The authors analyze von Neumann's algorithm for solving the initial value problem for Lagrangian equations of compressible flow. Von Neumann's difference scheme consists of replacing the time and space derivatives by central difference quotients. The scheme is linearly stable in \(\ell^ 2\) sense when GFL condition is satisfied. The authors study the convergence of approximations produced by the scheme as \(\Delta\xi\to 0\), which \(\Delta t/\Delta \xi =\lambda\) is kept fixed. Classical result of \textit{G. Strang} [Numer. Math. 6, 37-46 (1964; Zbl 0143.382)] for solution of smooth initial value problem for nonlinear hyperbolic equation by a difference scheme is applicable to the Lagrange equations of compressible flow and to von Neumann's difference scheme. The calculations given in the paper demonstrate admirably the uniform convergence of the approximations produced by von Neumann's scheme in these regions of \(\xi>t\) space where Strang's theorem guarantees their convergence. The calculations also indicate uniform convergence to the smooth solutions in a region decidedly larger than the domain of determinacy of the schemes although contained within the domain of determinacy of the differential equation. This result is very surprising since outside this region the solutions of difference equations are highly oscillating with a wavelength approximately but not exactly equal to \(\Delta\xi\), yet the boundary of the oscillatory region propagates with speed much smaller than \(\lambda\). The amplitude of the oscillations offers to be bounded away from zero and infinity, uniformly for all \(\Delta\xi\). As \(\Delta\xi\to 0\), the oscillatory regions seem to converge to a limiting region; within this region, the oscillatory functions u,p, and v appear to converge weakly to limits that are denoted by \(\bar u,\bar p\), and \(\bar v,\) respectively which contain shock-like structures i.e. discontinuities or at least regions of very rapid change. The Lagrange's laws of conservation of momentum and mass are linear in u, p, and v, so are their difference analogues and, therefore, the weak limits of these equations as \(\Delta\xi\), \(\Delta t\) tend to zero, can be taken to obtain the laws of conservation of momentum and mass for \(\bar u,\bar p,\bar v.\) It is quite otherwise with the law of conservation of energy; not only derivations of all kinds fail, but the numerical calculations indicate unequivocally that \(\bar u,\bar p,\bar v\) do not satisfy the law of conservation of energy. Some speculations about the equations that \(\bar u,\bar p,\bar v\) satisfy in such a situation are given in the paper. Since \(\lambda\) offers as a parameter, in the investigation and it is kept constant in the limiting procedure, one would expect the weak limits to depend on \(\lambda\); yet the numerical results suggest strongly that \(\bar u,\bar p,\bar v\) are independent of \(\lambda\) even in the oscillatory region. Some speculations on why this might be so are given herein. The authors have strong numerical evidence for the following: if the initial data have discontinuities whose time evolution leads to rarefaction waves, or contact discontinuities, not shocks, then the approximations constructed by von Neumann's method converge uniformly to the solution containing the rarefunction waves. Finally the limiting case \(\lambda=0\) i.e. a semidiscrete approximation is discussed. Using the conservation law for energy and entropy the authors find a plausible reason for the development of oscillations. When the initial data are isentropic, it is found that so is the flow for all later times, in this case, it is possible to interpret the semidiscrete equations as describing lattice vibrations, about which there is vast literature. The special case when p depends exponentially on v is the Toda lattice and it is known to be completely integrable. The authors believe that it is possible in case of complete integrability of Toda lattice to determine the weak limits \(\bar u,\bar p\) etc. as the limits of the solutions of the semidiscrete von Neumann equations \[ du_ k/dt=-(1/\Delta)(p_{k+}-p_{k-}),\;d/dtv_{k+}=(1/\Delta)(u_{k+1}-u_ k), \] \(h=\bar e^ v\) for arbitrary initial data as \(\Delta\to 0\). The problem dealt herein is original and is of great interest to people working in numerical treatment of hydrodynamical shocks and design of atomic bomb.