Thermodynamical Extension of a Symplectic Numerical Scheme with Half Space and Time Shifts Demonstrated on Rheological Waves in Solids On the example of the Poyntingxe2x80x93Thomsonxe2x80x93Zener rheological model for solids, which exhibits both dissipation and wave propagation, with nonlinear dispersion relation, we introduce and investigate a finite difference numerical scheme. Our goal is to demonstrate its properties and to ease the computations in later applications for continuum thermodynamical problems. The key element is the positioning of the discretized quantities with shifts by half space and time steps with respect to each other. The arrangement is chosen according to the spacetime properties of the quantities and of the equations governing them. Numerical stability, dissipative error, and dispersive error are analyzed in detail. With the best settings found, the scheme is capable of making precise and fast predictions. Finally, the proposed scheme is compared to a commercial finite element software, COMSOL, which demonstrates essential differences even on the simplestxe2x80x94elasticxe2x80x94level of modeling.

The Approach to the Construction of Difference Schemes with a Consistent Approximation of the StressStrain State and the Energy Balance of the Medium in Cylindrical Geometry In the paper, on unstructured metric grids of the theory of the support operator method, on the topological and geometric structure of which minimal reasonable restrictions are imposed, applicable to the symmetricized displacement tensor (t_mathbf u), discrete analogs of selfadjoint and signdefinite operations (mathop mathrm div(t_mathbf u)) and (mathop mathrm div(mathop mathrm tr(t_mathbf u)delta )) which are invariants to solid rotations, were obtained for modeling of force fields of elastic processes, as well as approximation of integrals of the form (int _varOmega mathrmtrleft( mathrmt_mathbf u2 right) dV ) and (int _varOmega mathrmtr2 left( mathrmt_mathbf u right) dV ), sufficient to simulate elasticdependent energy balances of the medium taking into account the curvature of space of the cylindrical geometry of the system.