Gunter H. Meyer

Multidimensional Stefan Problems

An implicit finite difference method for the multidimensional Stefan problem is discussed. The classical problem with discontinuous enthalpy is replaced by an approximate Stefan problem with continuous piecewise linear enthalpy. An implicit time approximation reduces this formulation to a sequence of monotone elliptic problems which are solved by finite difference techniques. It is shown that the resulting nonlinear algebraic equations are solvable with a Gauss-Seidel method and that the discretized solution converges to the unique weak solution of the Stefan problem as the time and space mesh size approaches zero.

On Solving Nonlinear Equations with a One-Parameter Operator Imbedding

One parameter operator imbedding to modify Newton method for solution of nonlinear equations

Free boundary problems with nonlinear source terms

SummaryThe method of lines is used to semi-discretize the non-linear Poisson equation over a domain with a free boundary. The resulting multipoint free boundary problem is solved with a line Gauss-Seidel method which is shown to converge monotonically. The method of lines solution is then shown to converge to the continuous solution of the variational inequality form of the obstacle problem. Some numerical results for the diffusion-reaction equation indicate that the method is applicable to more general free boundary problems for nonlinear elliptic equations.

The Method of Lines and Invariant Imbedding for Elliptic and Parabolic Free Boundary Problems

A combination of the method of lines and invariant imbedding is suggested as a general purpose numerical algorithm for free boundary problems. Its effectiveness is illustrated by computing the solidification of a binary alloy in one dimension, electrochemical machining and Hele–Shaw flow in two dimensions, and a Stefan and ablation problem in three dimensions.

Continuous orthonormalization for boundary value problems

Abstract Continuous orthonormalization describes an initial value method for linear 2-point boundary value problems which provides an orthogonal basis for the solution space at all points of the interval. In this paper the equations of continuous orthonormalization are derived with elementary projection arguments to provide geometric insight and motivate some modifications of an earlier algorithm. The method is then applied to some oscillatory and stiff boundary value problems to demonstrate that it is simple to use, problem independent, and as adaptive as the initial value code which is used to integrate the equations of continuous orthonormalization.

A least squares method for finding the preisach hysteresis operator from measurements

SummaryA finite element like least squares method is introduced for determining the density function in the Preisach hysteresis model from overdeterined measured data. It is shown that the least squares error depends on the quality of the data and the best approximations to the analytic density. For consistent data criteria are given for convergence of the approximate density and Preisach operator with increasing measurements.

The evaluation of barrier option prices under stochastic volatility

This paper considers the problem of numerically evaluating barrier option prices when the dynamics of the underlying are driven by stochastic volatility following the square root process of Heston (1993) [7]. We develop a method of lines approach to evaluate the price as well as the delta and gamma of the option. The method is able to efficiently handle both continuously monitored and discretely monitored barrier options and can also handle barrier options with early exercise features. In the latter case, we can calculate the early exercise boundary of an American barrier option in both the continuously and discretely monitored cases.

The method of lines for Poisson's equation with nonlinear or free boundary conditions

SummaryThe method of lines is used to solve Poissons equation on an irregular domain with nonlinear or free boundary conditions. The partial differential equation is approximated by a system of second order ordinary differential equations subject to multi-point boundary conditions. The system is solved with an SOR iteration which employs invariant imbedding for each one dimensional problem. An application of the method to a boundary control problem and to a free surface problem arising in electrochemical machining is described. Finally, some theoretical convergence results are presented for a model problem with radiative boundary conditions on fixed boundaries.

Parabolic PDEs with hysteresis and quasivariational inequalities

w0 being prescribed as an initial output of w; its input–output relation is illustrated in Fig. 1.1 (see [1, Part I, III-2] for the precise de nition); in the gure fa and fd are continuous and nondecreasing functions on R such that fa≤fd on R. We refer for some results on this sort of model to [1–4]. As is well known [1, Part I, III-2], (1.2) is equivalent to the following variational inequality wt(x; t) + @Iu(x; t)(w(x; t))3 0; (x; t)∈QT ; (1.3)

A NUMERICAL METHOD FOR TWO-PHASE STEFAN PROBLEMS*

A numerical technique for the solution of one-dimensional parabolic free interface (Stefan) problems is discussed. The method of lines is employed to approximate the partial differential equations at discrete time levels by free interface problems for ordinary differential equations which are solved by conversion to initial value problems. Comments on multiphase problems and a numerical example round out the discussion.