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Showing papers by "Stanley Osher published in 1986"


Journal ArticleDOI
TL;DR: This paper constructs an hierarchy of uniformly high-order accurate approximations of any desired order of accuracy which are tailored to be essentially nonoscillatory.

420 citations


Book ChapterDOI
01 Jan 1986
TL;DR: Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case, and hence convergent.
Abstract: A systematic procedure for constructing semi-discrete families of 2m - 1 order accurate, 2m order dissipa-tive, variation diminishing, 2m + 1 point band width, conservation form approximations to scalar conservation laws is presented. Here m is an integer between 2 and 8. Simple first order forward time discretization, used together with any of these approximations to the space derivatives, also results in a fully discrete, variation diminishing algorithm. These schemes all use simple flux limiters, without which each of these fully discrete algorithms is even linearly unstable. Extensions to systems, using a nonlinear field-by-field decomposition are presented, and shown to have many of the same properties as in the scalar case. For linear systems, these nonlinear approximations are variation diminishing, and hence convergent. A new and general criterion for approximations to be variation diminishing is also given. Finally, numerical experiments using some of these algorithms are presented.

160 citations


Proceedings ArticleDOI
09 Jun 1986
TL;DR: In this paper, an entropy correction method for the unsteady full potential equation is presented, which is modified to account for entropy jumps across shock waves, and solved in generalized coordinates using an implicit, approximate factorization method.
Abstract: An entropy correction method for the unsteady full potential equation is presented. The unsteady potential equation is modified to account for entropy jumps across shock waves. The conservative form of the modified equation is solved in generalized coordinates using an implicit, approximate factorization method. A flux-biasing differencing method, which generates the proper amounts of artificial viscosity in supersonic regions, is used to discretize the flow equations in space. Comparisons between the present method and solutions of the Euler equations and between the present method and experimental data are presented. The comparisons show that the present method more accurately models solutions of the Euler equations and experiment than does the isentropic potential formulation.

20 citations


Journal ArticleDOI
TL;DR: In this article, the authors derived necessary and sufficient conditions for an approximate Riemann solver to be consistent with any entropy inequality, and obtained the corresponding solution via a Legendre transform and showed that it is consistent with all entropy inequalities.
Abstract: Given a monotone function $z(x)$ which connects two constant states, $u_L u_R )$, we find the unique (up to a constant) convex (concave) flux function, $\hat f(u)$, such that $z({x / t})$ is the physically correct solution to the associated Riemann problem. For $z({x / t})$, an approximate Riemann solver to a given conservation law, we derive simple necessary and sufficient conditions for it to be consistent with any entropy inequality. Associated with any member of a general class of consistent numerical fluxes, $h_f (u_R ,u_L )$, we have an approximate Riemann solver defined through $z(\zeta ) = ({{ - d} / {d_\zeta }})h_{f_\zeta } (u_R ,u_L )$, where $f_\zeta (u) = f(u) - \zeta u$. We obtain the corresponding $\hat f(u)$ via a Legendre transform and show that it is consistent with all entropy inequalities iff $h_{f_\zeta } (u_R ,u_L )$ is an E flux for each relevant $\zeta $. Examples involving commonly used two point numerical fluxes are given, as are comparisons with related work.

7 citations