Wayne Enright
University of Toronto, Computer Science, Faculty Member
... 19970849 Paper first received 20th May and in revised form 20th August 1996 W. Enright is with the Electricity Corporation of New Zealand J ... A lOkW sine-wave generator suplies the test transformer, which is loaded with a series... more
... 19970849 Paper first received 20th May and in revised form 20th August 1996 W. Enright is with the Electricity Corporation of New Zealand J ... A lOkW sine-wave generator suplies the test transformer, which is loaded with a series resonant circuit tuned to the fifth harmonic. ...
Research Interests:
In recent years we have developed reliable order $p$ methods for the approximate solution of general systems of IVPs and DDEs. We have used these methods to implement effective techniques for parameter estimation and sensitivity analysis... more
In recent years we have developed reliable order $p$ methods for the approximate solution of general systems of IVPs and DDEs. We have used these methods to implement effective techniques for parameter estimation and sensitivity analysis for IVPs and DDEs. The two techniques we have investigated are the "variational approach" and the "adjoint approach". We will discuss the cost and reliability of each approach and identify several factors that contribute to the performance of each. In particular we will discuss why, for each technique, the underlying numerical IVP or DDE solver must exploit inherent parallelism if the number of state variables, the number of parameters, or the number of data points is large.
Research Interests:
Research Interests:
... the error at the endpoint, since it depends on the stability of the differential equation and the ... As a rule, algorithms for solving problems in linear algebra are compared on how well they ... on how well they control the... more
... the error at the endpoint, since it depends on the stability of the differential equation and the ... As a rule, algorithms for solving problems in linear algebra are compared on how well they ... on how well they control the difference between the true solution and the computed solution. ...
Research Interests:
Research Interests:
A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order... more
A long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h2k), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C0 continuous solution approximation that has a global error of O(hk+1). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C1 continuous "superconvergent" interpolants whose global errors are O(h2k). The key ideas are to use the theoretical framework ...
Research Interests:
The equal width wave (EW) equation is a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. The EW-Burgers equation models the propagation of nonlinear... more
The equal width wave (EW) equation is a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. The EW-Burgers equation models the propagation of nonlinear and dispersive waves with certain dissipative eects. In this work, we derive exact solitary wave solutions for the general form of the EW equation and the generalized EW-Burgers equation with nonlinear terms of any order. We also derive analytical expressions of three invariants of motion for solitary wave solutions of the generalized EW equation.
In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or... more
In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or linear multi-step methods, and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. Using this interpretation we develop an efficient technique to solve the resulting discontinuous IVP. We also give a more clear process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes. The new modular design for the resulting simulator we introduce, helps to accelerate the utilization of advances in the different components of an effective numerical method. Such components include the underlying discrete formula, the interpolant for dense output, the strategy for handling discontinuities and ...
Research Interests:
Using a Difierential Equation Interpolant (DEI), one can accurately approx- imate the solution of a Partial Difierential Equation (PDE) at ofi-mesh points. The idea is to allocate a multi-variate polynomial to each mesh element and... more
Using a Difierential Equation Interpolant (DEI), one can accurately approx- imate the solution of a Partial Difierential Equation (PDE) at ofi-mesh points. The idea is to allocate a multi-variate polynomial to each mesh element and consequently, the collection of such polynomials over all mesh elements will deflne a piecewise polynomial approximation. In this paper we will investigate such interpolants on a three-dimensional unstructured mesh. As reported in (1), for a tetrahedron mesh in three dimensions, tensor product tri-quadratic and pure tri-cubic interpolants are the most appropriate candidates. We will report on the efiectiveness of these alternatives on some typical PDEs.
Research Interests:
Research Interests:
Scientific computation is often carried out in general purpose problem solving environments (PSEs) such as MATLAB or MAPLE or in application-specific PSEs such as those associated with large scale, high performance simulations. When... more
Scientific computation is often carried out in general purpose problem solving environments (PSEs) such as MATLAB or MAPLE or in application-specific PSEs such as those associated with large scale, high performance simulations. When practitioners work in these environments it is essential that they have access to state-of-the-art numerical software and tools to visualize approximate solutions produced by such software. It is equally essential (although not yet standard practice) to have available tools that can be used to verify (or validate) that the approximate solution is consistent with the true solution of the underlying mathematical model. In this presentation we will identify a suite of tools that are being developed to address this need for the case where the underlying model is a system of ODEs or PDEs.
