The hydrocode convergence problem — part 1

Abstract

Abstract Assume that the following are given: (i) a hydrocode producing numerical solutions to some discrete analog of the conservation laws of continuum mechanics, (ii) a p-law (relation between the pressure p and other variables), (iii) initial and boundary value data and (iv) a mesh of points in space-time where the discrete solution is to be found by the hydrocode. In this paper the problem studied is whether or not the sequence of discrete solutions converges as the mesh is refined. Convergence can be proved in some cases as follows. For one-dimensional flows the problem can be split into three cases; in case n (n = 1, 2, 3) there is a system of n conservation laws. Further, case n divides into case nA and case nB — case nA assumes the problem well posed and the solution smooth, and convergence can be proved in this case; case nB does not make these assumptions, and convergence has not been proved in general in this case. Thus in case nB weak solutions (e.g. shock waves) are allowed. If the p-law allows p to depend only on the specific momentum u, then the u-law (conservation of momentum) is the only one of the three conservation laws needed to determine the motion and case 1 results; if the p-law allows p to depend only on the specific volume V, then the u-law and V-law (conservation of volume) are needed and case 2 results; if p depends only on V and the specific internal energy ζ, then the u-law, V-law and E-law (conservation of energy) are needed and case 3 results. Solutions to the problem for cases nA (n = 1, 2, 3) and cases nB (n = 1,2) are surveyed; then case 3B is studied and some results presented. The results are referred to as motion compactness theorems. They follow from the embedding theorem of Sobolev spaces and show that the sequence of discrete motions is sequentially compact under assumptions appropriate to case 3B and, in particular, with the case when the p-law is the ideal gas law.