total variation diminishing (TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy at smooth extrema [8], thus TVD schemes are at most second order accurate in the L1 norm for general smooth and non-monotone solutions. However, Sanders [12] introduced a third order accurate finite volume scheme w...

In this paper, we propose monotonicity preserving and total variation diminishing (TVD) multigrid methods for solving scalar conservation laws. We generalize the upwind-biased residual restriction and interpolation operators for solving linear wave equations to nonlinear conservation laws. The idea is to define nonlinear restriction and interpolation based on localRiemann solutions. Theoretical...

We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stii lower order term that approximates the original system with a small dissipative correction. The new system can be solved by under-resolved stable numerical di...

In [4], Jin and Xin developed a class of firstand second-order relaxing schemes for nonlinear conservation laws. They also obtained the relaxed schemes for conservation laws by using a Hilbert expansion for the relaxing schemes. The relaxed schemes were proved to be total variational diminishing (TVD) in the zero relaxation limit for scalar equations. In this paper, by properly choosing the num...

We present a discontinuous Galerkin (DG) scheme with suitable quadrature rules [15] for ideal compressible magnetohydrodynamic (MHD) equations on structural meshes. The semi-discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by Godunov [32], the entropy stable DG framework with suitable quadrature rules [15], the entropy conservati...

The objective of this work is to present a numerical study of contaminant migration in saturated porous media. For this purpose, the advection-difussion equation describing the evolution of contaminant plumes in a vertical cross section of an aquifer is discretized employing a Total Variation Diminishing method (TVD), which provides an efficient way to eliminate spurious numerical oscillations ...

The technique of obtaining high resolution, second order, oscillation free (TVD), explicit scalar difference schemes, by the addition of a limited antidiffusive flux to a first order scheme is explored and bounds derived for such limiters. A class of limiters is presented which includes a very compressive limiter due to Roe, and various limiters are compared both theoretically and numerically.

In recent years finite volume computations of two-fluid hydrodynamic models have been used in a number of numerical investigations of gas-solids fluidized bed and circulating fluidized bed processes. To avoid numerical diffusion from dominating simulation results, proper discretization of convective terms is critical i e higher-order methods should be used in the computation of convective terms...

An algorithm is proposed to study two-dimensional vortices in thin, polytropic accretion disks. It is based on a method which has been used to study vortices in planetary atmospheres. It is special in that it conserves both energy and potential enstrophy in the absence of dissipation. This leads to desirable stability properties which permit calculations at relatively high Reynolds numbers. The...