A geometric approach to high resolution TVD schemes

  title={A geometric approach to high resolution TVD schemes},
  author={Jonathan B. Goodman and Randall J. LeVeque},
  journal={SIAM Journal on Numerical Analysis},
We use a geometric approach, similar to van Leer’s MUSCL schemes, to construct a second-order accurate generalization of Godunov’s method for solving scalar conservation laws. By making suitable approximations we obtain a scheme which is easy to implement and total variation diminishing. We also investigate the entropy condition from the standpoint of the spreading of rarefaction waves. For Godunov’s method we obtain quantitative information on the rate of spreading which explains the kinks in… 

Figures from this paper

A Local Extrapolation Method for Hyperbolic Conservation Laws. I. The ENO Underlying Schemes

  • H. Yang
  • Mathematics
    J. Sci. Comput.
  • 2000
A local extrapolation method (LEM) for the essentially non-oscillatory (ENO) schemes solving nonlinear hyperbolic conservation laws and proposes a new balancing technique that preserves the symmetry of a symmetric wave that works well for a wide range of CFL numbers.

MUSCL reconstruction and Haar wavelets

MUSCL extensions (Monotone Upstream-centered Schemes for Conservation Laws) of the Godunov numerical scheme for scalar conservation laws are shown to admit a rather simple reformulation when recast

A class of high resolution shock capturing schemes for hyperbolic conservation laws

A Class of High Resolution Shock Capturing Schemes for Non-linear Hyperbolic Conservation Laws

In the present work the numerical fluxfunction for space discretization is constructed as a combination of numerical flux function of any entropy satisfying first order accurate scheme and second order accurate upstream scheme using the flux limiter function.

Discretization of Unsteady Hyperbolic Conservation Laws

  • K. Morton
  • Computer Science
    SIAM J. Numer. Anal.
  • 2002
A basic target algorithm for approximating unsteady hyperbolic conservation laws uses a finite volume formulation in three steps: recovery or reconstruction of a more accurate approximation from a

Stability of reconstruction schemes for scalar hyperbolic conservations laws

A new stability condition, derived from an analysis of the spatial convolutions of entropy solutions with characteristic functions of intervals, is exhibited and a criterion is proposed that ensures the existence of some numerical entropy fluxes.

An entropy satisfying MUSCL scheme for systems of conservation laws

This work proposes to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by Harten for scalar equations to derive entropy diminishing projections that ensure, both, the second order of accuracy and all of the classical discrete entropy inequalities.

A General Approach to Enhance Slope Limiters in MUSCL Schemes on Nonuniform Rectilinear Grids

This paper presents a general approach to study and enhance the slope limiter functions for highly nonuniform grids in the MUSCL-MOL framework and extends the classical reconstruct-evolve-project framework.

A space-time reconstruction algorithm for steady and unsteady and Euler equations

In this paper, a second-order characteristic based scheme is proposed using the control-volume method on unstructured meshes. The second-order accuracy is achieved by linearly reconstructing the



Convergence of Generalized MUSCL Schemes

  • S. Osher
  • Mathematics, Computer Science
  • 1985
Semidiscrete generalizations of the second order extension of Godunov’s scheme, known as the MUSCL scheme, are constructed, starting with any three point “E” scheme. They are used to approximate

High Resolution Schemes and the Entropy Condition

A systematic procedure for constructing semidiscrete, second order accurate, variation diminishing, five-point band width, approximations to scalar conservation laws, is presented. These schemes are

On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws

This paper reviews some of the recent developments in upstream difference schemes through a unified representation, in order to enable comparison between the various schemes. Special attention is

Numerical Viscosity and the Entropy Condition for Conservative Difference Schemes

It is shown that difference schemes containing more numerical viscosity will necessarily converge to the unique, physically relevant weak solution of the approximated conservation equation, and entropy satisfying convergence follows for E schemes - those containing more Numbers than Godunov's scheme.

Riemann Solvers, the Entropy Condition, and Difference

A condition on the numerical flux for semidiscrete approximations to scalar, nonconvex conservation laws is introduced, and shown to guarantee convergence to the correct physical solution. An equal...

Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves

Quasi-linear Hyperbolic Equations Conservation Laws Single Conservation Laws The Decay of Solutions as t Tends to Infinity Hyperbolic Systems of Conservation Laws Pairs of Conservation Laws Notes