Stefka Dimova

Numerical analysis of radically nonsymmetric blow-up solutions of a nonlinear parabolic problem

Abstract The process of combustion for a nonlinear heat conducting medium with a nonlinear volume source is considered. Blow-up self-similar solutions which describe the evolution of radially nonsymmetric waves — with complex symmetry and spiral waves — are realized numerically. Their asymptotic behaviour is analysed and their metastability is established. To solve the self-similar nonlinear problem the continuous analog of the Newton method and the finite element method are used. The semidiscrete Galerkin finite element method and an explicit difference scheme are used to solve the nonlinear parabolic problem. Special adaptive grids, consistent with the structure of the self-similar solutions, are utilized.

Phase-field versus level set method for 2D dendritic growth

The goal of the paper is to review and compare two of the most popular methods for modeling the dendritic solidification in 2D, that tracks the interface between phases implicitly, e.g. the phase-field method and the level set method. We apply these methods to simulate the dendritic crystallization of a pure melt. Numerical experiments for different anisotropic strengths are presented. The two methods compare favorably and the obtained tip velocities and tip shapes are in good agreement with the microscopic solvability theory.

Numerical Methods and Applications

In this note we propose a grid refinement procedure for direction splitting schemes for parabolic problems that can be easily extended to the incompressible Navier-Stokes equations. The procedure is developed to be used in conjunction with a direction splitting time discretization. Therefore, the structure of the resulting linear systems is tridiagonal for all internal unknowns, and only the Schur complement matrix for the unknowns at the interface of refinement has a four diagonal structure. Then the linear system in each direction can be solved either by a kind of domain decomposition iteration or by a direct solver, after an explicit computation of the Schur complement. The numerical results on a manufactured solution demonstrate that this grid refinement procedure does not alter the spatial accuracy of the finite difference approximation and seems to be unconditionally stable.

LUMPED-MASS FINITE-ELEMENT METHOD WITH INTERPOLATION OF THE NONLINEAR COEFFICIENTS FOR A QUASILINEAR HEAT TRANSFER EQUATION

Abstract A unified numerical technique for computing single-point, regional, and total blow-up solutions of quasilinear heat transfer equations in the radial symmetric case is proposed and realized. It is based on the lumped-mass finite-element method in space, with interpolation of the nonlinear coefficients and explicit methods for solving the system of ordinary differential equations. A special mesh adaptation, consistent with the structure of the known self-similar solution, is realized for the cases of single point and total blow-up. This gives a possibility to calculate the solution close to the blow-up time and so to analyze its asymptotic behavior.

Numerical realization of blow‐up spiral wave solutions of a nonlinear heat‐transfer equation

The problem of finding the possible classes of solution of different nonlinear equations seems to be of a great importance for many applications. In the context of the theory of self‐organization it is interpreted as finding all possible structures which arise and preserve themselves in the corresponding unbounded nonlinear medium. First, results on the numerical realization of a class of blow‐up invariant solutions of a nonlinear heat‐transfer equation with a source are presented in this article. The solutions considered describe a spiral propagation of the inhomogeneities in the nonlinear heat‐transfer medium. We have found initial perturbations which are good approximations to the corresponding eigen functions of combustion of the nonlinear medium. The local maxima of these initial distributions evolve consistent with the self‐similar law up to times very close to the blow‐up time.

Numerical simulation of critical dependences for symmetric two-layered Josephson junctions

Partial critical dependences of the form current-magnetic field in a two-layered symmetric Josephson junction are modeled. A numerical experiment shows that, for the zero interaction coefficient between the layers of the junction, jumps of the critical currents corresponding to different distributions of the magnetic fluxes in the layers may appear on the critical curves. This fact allows a mathematical interpretation of the results of some recent experimental results for two-layered junctions as a consequence of discontinuities of partial critical curves.

Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

The paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach is consistent with the peculiarities of the problems — multiple solutions of the elliptic problem and blow-up solutions of the parabolic one.

Phase-field method for 2D dendritic growth

The phase field method is used to model the free dendritic growth into undercooled melt in 2D. The pair of two nonlinear reaction-diffusion equations – for the temperature and for the phase-field function – are solved numerically by using the finite element method in space. A second order modification of the Runge-Kutta method with extended region of stability and automatic choice of the time step is used to solve the resulting system of ordinary differential equations in time. Numerical experiments are made to analyze the evolution in time of a spherical seed in the isotropic case and for different kinds of anisotropy. The results for the dimensionless dendritic tip velocity, found in the different cases, are compared with the results of the microscopic solvability theory and with numerical results found by the finite difference method.

Structures and Waves in a Nonlinear Heat-Conducting Medium

This paper is an overview of the main contributions of a Bulgarian team of researchers to the problem of finding the possible structures and waves in the open nonlinear heat-conducting medium, described by a reaction–diffusion equation. Being posed and actively worked out by the Russian school of A.A. Samarskii and S.P. Kurdyumov since the seventies of the last century, this problem still contains open and challenging questions.

Numerical investigation of spiral structure solutions of a nonlinear elliptic problem

The nonlinear elliptic problem considered arises when investigating a class of self-similar solutions of a reaction-diffusion equation. We focus our study on the solutions of spiral structure. The proposed approach is based on the continuous analog of the Newtons method and on the Galerkin finite element method. To reveal solutions of spiral structure appropriate initial approximations are used. The last ones are expressed by the confluent hypergeometric function 1F1(a, b; z). Algorithms for accurate, fast and reliable computation of its values for broad ranges of the parameters a and b and of the variable z are worked out. A detailed numerical analysis of the evolution of the spiral structure solutions with respect to the medium parameters, including critical values, is carried out.