Energy Estimates for Nonlinear Conservation Laws with Applications to Solutions of the Burgers Equation and One-Dimensional Viscous Flow in a Shock Tube by Central Difference Schemes

摘要

With careful treatment of the boundary conditions, this provides a path to the construction of non-dissipative stable discretizations of the governing equations. If shock waves appear in the solution, the discretization must be augmented by appropriate shock operators to account for the dissipation of energy by the shock waves. In the case of the viscous Burgers equation, it is also shown that shock waves can be fully resolved by non-dissipative discretizations of this type with a fine enough mesh, such that the cell Reynolds number < 2. The results are extended to the equations of gas dynamics. Two schemes are proposed, entropy preserving (EP) and kinetic energy preserving (KEP). Both are applied to the direct numerical simulation of one-dimensional viscous flow in a shock-tube.