Niel K. Madsen

Algorithm 540: PDECOL, General Collocation Software for Partial Differential Equations [D3]

PDECOL, new computer software package for numerically solving coupled systems of nonlinear partial differential equations (PDEs) in one space and one time dimension, is discussed. The package implements finite element collocation methods based on piecewise polynomials for the spatial discretization techniques. The time integration process is then accomplished by widely acceptable procedures that are generalizations of the usual methods for treating time-dependent partial differental equations. PDECOL is unique because of its flexibiility both in the class of problems it addresses and in the variety of methods it provides for use in the solution process. High-order methods (as well as low-order ones) are readily available for use in both the spatial and time discretization procedures. The time integration methods used feature automatic time step size and integration formula order selection so as to solve efficiently the problem at hand and yet achieve a user-specific time integration error level. PDECOL consists of a collection of 19 subroutines written in reasonably standard Fortran, and therefore is quite portable. No special hardware features are required. PDECOL is designed to solve broad classes of difficult systems of partial differential equations that descrobe physical processes. 4 figures, 1 table. (RWR)

Software for Nonlinear Partial Differential Equations

The numerical solution of physically realistic nonlinear partial differential equations (PDEs) is a complicated and highly problem-dependent process which usually requires the scientist to undertake the difficult and time-consuming task of developing his own computer program to solve his problem. This paper presents a software interface which can eliminate much of the expensive and time-consuming effort involved in the solution of nonlinear PDEs. The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators. Besides being portable, efficient, and easy to use, the software interface along with an ODE integrator will discretize the problem, select the time step and order, solve the nonlinear equations (checking for convergence, etc.), and maintain a user-specified time integration accuracy, all automatically and reliably. Physically realistic examples are given to illustrate the use and capability of the software.

A Three-Dimensional Modified Finite Volume Technique for Maxwell's Equations

Abstract A modified finite volume method for solving Maxwells equations in the time-domain is presented. This method, which allows the use of general nonorthogonal mixed-polyhedral grids, is a direct generalisation of the canonical staggered-grid finite difference method. Employing mixed polyhedral cells, (hexahedral, tetrahedral, etc.) this method allows more accurate modeling of non-rectangular structures. The traditional “stair-stepped” boundary approximations associated with the orthogonal grid based finite difference methods ate avoided. Numerical results demonstrating the accuracy of this new method are presented.

Numerical solution of Maxwell's equations in the time domain using irregular nonorthogonal grids

Abstract Several different methods for solving Maxwells equations in the time-domain through the use of irregular nonorthogonal grids are presented. Employing quadrilateral and/or triangular elements, these methods allow more accurate modeling of nonrectangular structures. The traditional “stair-stepping” boundary approximations associated with standard orthogonal-grid finite-difference methods are avoided. Numerical results comparing all of the methods are given. A modified finite-volume method, which is a direct generalization of the standard finite-difference method to arbitrary polygonal grids, is shown to be the most accurate.

Matrix multiplication by diagonals on a vector/parallel processor☆

The advent of vector and parallel computers has forced the reexamination, reformulation, and 9 rethinking of essentially all of the basic mathematical algorithms. In this paper, we will bonsider the seemingly straightforward process of matrix multiplication. We will be primarily concerned with this process on a vector processor, the CDC STAR-100. For large full matrices, matrix *multiplication is easily “vectorized” when the matrix is stored by columns in the typical Fortran fashion. However, there are at least two disadvantages to this normal approach. First, it becomes quite inefficient for banded matrices with relatively narrow bandwidths. Second, when a matrix is stored by columns (or rows), the transpose of the matrix is not as readily available for use in a vector form (this is particularly a problem on the STAR-100). The purpose of this paper is to present a new algorithm for matrix multiplication which: is readily “‘vectorizetl”, is very efficient for narrow banded matrices, and allows for the transpose to be easily accessible in a vector form.

A mixed finite element formulation for Maxwell's equations in the time domain

Abstract A Galerkin finite element solution technique for the Maxwells equations is discussed. This new formulation can be viewed as a generalization of certain staggered-grid finite difference schemes to arbitrary meshes. It is shown that this technique is simple to implement and is more accurate as well as more cost effective than the standard equal-order finite element approach. Numerical are presented to evaluate the performance of this new element relative to the standard element.

Three-dimensional computer modeling of electromagnetic fields: A global lookback lattice truncation scheme☆

Abstract A new lattice truncation scheme for the finite difference time domain approach to the solution of Maxwells equations has been developed. The problem space is truncated near the sources and the field components on its boundary are generated from those field values known at retarded times on an interior surface one cell from it with an integral representation of the electromagnetic field. The numerical implementation of this global lookback scheme is discussed. Examples which have been used to determine its characteristics and its validity are given.

Algorithm 494: PDEONE, Solutions of Systems of Partial Differential Equations [D3]

SUBROUTINE PDEONE(T, U, UDOT, NPDE, NPTS) DIMENSION U(NPDE,NPTS}, UDOT(NPD£,NPTS) C PDEONE IS AN INTERFACE SUBROUTINE WHICH USES CENTERED DIFC ~ERENCE APPROXIMATIONS TO CONVERT ONE-DIMENSIONAL SYSTEMS C OF PARTIAL DIFFERENTIAL EQUATIONS INTO A SYSTEM OF ORDINARY C DIFFERENTIAL EQUATIONS, UDOT = F(T,X,U). THIS ROUTINE IS C INTENDED TO BE USED WITH A ROBUST ODE INTEGRATOR. C INPUT.. C NPDE = NUMBER OF PARTIAL DIFFERENTIAL EQUATIONS. C NPTS = NUMBER OF SPATIAL GRID POINTS. C ~ = CURRENT VALUE OF TIME. C U = AN NPDE BY NPTS ARRAY CONTAINING THE COMPUTED C SOLUTION AT ~IME 5. C OUTPUT.. C UDOT = AN NPDE BY NPTS ARRAY CONTAINING THE RIGHT HAND C SIDE OF THE RESULTING SYSTEM OF ODE*S, F(T,X,U), C OBTAINED BY DISCREqlZING THE GIVEN PDE*S. PDE ig PDE 20 PDE 30 PDE 40 PDE 50 PDE 6@ PDE 70 PDE 80 PDE 90 PDE 10g PDE 110 PDE 120 PDE 130 PDE 14g PDE 150 PDE 16g PDE 170

General Software for Partial Differential Equations

Several general purpose digital computer programs for numerically solving partial differential equations (PDEs) have been developed. This chapter discusses the possible advantages of PDECOL, which is a new general purpose digital computer program for numerically solving partial differential equations. The most significant feature of PDECOL is the spatial discretization technique that is implemented. The discretization technique can best be described as a finite element collocation method that uses piecewise polynomials for the trial function space. The chapter highlights the use of PDECOL to solve four different PDE problems illustrating few advantages and disadvantages of the use of higher order methods. PDECOL is an extremely versatile and unique software package. It can reliably solve a very broad class of nontrivial and nonlinear systems of partial differential equations. Its higher order methods can produce extremely accurate solutions quite efficiently when compared to lower order methods. The package is quite portable and very easy to use.

Odd-Even Reduction for Banded Linear Equations

The method of odd-even reduction for tridiagonal systems is generalized to banded systems. The method is developed so that it can be easily implemented on a vector processor such as the CDC STAR-100. Results are presented which describe when this odd-even reduction can be performed on a pentadiagonal system. A computational example is given. 1 table.