On one approach for the numerical solving of hyperbolic initial-boundary problems on an adaptive moving grids

Abstract

An approach for the quality improvement of numerical modeling of the processes described by nonlinear hyperbolic equations, which have conservative form is proposed. The transition to synchronous moving coordinate system along with Godunov type numerical scheme for the problem on moving mesh is applied. The grid motion law relies heavily on Rankine–Hugoniot relations. In order to prevent the degeneration of moving computational mesh the regularization mechanism is implemented. It allows to almost automatically fit the computational mesh to the admissible width of the solution singularity resolution. The proposed algorithm is tested on several initial-boundary problems for the Hopf equation with well known exact solutions. The hyperbolic Buckley–Leverett equation is also considered as an example of application of the proposed approach.