Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave   Equation
  We propose a fully discretised numerical scheme for the hyperelastic rod wave equation on the line. The convergence of the method is established. Moreover, the scheme can handle the blow-up of the derivative which naturally occurs for this equation. By using a time splitting integrator which preserves the invariants of the problem, we can also show that the scheme preserves the positivity of the energy density. 
David Cohen and Xavier Raynaud
math.NA