The Convergence of Numerical Solutions of Hydrodynamic Shock Problems.
Technical rept. May 68-Jan 69,
AIR FORCE WEAPONS LAB KIRTLAND AFB NM
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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 Neumanns, 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
- Theoretical Mathematics
- Fluid Mechanics