Symplectic numerical schemes for reversible dynamical systems predict the solution
reliably over large times as well, and are a good starting point for extension to
schemes for simulating irreversible situations like viscoelastic wave propagation
and heat conduction coupled via thermal expansion occuring in rocks, plastics, biological
samples etc. Dissipation error (artificial nonpreservation of energies and amplitudes)
of the numerical solution should be as small as possible since it should not be confused
with the real dissipation occurring in the irreversible system. In addition, the other
well-known numerical artefact, dispersion error (artificial oscillations emerging
at sharp changes), should also be minimal to avoid confusion with the true wavy behavior.
The continuum thermodynamical aspects (respect for balances with fluxes, systematic
constitutive relationships between intensive quantities and fluxes, the second law
of thermodynamics with positive definite entropy production, and the spacetime-based
kinematic viewpoint) prove valuable for obtaining such extended schemes and for monitoring
the solutions. Generalizing earlier works in this direction, here, we establish and
investigate such a numerical scheme for one-dimensional viscoelastic wave propagation
in the presence of heat conduction coupled via thermal expansion, demonstrating long-term
reliability and the applicability of thermodynamics-based quantities in supervising
the quality of the solution.
