Undercompressive Discontinuities of a Hyperbolic System of Conservation Law Equations: Finite-Difference Schemes

A class of finite-difference schemes with well-controlled dissipation is used to solve equations describing long longitudinal–torsional waves in elastic rods. The governing system of equations is a hyperbolic system of conservation laws whose solutions may include undercompressive discontinuities (nonclassical discontinuities). It is well known that such solutions depend on the choice of a regularizing dissipative operator distinguishing a unique solution of the problem. In the scheme with well-controlled dissipation, the dissipative operator defined by its first differential approximation coincides up to small higher order terms with the operator used to define the solution in the continual formulation. The class of schemes under discussion has been poorly studied to date. Numerical experiments are presented that demonstrate the efficiency of this approach.