John W. Stephenson

Single cell discretizations of order two and four for biharmonic problems

Abstract New difference formulas are derived for solving the biharmonic problem in two dimensions over a rectangular domain. These methods use only the nine grid points of a single mesh cell and do not require fictitious points in order to approximate the boundary conditions. Derivatives of the solution are obtained as a by-product of the methods. Second order formulas are derived for both the first and second biharmonic problems. In numerical experiments, the new second order formulas compare favourably with the standard second order methods. Extensions to fourth order formulas are given. The method of deriving these formulas can be used to derive similar formulas for arbitrarily shaped regions.

A two-dimensional adaptive mesh generation method

Abstract An automatic two-dimensional adaptive mesh generation method is persented. The method is designed so that a small portion of the mesh can be modified without disturbing a large number of adjacent mesh points. The method can be used with or without boundary-fitted coordinate generation procedures. On the generated mesh a differential equation can be discretized by using classical difference formulas designed for uniform meshes as well as the difference formulas developed in this work. Both cases are illustrated by applying the method to the Hiemenz flow for which the exact solution of the Navier-Stokes equation is known [1] and to a two-dimensional viscous internal flow model problem.

Single cell high order difference methods for the Helmholtz equation

Abstract A procedure for deriving certain high order difference formulas for the Helmholtz equation is given. Families of formulas of order 2, 4, and 6 are derived. The distinction between “atomic” and “nonatomic” formulas is made, and a nonatomic formula of order 4 is given for a similar equation with variable coefficients. Numerical results using these formulas are given.

Relationships between the solutions of Feigenbaum's equation

Abstract In [1] we gave a numerical algorithm to solve Feigenbaums Equation which is g(x) = − α g(g(−x/α)), g(0) = 1, where g(x) is an even function and α is a constant. With this algorithm we discovered many solutions in the form of 1 + ∑Ni=1 gi(x2n)i, where n = 1,2,…. We also found that the corresponding constant α can be positive or negative. M. Campanino, H. Epstein and D. Ruelle [2] suggested that if g(x) is a solution of Feigenbaums Equation on [−1, 1], then ψ(x) = (g( |x| )) 2 is also a solution. We generalized Campanino, Epstein and Ruelles suggestion and discovered analytic relationships between the solutions. In the following theorems we will establish these relationships.

Relationships between eigenfunctions associated with solutions of Feigenbaum's equation

Abstract In [1] we have given a numerical algorithm to solve Feigenbaums Equation and the corresponding eigenvalue problem. With our algorithm we found many solutions to Feigenbaums Equation. In [2] we gave some analytic relationships between these solutions. In this paper we will give some analytic relationships between eigenfunctions corresponding to related solutions of Feigenbaums Equation.

Existence of Second Order Discretizations on Irregular Mesh

Abstract We give the necessary and sufficient conditions to obtain a difference formula of order two on an arbitrary distribution of mesh points. We also show how to form a 6 point computational cell in the physical plane if the mesh is generated by a boundary fitted coordinate generation procedure.