Finite difference method

Abstract: A mathematical model for low‐pressure chemical vapor deposition (LPCVD) in a single wafer reactor of the impinging jet type has been developed. The model includes the partial differential equations describing the balance of mass, momentum, heat, and species concentration, Stefan‐Maxwell equations for multicomponent diffusion, multicomponent thermodiffusion, multiple surface reactions, and variable gas properties. Gas‐phase chemistry is neglected. The equations are solved numerically in two‐dimensional, axisymmetric form, using a control‐volume‐based finite difference method. The model is applied to silicon LPCVD from silane. It is shown that in single wafer LPCVD modeling, the coupling of the flow equations to the species concentrations is very important, as are thermodiffusion effects. Multicomponent diffusion phenomena can be modeled accurately using Wilke's approximation in many cases. In some situations, however, the Stefan‐Maxwell equations should be used. The model is used to optimize both reactor geometry and process conditions, in order to obtain uniform deposition on large wafers at high growth rates. A series of parameter variations is presented, illustrating the power of the model as an aid in such an optimization study.

Abstract: The multi-dimensional Black-Scholes equation is solved numerically for a European call basket option using a priori-a posteriori error estimates The equation is discretized by a finite difference method on a Cartesian grid The grid is adjusted dynamically in space and time to satisfy a bound on the global error The discretization errors in each time step are estimated and weighted by the solution of the adjoint problem Bounds on the local errors and the adjoint solution are obtained by the maximum principle for parabolic equations Comparisons are made with Monte Carlo and quasi-Monte Carlo methods in one dimension, and the performance of the method is illustrated by examples in one, two, and three dimensions