Equations of fast phase transitions, in which the phase boundaries move with velocities comparable with the atomic diffusion speed or with the speed of local structural relaxation, are analyzed. These equations have a singular perturbation due to the second derivative of the order parameter with respect to time, which appears due to phenomenologically introduced local nonequilibrium. To develop unconditionally stable computational schemes, the Eyre theorem [D. J. Eyre, unpublished] proved for the classical equations, based on hypotheses of local equilibrium, is used. An extension of the Eyre theorem for the case of equations for fast phase transitions is given. It is shown that the expansion of the free energy on contractive and expansive parts, suggested by Eyre for the classical equations of Cahn-Hilliard and Allen-Cahn, is also true for the equations of fast phase transitions. Grid approximations of these equations lead to gradient-stable algorithms with an arbitrary time step for numerical modeling, ensuring monotonic nonincrease of the free energy. Special examples demonstrating the extended Eyre theorem for fast phase transitions are considered.
The analysis of the application of the variational method for solving the boundary value problem of hydrodynamics is carried out. From the point of view of numerical research of mathematical physics problems, these variation formulations are considered as the basis of projection methods (the Ritz method). The article presents the main techniques that allow reducing the cost of machine time and speed up the convergence of the computational process when calculating the hydrodynamic characteristics of cavities of various configurations. Using the Trefts method allows you to reduce the calculation time of the boundary value problem. The green transformation allows us to reduce the three dimensional integral to a one-dimensional. This creates a universal method for determining hydrodynamic coefficients for rotation cavities with an arbitrary contour of the Meridian section. However, for most configurations of cavities, the convergence rate is satisfactory and provides numerical values with a high degree of accuracy.
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