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On the example of the Poynting-Thomson-Zener rheological model for solids,which exhibits both dissipation and wave propagation - with nonlineardispersion relation -, we introduce and investigate a finite differencenumerical scheme. Our goal is to demonstrate its properties and to ease thecomputations in later applications for continuum thermodynamical problems. Thekey element is the positioning of the discretized quantities with shifts byhalf space and time steps with respect to each other. The arrangement is chosenaccording to the spacetime properties of the quantities and of the equationsgoverning them. Numerical stability, dissipative error and dispersive error areanalysed in detail. With the best settings found, the scheme is capable ofmaking precise and fast predictions. Finally, the proposed scheme is comparedto a commercial finite element software, COMSOL, which demonstrates essentialdifferences even on the simplest - elastic - level of modelling.

Thermodynamical extension of a symplectic numerical scheme with half space and time shift demonstrated on rheological waves in solids