Grid-characteristic difference scheme for solving the Hopf equation based on two different divergent forms

A new two-parameter family of difference schemes for the numerical solution of the Hopf equation is constructed. The original problem was replaced by a problem for a system of two differential equations based on various divergent forms of the Hopf equation. The flux terms were expressed as linear combinations of the variables included in the different divergent forms. In contrast to most works that use uncertain coefficient methods to construct difference schemes, in this approach uncertain coefficients arise in the formulation of the differential problem. The system of equations retains the hyperbolic type for any parameter values. For the numerical implementation, the well-known grid-characteristic scheme in Riemann invariants is chosen as the basis, which in the case of a linear equation with constant coefficients passes into a Lax–Wendroff scheme. Calculations of two test problems – on the evolution of a smooth initial condition and the formation of a discontinuous solution and on the propagation of a “shock wave” – have been performed. Based on the results of the test calculations, we selected extrapolation coefficients that allow us to obtain a good agreement with the exact solution. The a posteriori order of convergence to the limit function for discontinuous solutions was investigated. When the extrapolation coefficients are well chosen, it insignificantly exceeds 1 at the moment of a gradient catastrophe. The order of convergence decreases to 0,76 when the strong discontinuity propagates at large times. The question of setting an optimization problem that allows one to choose the extrapolation coefficients in the best way, possibly depending on the local properties of the solution, remains open. The question of creating hybrid difference schemes with variable extrapolation coefficients depending on the smoothness of the solution also remains open.