Dispersion relations for continuum and numerical models of multiscale mechanics of compressible two-phase media.

Mathematical models of processes in continuum mechanics, plasma physics, astrophysics and other applications are often described by partial differential equations (PDE). When creating computer models, the calculation methods are replaced by discrete equations that are solved numerically. For the study of continuum and numerical models, a technique has been developed that allows us to match the dispersion relation (classical or approximate, respectively) to the model. The talk will show the application of this technique to the study of continuous and numerical macro-level models of gas-dust polydisperse media (GPS), in which the characteristic times of high-speed and thermal relaxation (mutual exchange of momentum and thermal energy) can be several orders of magnitude less than the simulation time. Two types of partial solutions are constructed for continuous models: (1) for arbitrary values of relaxation times in the form of monochromatic sound waves, (2) for infinitesimal relaxation times in the form of known solutions for single-phase models with modified viscosity or sound velocity. Small relaxation times make the problem tough, that is, requiring special numerical solution methods. Methods in which, at a fixed numerical resolution (the magnitude of the sampling steps in space and time), the error decreases as the "small" physical parameters tend to zero, we will call methods that preserve asymptotics (also known as uniformly converging methods). Based on the Lagrangian method of smoothed particle hydrodynamics (SPH), the SPH-IDIC method has been developed for modeling GPS with different scale parameters. Approximate dispersion relations are constructed for the developed SPH-IDIC method and its classical analogues, which make it possible to justify the preservation or non-preservation of asymptotics, to investigate the stability of numerical solutions in a linear approximation, and also to determine the order of approximation of numerical models.