Wayne R. Dyksen

A Population of Linear, Second Order, Elliptic Partial Differential Equations on Rectangular Domains. Part 1

Abstract : We present a population of 56 linear, two-dimensional elliptic partial differential equations (PDEs) suitable for evaluating numerical methods and software. Forty two of the PDEs are parameterized which allows much larger populations to be made; 189 specific cases are presented here along with solutions (some are only approximate). Many of the PDEs are artificially created so as to exhibit various mathematical behaviors of interest; the others are taken from real world problems in various ways. The population has been structured by introducing measures of complexity of the operator, boundary conditions, solution, and problem. The PDEs are first presented in mathematical terms along with contour plots of the 189 specific solutions. Machine readable descriptions are given in Part 2, MRC Technical Summary Report no. 2079; many of the PDEs involve lengthy expressions and about a dozen involve extensive tabulations of approximate solutions.

Tensor product generalized ADI methods for separable elliptic problems

We consider solving separable, second order, linear elliptic partial differential equations. If an elliptic problem is separable, then, for certain discretizations, the matrices involved in the corresponding discrete problem can be expressed in terms of tensor products of lower order matrices. In the most general case, the discrete problem can be written in the form

Interactive ELLPACK: an interactive problem-solving environment for elliptic partial differential equations

(A_1 \otimes B_2 + B_1 \otimes A_2 )C = F

A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems

. We present a new Tensor Product Generalized Alternating Direction Implicit (TPGADI) iterative method for solving such discrete problems. We prove convergence and establish computational efficiency. The TPGADI method is applied to the Hermite bicubic collocation equations. We conclude that the TPGADI method is an effective tool for solving the discrete elliptic problems arising from a large class of elliptic problems.

Ellipic expert: an expert system for elliptic partial differential equations

ELLPACK is a versatile, very high-level language for solving elliptic partial differential equations.Solving elliptic problems with ELLPACK typically involves a process in which one repeatedlycomputes a solution, analyzes the results, and modifies the solution technique. Although this processis best suited for an interactive environment, ELLPACK itself is batch oriented. With this in mind,we have developed Interactive ELLPACK, an extension of ELLPACK that provides true interactiveelliptic problem solving by allowing the user to interactively build grids, choose solution methods,and analyze computed results. Interactive ELLPACK features a sophisticated interface with window-ing, color graphics output, and graphics input.

The importance of scaling for the Hermite bicubic collocation equations

We present a new decoupling technique for solving the linear systems arising from Hermite cubic collocation solutions to boundary value problems with both Dirichlet and Neumann boundary conditions. While the traditional approach yields a linear system of order 2N×2N with bandwidth 2, our technique decouples this system into two systems, one with a tridiagonal system of order N−1×N−1 and the other with the identity matrix of order N×N. Besides cutting the work in half, our new approach results in a new tridiagonal system that exhibits the same desirable properties (e.g. symmetric, positive definite) as in the case of finite difference approximations. We validate our theoretical work with a number of experimental results, demonstrating both accuracy and stability.

Pipelined iterative methods for shared memory machines

Problem oriented, very high level languages represent a first step toward the modernization of scientific mathematical software. An example of such a system is XELLPACK, an environment for solving elliptic partial differential equations (PDEs). XELLPACK is based on ELLPACK and the X Window System, making full use of interactive color graphics output and input.

A tensor product generalized ADI method for elliptic problems on cylindrical domains with holes

It is well known that improper scaling of linear equations can result in catastrophic loss of accuracy from Gauss elimination. The scaling process is not well understood and the commonly used “scaling rules” can fail. We study the scaling problem for the linear equations that arise from solving elliptic partial differential equations by collocation using Hermite bicubics. We present an a priori scaling rule that is effective but not foolproof. We conclude that one should use scaled partial pivoting for such equations. We also explore the relationship between the ordering used during Gauss elimination and the underlying geometry of the elliptic problem; we conjecture that this ordering must maintain the geometric integrity of the problem in order to avoid severe round-off problems.

Symmetric Versus Nonsymmetric Differencing

Abstract In this paper we describe a new parallel iterative technique to solve a set of linear equations. The technique can be applied to any serial iterative scheme and involves pipelining successive iterations. We give an example of this technique by modifying the classical successive overrelaxation method (SOR). The algorithm is implemented on a Sequent Symmetry multiprocessor machine and the experimental results are presented.

Editorial: special issue in honor of John Rice's 65th birthday

Abstract We consider solving second order linear elliptic partial differential equations together with Dirichlet boundary conditions in three dimensions on cylindrical domains (nonrectangular in x and y) with holes. We approximate the partial differential operators by standard partial difference operators. If the partial differential operator separates into two terms, one depending on x and y, and one depending on z, then the discrete elliptic problem may be written in tensor product form as (Tz⊗I + I⊗Axy) U=F. We consider a specific implementation which uses a Method of Planes approach with unequally spaced finite differences in the xy direction and symmetric finite difference in the z direction. We establish the convergence of the Tensor Product Generalized Alternating Direction Implicit iterative method applied to such discrete problems. We show that this method gives a fast and memory efficient scheme for solving a large class of elliptic problems.