Application of Higher Order BDF Discretization of the Boussinesq Equation and the Heat Transport Equation

Abstract

During the growth of crystals there were observed crystal defects under some conditions of the growth device. As a result of experiments a transition from the twodimensional flow regime of a crystal melt in axisymmetric zone melting devices to an unsteady threedimensional behavior of the velocity and temperature field was found. This behavior leads to striations as undesirable crystal defects. For the investigation of this symmetry break a mathematical model of the crystal melt was formulated for

i) the theoretical description of the experimentally observed behavior and

ii) the identification of critical parameters of the growth device, i.e. the evaluation of bifurcation points

To describe and to avoid such a behavior it is necessary to solve the unsteady three-dimensional Boussinesq equation coupled with the heat transport equation efficiently

To improve first and second Euler, leapfrog and Adams-Bashforth methods higher order explicit and BDF (Backward Differenciation Formulas) methods are applied and constructed for time dependend calculations and a Newton method is discussed for the resulting nonlinear equation systems for implicit integration methods and the steady state solution