The boundary-value problem is discretized on several grids (or finite-element spaces) of widely different mesh sizes. Interactions between these levels enable us (i) to solve the possibly nonlinear system of n discrete equations in 0(n) operations (40n additions and shifts for Poisson problems); (ii) to conveniently adapt the discretization (the local mesh size, local order of approximation, etc.) to the evolving solution in a nearly optimal way, obtaining \"°°-order\" approximations and low n, even when singularities are present. General theoretical analysis of the numerical process. Numerical experiments with linear and nonlinear, elliptic and mixed-type (transonic flow) problemsconfirm theoretical predictions. Similar techniques for initial-value problems are briefly

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Multi-level adaptive solutions to boundary-value problems

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 expl...

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High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws

We derive a new upwind finite difference approximation to systems of nonlinear hyperbolic conservation laws. The scheme has desirable properties for shock calculations. Under fairly general hypotheses we prove that limit solutions satisfy the entropy condition and that discrete steady shocks exist which are unique and sharp. Numerical examples involving the Euler and Lagrange equations of compressible gas dynamics in one and two space dimensions are given.

https://www.ams.org/mcom/1982-38-158/S0025-5718-1982-0645656-0/S0025-5718-1982-0645656-0.pdf

Upwind difference schemes for hyperbolic systems of conservation laws

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...

Riemann Solvers, the Entropy Condition, and Difference

The upwind-differencing first-order schemes of Godunov, Engquist–Osher and Roe are discussed on the basis of the inviscid Burgers equations. The differences between the schemes are interpreted as differences between the approximate Riemann solutions on which their numerical flux-functions are based. Special attention is given to the proper formulation of these schemes when a source term is present. Second-order two-step schemes, based on the numerical flux-functions of the first-order schemes are also described. The schemes are compared in a numerical experiment and recommendations on their use are included.

/pdf/on-the-relation-between-the-upwind-differencing-schemes-of-1l3shfg9wf.pdf

On the Relation Between the Upwind-Differencing Schemes of Godunov, Engquist–Osher and Roe

The requirements for uniqueness of the calculated jump conditions across embedded shock waves are investigated for type-dependent difference systems used in transonic flow studies. A mathematical analysis shows that sufficient conditions are (1) the equations should be differenced in conservative form and (2) a special difference operator should be used when switching from a hyperbolic to an elliptic operator. The latter results in a consistency condition on the integral equations, rather than the differential, at these points. Calculated jump conditions for several embedded and detached shock waves are analyzed in the physical and hodograph planes. Comparisons are made with previous results, a time-dependent calculation, and data.

Earll Teske Murman

Papers

Analysis of Embedded Shock Waves Calculated by Relaxation Methods