On dispersive difference schemes (Q1819557)

In 1944, J. von Neumann proposed a difference scheme for calculating one- dimensional compressible flows with shocks. He conjectured that the approximations converge in the weak sense to the appropriate discontinuous solution of the differential equations. In this note, the author analyzes a special discrete form of the energy equation in the von Neumann scheme, the limit of the solutions of the KdV equation \(U_ t+UU_ x+\epsilon U_{xxx}=0\) as \(\epsilon\) tends to zero and a semidiscrete approximation of the model equation \(U_ t+(0.5U^ 2)_ x=0\) to get reasons that the assertion of von Neuman is wrong.