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.

可逆动力系统的辛数值方案也能在长时间内可靠地预测解，并且是扩展模拟不可逆情况的方案的良好起点，例如通过岩石、塑料、生物样品中发生的热膨胀耦合的粘弹性波传播和热传导数值解的耗散误差（能量和振幅的人为不守恒）应尽可能小，因为它不应与不可逆系统中发生的实际耗散相混淆。此外，另一个众所周知的数值伪影——色散误差（急剧变化时出现的人为振荡）也应该最小化，以避免与真实的波浪行为混淆。连续热力学方面（考虑通量平衡、密集量和通量之间的系统本构关系、具有正定熵产生的热力学第二定律以及基于时空的运动学观点）对于获得此类扩展方案和监测解决方案。概括了这个方向的早期工作，在这里，我们建立并研究了在存在通过热膨胀耦合的热传导的情况下一维粘弹性波传播的数值方案，证明了基于热力学的量在监督中的长期可靠性和适用性解决方案的质量。