Convergent Numerical Schemes for the Compressible Hyperelastic Rod Wave Equation

TL;DR

This paper introduces a fully discretized numerical scheme for the hyperelastic rod wave equation, proving its convergence, ability to handle blow-up scenarios, and preservation of energy positivity through invariant-preserving time splitting.

Contribution

It presents a novel numerical scheme that guarantees convergence, handles blow-up, and preserves invariants and positivity for the hyperelastic rod wave equation.

Findings

The scheme converges for the hyperelastic rod wave equation.

It can handle blow-up of derivatives naturally occurring in the equation.

The scheme preserves energy positivity through invariant-preserving integrators.

Abstract

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.