|Source:||C. Yang, Rice University|
This is a large nonsymmetric standard eigenvalue problem that arises from the stability analysis of a crystal growth problem. To determine the stability of the interfacial crystallization of a piece of solid crystal solidifying into some undercooled melt, solutions of the following equations are sought.
Zero Dirichlet boundary condition is imposed at infinity. The variable U in the above equations represents the temperature perturbation of the liquid, and N describes the interface perturbation in a transformed (parabolic) coordinate system. The constant P is the Peclet number. The second equation is satisfied only at the interface eta=1. Eigenvalues with largest real parts are of interest. They indicate the growth or decay of the initial disturbance at the solid-liquid interface. A change of variable is used to map the partial differential equation from an infinite domain to a finite box [0,1] × [0,1]. The matrix eigenvalue problem follows from discretization using the standard second order finite difference formulae. The Peclet number used here is P=0.05.
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