The numerical solution of parabolic differential equations using variational methods

Summary The question of constructing stable numerical representations for the solutions of initial-boundary value problems for parabolic differential equations is examined.An earlier formulation and discussion of this work can be found in the Author's Ph.D. Thesis (University of Adelaide, South Australia, 1967).     

Two-step almost collocation methods for ordinary differential equations

A new class of two-step Runge-Kutta methods for the numerical solution of ordinary differential equations is proposed. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. Local error estimation for these methods is also discussed.     

Two-step fourth orderP-stable methods for second order differential equations

A family of symmetric (hybrid) two-step fourth order methods is derived fory'=f(x,y). We then show the existence of a sub-family of these methods which when applied toy'=– ² y, real, areP-stable. We also note that a general (order) symmetric two-step method isP-stable iff it is unconditionally stable.     

Two-step fourth orderP-stable methods for second order differential equations

A family of two-step fourth order methods, which requires two function evaluations per step, is derived fory=f(x,y). We then show the existence of a sub-family of these methods which when applied toy=–k ² y,k real, areP-stable.     

Trees and numerical methods for ordinary differential equations

This paper presents a review of the role played by trees in the theory of Runge–Kutta methods. The use of trees is in contrast to early publications on numerical methods, in which a deceptively simpler approach was used. This earlier approach is not only non-rigorous, but also incorrect. It is now known, for example, that methods can have different orders when applied to a single equation and when applied to a system of equations; the earlier approach cannot show this. Trees have a central role in the theory of Runge–Kutta methods and they also have applications to more general methods, involving multiple values and multiple stages.     

Higher order numerical methods for solving fractional differential equations

In this paper we introduce higher order numerical methods for solving fractional differential equations. We use two approaches to this problem. The first approach is based on a direct discretisation of the fractional differential operator: we obtain a numerical method for solving a linear fractional differential equation with order 0<α<1. The order of convergence of the numerical method is O(h ^3?α ). Our second approach is based on discretisation of the integral form of the fractional differential equation and we obtain a fractional Adams-type method for a nonlinear fractional differential equation of any order α>0. The order of convergence of the numerical method is O(h ³) for α≥1 and O(h ^1+2α ) for 0<α≤1 for sufficiently smooth solutions. Numerical examples are given to show that the numerical results are consistent with the theoretical results.     

Stability analysis of numerical methods for delay differential equations

Summary This paper deals with the stability analysis of step-by-step methods for the numerical solution of delay differential equations. We focus on the behaviour of such methods when they are applied to the linear testproblemU(t)=U(t)+U(t–) with >0 and , complex. A general theorem is presented which can be used to obtain complete characterizations of the stability regions of these methods.     

Collocation methods for parabolic partial differential equations in one space dimension

Summary Collocation at Gaussian points for a scalarm-th order ordinary differential equation has been studied by C. de Boor and B. Swartz. J. Douglas, Jr. and T. Dupont, using collocation at Gaussian points, and a combination of energy estimates and approximation theory have given a comprehensive theory for parabolic problems in a single space variable. While the results of this report parallel those of Douglas and Dupont, the approach is basically different. The Laplace transform is used to lift the results of de Boor and Swartz to linear parabolic problems. This indicates a general procedure that may be used to lift schemes for elliptic problems to schemes for parabolic problems. Additionally there is a section on longtime integration and A-stability.Supported by the Office of Naval Research under contract N-00014-67-A-0128-0004     

High-order adaptive finite element-singly implicit Runge-Kutta methods for parabolic differential equations

We describe an adaptive mesh refinement finite element method-of-lines procedure for solving one-dimensional parabolic partial differential equations. Solutions are calculated using Galerkin's method with a piecewise hierarchical polynomial basis in space and singly implicit Runge-Kutta (SIRK) methods in time. A modified SIRK formulation eliminates a linear systems solution that is required by the traditional SIRK formulation and leads to a new reduced-order interpolation formula. Stability and temporal error estimation techniques allow acceptance of approximate solutions at intermediate stages, yielding increased efficiency when solving partial differential equations. A priori energy estimates of the local discretization error are obtained for a nonlinear scalar problem. A posteriori estimates of local spatial discretization errors, obtained by order variation, are used with the a priori error estimates to control the adaptive mesh refinement strategy. Computational results suggest convergence of the a posteriori error estimate to the exact discretization error and verify the utility of the adaptive technique.This research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR-90-0194; the U.S. Army Research Office under Contract Number DAAL 03-91-G-0215; by the National Science Foundation under Grant Number CDA-8805910; and by a grant from the Committee on Research, Tulane University.     

Order conditions for numerical methods for partitioned ordinary differential equations

Summary Motivated by the consideration of Runge-Kutta formulas for partitioned systems, the theory of P-series is studied. This theory yields the general structure of the order conditions for numerical methods for partitioned systems, and in addition for Nyström methods fory=f(y,y), for Rosenbrock-type methods with inexact Jacobian (W-methods). It is a direct generalization of the theory of Butcher series [7, 8]. In a later publication, the theory ofP-series will be used for the derivation of order conditions for Runge-Kutta-type methods for Volterra integral equations [1].     

Monotone iterative methods for infinite systems of parabolic functional differential equations

Two monotone iterative methods for an infinite system of parabolic functional differential equations with initial boundary conditions are constructed: the method of direct iterations and the Chaplygin method. By using the first one the existence theorem is proved. Next, it is shown that the Chaplygin sequences converge quadratically to the unique solution of the original problem.     

A Robust numerical approach for singularly perturbed time delayed parabolic partial differential equations

We consider the numerical solution of a initial boundary value problem with a time delay. The problem under consideration is singularly perturbed from the mathematical perspective. Assuming that the coefficients of the differential equation are smooth, we construct and analyze the finite difference method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameter. The method permits its extension to the case of adaptive meshes, which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practice satisfies the theoretical predictions.     

Convergence of numerical schemes for the solution of parabolic stochastic partial differential equations

We consider the numerical solution of the stochastic partial differential equation , where is space-time white noise, using finite differences. For this equation Gyöngy has obtained an estimate of the rate of convergence for a simple scheme, based on integrals of over a rectangular grid. We investigate the extent to which this order of convergence can be improved, and find that better approximations are possible for the case of additive noise ( ) if we wish to estimate space averages of the solution rather than pointwise estimates, or if we are permitted to generate other functionals of the noise. But for multiplicative noise ( ) we show that no such improvements are possible.

     

The complexity of numerical methods for elliptic partial differential equations

We consider three Ritz-Galerkin procedures with Hermite bicubic, bicubic spline and linear triangular elements for approximating the solution of self-adjoint elliptic partial differential equations and a collocation with Hermite bicubics method for general linear elliptic equations defined on general two dimensional domains with mixed boundary conditions. We systematically evaluate these methods by applying them to a sample set of problems while measuring various performance criteria. The test data suggest that collocation is the most efficient method for general use.     

Mean-square convergence of numerical methods for random ordinary differential equations

Numerical Algorithms - The aim of this work is to analyze the mean-square convergence rates of numerical schemes for random ordinary differential equations (RODEs). First, a relation between the...     

Stability properties of numerical methods for solving delay differential equations

Stability properties of numerical methods for delay differential equations are considered. Some suitable definitions for the stability of the numerical methods are included and Runge-Kutta type methods satisfying these properties are tested on a numerical example.