Accession Number : AD0849487

Title :   The Convergence of Numerical Solutions of Hydrodynamic Shock Problems.

Descriptive Note : Technical rept. May 68-Jan 69,

Corporate Author : AIR FORCE WEAPONS LAB KIRTLAND AFB NM

Personal Author(s) : Hicks, Darrell Lee

Report Date : MAR 1969

Pagination or Media Count : 121

Abstract : In 1944 von Neumann conjectured that only the averages of numerical solutions, produced by a discrete scheme he had proposed to solve hydrodynamic shock problems, would converge as the meshes were refined. The hydrodynamic shock problem is defined here as follows: find a solution, allowing discontinuous solutions, to the three conservation laws, the increasing-entropy law, and the ideal-gas law when given physically acceptable initial and boundary values. For discrete schemes similar to von Neumann's, it is verified that the averages do converge and it appears that von Neumann was also correct in surmising that only the averages converge. A smoothing device named 'conservative smoothing' is developed and has been successfully introduced into one-, two-, and three-dimension hydrocodes. A conservative discrete scheme with conservative smoothing, in the one-dimension Lagrangian formulation, produces in the limit, for a suitably chosen sequence of meshes, generalized functions which satisfy the conservation laws. A priori bounds and convergence and compactness lemmas for sequences of solution refinements in LP-spaces are developed. (Author)

Descriptors :   *NUMERICAL ANALYSIS), (*HYDRODYNAMICS, SHOCK(MECHANICS), CONVERGENCE, BOUNDARY VALUE PROBLEMS, BOUNDARY VALUE PROBLEMS, INEQUALITIES, THEOREMS, ENTROPY.

Subject Categories : Theoretical Mathematics
      Fluid Mechanics

Distribution Statement : APPROVED FOR PUBLIC RELEASE