On the Structure of Alternating Direction Implicit (A.D.I.) and Locally One Dimensional (L.O.D.) Difference Methods

The general structure of A.D.I. and L.O.D. difference schemesis considered with regard to their construction for time dependentproblems in two and three space dimensions. By considering approximationsto exp {k(L+M)} where L and M are differential operators inthe space variables and k is the time step, we show how severalknown schemes can be viewed as having come from this type ofapproximation. In addition several new schemes based on thistype of approximation are suggested. The arguments used areentirely informal and no attempt is made to prove the stabilityor convergence of the various schemes. Our aim is merely topoint out a possible structure for the generation of A.D.I.and L.O.D. difference schemes.     

Modified Locally One Dimensional Methods for Parabolic Partial Differential Equations in Two Space Variables

In this note we consider the locally one dimensional methodas discussed in Gourlay & Mitchell (1969, 1972) and indicatehow this scheme can be used to solve accurately a parabolicpartial differential equation in two space variables.     

Rank-one and Rank-two Corrections to Positive Definite Matrices Expressed in Product Form

It is shown that certain rank-one and rank-two corrections tosymmetric positive definite matrices may be expressed in theform of a product. This product form gives control over thepositive definiteness, determinant value and conditioning ofthe corrected matrix. An application to updating formulae ofquasi-Newton methods for unconstrained minimization is discussed.     

Linear Combinations of Generalized Crank Nicolson Schemes

The combination of two or more generalized Crank Nicolson schemesin order to obtain second, third and fourth order accurate discretizationsin time is considered. Particular attention is given to thestability properties of the methods proposed.     

Hopscotch: a Fast Second-order Partial Differential Equation Solver

Present address: IBM (United Kingdom) Ltd., Peterlee, Co. Durham, England. An idea of Gordon for the numerical solution of evolutionaryproblems is reformulated and shown to be equivalent to a Peaceman-Rachfordprocess. A fast computational process is then developed andapplied to parabolic and elliptic problems, both linear andnon-linear. This algorithm is very efficient with regard tocomputing time, storage requirements and ease of programming.Several fairly general conditions are given which ensure convergencefor parabolic and elliptic problems.     

The Solution of a Two-dimensional Time-dependent Diffusion Problem Concerned with Oxygen Metabolism in Tissues

An approximate numerical solution of the diffusion equationin a semi-infinite two-dimensional domain is ibtained usingfa hopscotch technique. The solution describes the movement ofoxygen in a metabolizing medium covered by a finite layer ofanother non-metabolizing material. The initial condition isdetermined by the steady state resulting from a uniform concertrationof oxygen concerntration within a circular area r R in thebounding surface, and zero oxygen gradient normal to the surfaceforr > R. The time-dependence of oxygen flux into the circularares is obtained by an interpolation and integration method.     

Block Hopscotch Procedures for Second Order Parabolic Differential Equations

The authors consider a general block implicit hopscotch procedurefor solving parabolic partial differencial equations. Stabilityand convergence analyses are given and an examination of thecomparative accuracy of various particular cases is given. Theusefulness of these algorithms in solving problems using (r)geometry is indicated and their extensions to general exteriorproblems using a peripheral approach is included.     

On Chebychev Acceleration Procedures for Alternating Direction Iterative Methods

The solution of large sparse systems of linear equations arising,for example, from the numerical solution of elliptic partialdifferential equations is considered, with reference to theacceleration technique commonly known as Chebychev acceleration.In particular its application to alternating direction iterative(A.D.I.) methods is compared with the more standard techniquessuch as successive overrelaxation. It is conjectured that inmost circumstances a suitable A.D.I. strategy is that of applyingChebychev semiiteration to an A.D.I. process with a single A.D.I.parameter. It is shown that under general conditions this procedure maysometimes produce faster convergence than the usual multiparameterA.D.I. procedure.