Showing papers in "Communications on Pure and Applied Mathematics in 1960"

Systems of conservation laws

Abstract: : In this paper a wide class of difference equations is described for approximating discontinuous time dependent solutions, with prescribed initial data, of hyperbolic systems of nonlinear conservation laws. Among these schemes we determine the best ones, i.e., those which have the smallest truncation error and in which the discontinuities are confined to a narrow band of 2-3 meshpoints. These schemes are tested for stability and are found to be stable under a mild strengthening of the Courant-Friedrichs-Levy criterion. Test calculations of one dimensional flows of compressible fluids with shocks, rarefaction waves and contact discontinuities show excellent agreement with exact solutions. In particular, when Lagrange coordinates are used, there is no smearing of interfaces. The additional terms introduced into the difference scheme for the purpose of keeping the shock transition narrow are similar to, although not identical with, the artificial viscosity terms, and the like of them introduced by Richtmyer and von Neumann and elaborated by other workers in this field.

Continuous dependence on data for solutions of partial differential equations with a prescribed bound

Abstract: Problems in partial differential equations usually require that a solution u be determined from certain data f. Ordinarily the data consist of the values of u and of a certain number of derivatives of u on a manifold Φ. For the classical problems which are well posed in the sense of Hadamard the solution u exists and is determined uniquely for all f of some class C s ; moreover u depends continuously on f if suitable norms are used. Usually no solution u exists in a problem which is not well posed, even for f in C∞. Moreover, even for those f for which u exists we have no continuous dependence of u on f. This is best illustrated by the classical example of Hadamard [1] of the Cauchy problem for the Laplace equation u xx +u=0, u=0, u y =f(x), for which no solution exists unless f is analytic. If f(x) = n -2 cos nx we do have a solution u = n -3 (cos nx)(sinh ny). However though f and its first and second derivatives tend to zero for n →∞ the corresponding solutions u tend to ∞ for x = 0, y ≠ 0.

On the problem of a shock wave arriving at the edge of a gas

Abstract: An investigation was made as to what kind of phenomena may be expected when a shock wave propagates through a nonuniform medium of decreasing density and reaches the boundary where the density vanishes. This situation may arise for shock waves moving in plasmas sustained by magnetic pressure. (T.R.H.)