Patrick Keast

Algorithm 688: EPDCOL: a more efficient PDECOL code

The software package PDECOL [7] is a popular code among scientists wishing to solve systems of nonlinear partial differential equations. The code is based on a method-of-lines approach, with collocation in the space variable to reduce the problem to a system of ordinary differential equations. There are three principal components: the basis functions employed in the collocation; the method used to solve the system of ordinary differential equations; and the linear equation solver which handles the linear algebra. This paper will concentrate on the third component, and will report on the improvement in the performance of PDECOL resulting from replacing the current linear algebra modules of the code by modules which take full advantage of the special structure of the equations which arise. Savings of over 50 percent in total execution time can be realized.

Remark on “Algorithm 603: COLROW and ARCECO: FORTRAN Packages for Solving Certain Almost Block Diagonal Linear Systems by Modified Alternate Row and Column Elimination”

In the numerical solution of boundary-value problems for ordinary differential equations, hnear systems with a partmular block structure are encountered. In this paper two packages of FORTRAN subroutines, COLROW and ARCECO, which use alternate row and column elimination for solving such systems, as discussed by Varah, are described Varahs procedure is stable, and introduces no fillin, tha t is, reqmres no additional storage To improve its efficiency, use is made of the fact tha t after each sequence of eliminations a reducible matrix is obtained. In addition, the packages presented treat systems with a more general structure than those considered by Varah. The results of numerical experiments, which demonstrate the effectiveness of the new packages and their superiority over a prevmusly pubhshed code, on a suitably restrmted class of problems, are presented

On the Structure of Fully Symmetric Multidimensional Quadrature Rules

One method for constructing fully symmetric quadrature rules of specified moderate polynomial degree for fully symmetric integration regions consists of solving directly the possibly large system of moment fitting equations which define the problem. A familiar hazard is to find these equations are inconsistent.In a recent fundamental paper, Mantel and Rabinowitz [4] (SIAM J. Numer. Anal., 1977) treated this problem in some detail in a three dimensional context. They defined and calculated a set of consistency conditions and using these, systematized to a significant extent the present state of the art for two and three dimensional fully symmetric quadrature rules.This paper is complementary to [4]. We introduce a set of null spaces. Using these, the calculations of the consistency conditions which is in general tedious, can be reduced to an automatic linear procedure in a way which makes human error significantly less likely. In addition, these spaces may be used in the actual organization of a calculatio...

The solution of almost block diagonal linear systems arising in spline collocation at Gaussian points with monomial basis functions

Numerical techniques based on piecewise polynomial (that is, spline) collation at Gaussian points are exceedingly effective for the approximate solution of boundary value problems, both for ordinary differential equations and for time dependent partial differential equations. There are several widely available computer codes based on this approach, all of which have at their core a particular choice of basis representation for the piecewise polynomials used to approximate the solutions. Until recently, the most popular approach was to use a B-spline representation, but it has been shown that the B-spline basis is inferior, both in operation counts and conditioning, to a certain monomial basis, and the latter has come more into favor. In this paper, we describe a linear algebraic equations which arise in spline collocation at Gaussian points with such a monomial basis. It is shown that the new package, which implements an alternate column and row pivoting algorithm, is a distinct improvement over existing packages from the points of view of speed and storage requirements. In addition, we describe a second package, an important special case of the first, for solving the almost block diagonal systems which arise when condensation is applied to the systems arising in spline collocation at Gaussian points, and also in other methods for solving two-point boundary value problems, such as implicit Runge-Kutta methods and multiple shooting.

Finite difference solution of the third boundary problem in elliptic and parabolic equations

SummaryFinite difference methods (including the Peaceman-Rachford method) are considered for the solution of the third boundary value problem for parabolic and elliptic equations. Conditions on the coefficients involved in the boundary conditions are obtained from the stability requirements of the difference methods and shown to coincide with those necessary for asymptotic stability of the differential system.

Onp-generator fully symmetric quadrature rules

SummaryWe consider fully symmetric quadrature rules for fully symmetricn-dimensional integration regions. When the region is a product region it is well known that product Gaussian rules exist. These obtain an approximation of polynomial degree 4p+1 based on (2p+1)n function values arranged on a rectangular grid. We term rules using such a grid,p-generator rules. In this paper we determine the necessary conditions on the region of integration forp-generator rules of degree 4p+1 to exist. Regions with this property are termed PropertyQ regions and besides product spaces, this class includes the hypersphere and other related regions.

The third boundary value problem for elliptic equations

In [I], the author discussed the numerical solution of the heat conduction equation in an open rectangular region, under boundary conditions involving a linear combination of the function and its normal derivative. It was shown that the instability which was observed in the difference methods examined, could be traced to the existence of solutions of the differential equation which grew, asymptotically, with time. This numerical instability was important in the solution of the heat equation only for large values of the time, and so did not affect calculations which were carried out over a few time steps. But, in the numerical solution of elliptic equations by iteration, such asymptotic instability will prevent convergence of the iterative process to the original system of equations. It is the purpose of this note to demonstrate this fact, and also to discuss the solution of Laplaces and Poissons equations, when the Laplacian operator is singular, in a sense to be defined. It wilt be shown that, for certain boundary conditions, the numerical solution of Laplaces equation is best obtained by direct methods, rather than by iterative methods. In addition, a condition for the existence of a solution of the singular problem is obtained. 2. The Difference Equations Let D be any closed region in (x, y) space, with boundary aD. The outward normal to D on aD is denoted by n. Consider the equation ~: + -~- = ¢ (x, y)

ON THE H-1-GALERKIN METHOD FOR SECOND-ORDER LINEAR TWO-POINT BOUNDARY VALUE PROBLEMS*

We discuss several aspects of the

The stability of difference approximations to a selfadjoint parabolic equation, under derivative boundary conditions

H^{ - 1}

Numerical integration : recent developments, software, and applications

-Galerkin method for the approximate solution of two-point boundary value problems. First we present a formulation of the method for second-order linear equations subject to general linear separated boundary conditions. it is shown that corresponding to such boundary conditions there are adjoint boundary conditions which must be satisfied by elements of the test space. Optimal error estimates for the new