Showing papers in "Communications on Pure and Applied Mathematics in 1974"

Formation of singularities in one‐dimensional nonlinear wave propagation

Abstract: The waves considered here are solutions of a first-order strictly hyperbolic system of differential equations, written in the form $${u_t} + a(u){u_x} = 0$$ (1) , where u = u(x, t) is a vector with n components u 1,· ·· , u n depending on two scalar independent variables x, t, and a = a(u) is an n-th order square matrix. The question to be discussed is the formation of singularities of a solution u of (1) corresponding to initial data (2) u(x, 0) = f(x). It will be shown that if the system (1) is “genuinely nonlinear” in a sense to be defined below, and if the initial data are “sufficiently small” (but not identically 0), the first derivatives of u will become infinite for certain (x, t) with t > 0. The result is well known for n = 1 and n = 2 (see the papers by Lax and by Glimm and Lax [1], [2], [10]). In many cases such a singular behavior can be identified physically with the formation of a shock, and considerable interest attaches to the study of the subsequent behavior of the solution. In the present paper we shall be content just to reach the onset of singular behavior, without attempting to define and to follow a solution for all times. For that we would need a physical interpretation of our system to guide us in formulating proper shock conditions.

Iterative solution of transonic flows over airfoils and wings, including flows at mach 1

Abstract: A new method of calculating transonic flows based on a 'rotated' difference scheme is described. It is suitable for the calculation of both two- and three-dimensional flows without restriction on the speed at infinity and is well adapted to computer use. The Murman procedure is modified to eliminate any assumptions about the direction of flow when constructing the difference scheme. The proper directional property is obtained by rotating the difference scheme to conform with the local stream direction. In the hyperbolic region retarded difference formulas are used for all contributions to the streamwise second derivative, producing a correctly oriented positive artificial viscosity. In the absence of a simple implicit scheme in the hyperbolic and elliptic regions, the concept of iterations as steps in artificial time is introduced. Computer testing of this procedure provides numerical confirmation of the existence and uniqueness of weak solutions of the potential equation when a suitable entropy inequality is enforced.

On the Structure of Magnetohydrodynamic Shock Waves.

Abstract: : Fast and slow magnetohydrodynamic shock waves of arbitrary strength, for general equations of state and for arbitrary non-zero viscosity matrix, are shown to possess structure. (Author)