Michal Beneš

Mathematical analysis of phase-field equations with numerically efficient coupling terms

This paper deals with the equations in a phase-field model with special terms coupling the heat equation and the equation of phase. A finer control of latent heat release together with a gradient coupling term in the phase equation are introduced as a consequence of an extensive numerical work with models of phase transitions within the context of the solidification of crystalline substances. We present a proof of the existence and uniqueness of the weak solution of the modified system of equations. Furthermore, we perform an asymptotic procedure to recover sharp-interface relations. Finally, several numerical studies demonstrate how the model behaves compared to its standard version.

Nonlinear Galerkin method for reaction-diffusion systems admitting invariant regions

Abstract The article presents an analysis of the nonlinear Galerkin method applied to a system of reaction–diffusion equations. If the system admits a bounded invariant region, it is possible to demonstrate the convergence of the approximate solutions to the weak solution of the system. The proof is based on the compactness technique. It is performed for arbitrary ratio of dimensions of the approximation space and of the correction space used in the nonlinear Galerkin method. This fact, generalizing the previously published results, is important for the practical use of the method and allows optimization of the CPU-time consumption of the algorithm. The method is applied to the well-known Brusselator system for which we present an overview of the computational results and our experience with the numerical method used.

Mechanisms controlling the cyclic saturation stress and the critical cross-slip annihilation distance in copper single crystals

The proposed model is inspired by Brown’s suggestion that the saturation stress in cycling is controlled by the stress required to separate two screw dislocations of opposite signs, which are just on the point of mutual annihilation by cross-slip. Cross-slip is treated as the deterministic, stress-activated process governed by the line tension, the applied stress and the interaction force between dislocations. The extension of the dislocation cores is neglected. The saturation stress and the critical cross-slip annihilation distance predicted simultaneously by the model agree with the available experimental data.

Second order numerical scheme for motion of polygonal curves with constant area speed

We study polygonal analogues of several moving boundary problems and their time discretization which preserves the constant area speed property. We establish various polygonal analogues of geometric formulas for moving boundaries and make use of the geometric formulas for our numerical scheme and its analysis of general constant area speed motion of polygons. Accuracy and efficiency of our numerical scheme are checked through numerical simulations for several polygonal motions such as motion by curvature and area-preserving advected flow etc.