Computational macromechanics as a generalization of the ideas of S.K. Godunov

Abstract

This article presents a critical analysis of two approaches to mathematical modeling of physical-and-chemical processes in the continuum approximation. The first (continuous) approach is based on finding solutions of initial-boundary value problems, while the second (discrete) approach generalizes S.K. Godunov's ideas: dividing a domain into finite volumes such that the medium in each volume can be considered continuous, and all volumes are in a thermodynamic equilibrium state. All macroparameters, as well as other functions, are assumed constant within each volume and discontinuous at its faces. The description of physical-and-chemical processes in each finite volume is based on the fundamental conservation laws, combined with phenomenological laws, equations of state, and additional hypotheses. The discontinuous approximation (computational macromechanics) allows one to construct difference schemes without approximating differential equations, using yet physical constraints on the minimum spatial and temporal scales of the modeled physical-and-chemical processes (the hypothesis of continuity and local thermodynamic equilibrium). The article formulates the fundamental principles of computational macromechanics, presents the results of computational experiments and performs comparative analysis of the errors of numerical solutions. The study demonstrates the advantages of this approach for computer modeling, particularly in black-box software and for problems with discontinuous solutions or boundary conditions. The results obtained are of interest to specialists in the field of computational continuum mechanics and software developers.