Accelerated convergence in numerical simulations of surface supersaturation for crystal growth in solution under steady-state conditions

Abstract

This is an investigative paper which reports the results of comparisons of two numerical techniques for the solution of the Burton Cabrera and Frank (BCF) equation for the growth on crystal surfaces under steady state conditions. A successive over-relaxation (SOR) scheme for the equivalent finite difference equation gives rapid convergence to the static solution. It is known that a suitable choice of scattering parameters in a transmission line matrix (TLM) network analogue of the Laplace equation yields ultra-fast convergence. The results of numerical experiments which are reported here suggests that a similar situation also applies to the solution of the Poisson equation with shunt losses (the BCF equation), although the choice of optimum conditions appears to be different for different spatial positions within the solution spac

A TLM method for steady‐state convection–diffusion: Some additions and refinements

“…Using Equations (12) and (13) to replace the scattered voltages in Equation (11) and rearranging give the node voltage at node n and time step k in terms of the incident voltages and k I C n k Vn n = 2 k Vil n +2P nk Vir n + Z nk I C n 1+ P n The current added at node n by the lumped current source, k I C n , is a function of the distributed current source, I C d (x), between nodes n −1 and n +1 at time step k for the TL being modelled. From Equation (5), this current has two components: one associated with S(x), and the other associated with K * (x)V (x, t).…”

“…The model is initiated by setting the two incident voltages at each node equal to half the desired initial node voltage distribution along the line. Equation (16) is used to calculate the resulting node voltages, then Equations (12), (13) give the scattered voltages, and Equations (17), (18) give the incident voltages for the next time step. The only difference between this and the standard lossy TLM method for diffusion is in the equation for the node voltage.…”

“…The only difference between this and the standard lossy TLM method for diffusion is in the equation for the node voltage. Traditionally, if conditions are not time-varying and a steady-state solution is required, then this procedure is repeated until the desired level of convergence is obtained [8,12,13]. It is shown below that a steady-state solution can, instead, be obtained directly.…”

“…Traditional lossy TLM is a straightforward explicit time-domain method [8]. To find a steadystate solution it must be run until transients reduce to an acceptable level [12,13]. It is shown here how steady-state solutions can be found directly, thus increasing the efficiency and avoiding difficulties associated with the time-domain scheme.…”

An alternative TLM method for steady‐state convection–diffusion

“…the length of the TL) and time are divided into finite increments. Traditionally steady-state solutions have been found by running the scheme until transients reduce to an acceptable level [7,10,11], but a recent paper has shown that they can also be found directly [5]. The first step in modelling a transmission line using TLM is to choose the locations of the nodes at which the solution will be calculated and a time step length, Δt.…”