The Convergence Of Numerical Solutions Of Hydrodynamic Shock Problems.

In 1944 von Neumann Conjectures that only 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: Find a solution, allowing discontinuous solutions, to the three conservation laws, the increasing-entropy law, and the ideal-gas law when giben 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 suitable 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.