Showing papers by "Stanley Osher published in 1982"

Upwind difference schemes for hyperbolic systems of conservation laws

Abstract: 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.

Monotone Difference Schemes for Singular Perturbation Problems

Abstract: We consider numerical solutions of the boundary value problem $\varepsilon y'' - (f(y))' - b(x,y) = 0$, $0 \leqq x \leqq 1$, $y(0) = A$, $y(1) = B$, $\varepsilon < 0$, $b_y \geqq \delta < 0$, with monotone difference schemes on nonuniform grids We prove general convergence results and show that the Engquist–Osher monotone scheme will reproduce essential properties of the true solution for any grid For the inversion of the nonlinear scheme, we suggest implicit time relaxation with variable time steps

Numerical experiments with the Osher upwind scheme for the Euler equations

Abstract: The Osher algorithm for solving the Euler equations is an upwind finite difference procedure that is derived by combining the salient features of the theory of conservation laws and the mathematical theory of characteristi cs for hyperbolic systems of equations. A first-order accurate version of the numerical method was derived by Osher circa 1980 for the one-dimensional non-isentropic Euler equations in Cartesian coordinates. In this paper, the extension of the scheme to arbitrary two-dimensional geometries is explained. Results are then presented for several example problems in one and two dimensions. Future work will include extension of the method to second-order accuracy and the development of implicit time differencing for the Osher algorithm.