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Research Interests: Applied Mathematics, Visualization, Software Development, Numerical Analysis, Numerical Method, and 14 moreVerification, Reliability, Applied Mathematics and Computational Science, Visualisation, Mathematical Model, Defect, Odes, Numerical Analysis and Computational Mathematics, Indexation, Large Scale, Differential equation, Electrical And Electronic Engineering, Ordinary Differential Equation, and Test Method
Research Interests:
Research Interests:
ABSTRACT Research in explicit Runge-Kutta methods is producing continual improvements to the original algorithms, and the aim of this survey is to relate the current state-of-the-art. In drawing attention to recent advances, we hope to... more
ABSTRACT Research in explicit Runge-Kutta methods is producing continual improvements to the original algorithms, and the aim of this survey is to relate the current state-of-the-art. In drawing attention to recent advances, we hope to provide useful information for those who apply numerical methods. We describe recent work in the derivation of Runge-Kutta coefficients: "classical" general-purpose formulas, "special" formulas for high order and Hamiltonian problems, and "continuous" formulas for dense output. We also give a thorough review of implementation details. Modern techniques are described for the tasks of controlling the local error in a step-by-step integration, computing reliable estimates of the global error, detecting stiffness, and detecting and handling discontinuities and singularities. We also discuss some important software issues. 1 Introduction Explicit Runge-Kutta (ERK) formulas are among the oldest and best-understood schemes in the numerical analyst's toolkit. H...
Research Interests:
Research Interests:
Research Interests:
SECOND DERIVATIVE MULTISTEP METHODS 323 Fig. 1. Region Rl U R2 associated with stiff stability possibly oscillatory components. This property can restrict the step size, especially if there are several transients. Formulas that do not... more
SECOND DERIVATIVE MULTISTEP METHODS 323 Fig. 1. Region Rl U R2 associated with stiff stability possibly oscillatory components. This property can restrict the step size, especially if there are several transients. Formulas that do not suffer from this property will be called " ...
Research Interests:
ABSTRACT A higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations. In this paper, we introduce an adaptive stepsize selection strategy resulting in an... more
ABSTRACT A higher level of realism can be achieved by incorporating distributed delays in the mathematical models described by differential equations. In this paper, we introduce an adaptive stepsize selection strategy resulting in an approximate solution whose associated defect (residual) satisfies certain properties that allow us to monitor the global error reliably and efficiently. In addition, a companion system of equations is introduced in order to estimate the mathematical conditioning of the problem. A side effect of introducing this companion system is that it provides a global error estimate, at a modest increase in cost. The significance of our method will be demonstrated through real problems from population dynamics and actuarial sciences.
Research Interests:
ABSTRACT A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the... more
ABSTRACT A popular approach to the numerical solution of boundary value ODE problems involves the use of collocation methods. Such methods can be naturally implemented so as to provide a continuous approximation to the solution over the entire problem interval. On the other hand, several authors have suggested as an alternative, certain subclasses of the implicit Runge--Kutta formulas, known as mono-implicit Runge--Kutta (MIRK) formulas, which can be implemented at a lower cost per step than the collocation methods. These latter formulas do not have a natural implementation that provides a continuous approximation to the solution; rather, only a discrete approximation at certain points within the problem interval is obtained. However, recent work in the area of initial value problems has demonstrated the possibility of generating inexpensive interpolants for any explicit Runge--Kutta formula. These ideas have recently been extended to develop continuous extensions of the MIRK formulas. In this paper, we describe our investigation of the use of continuous MIRK formulas in the numerical solution of boundary value ODE problems. A primary thrust of this investigation is to consider defect control, based on the continuous MIRK formulas, as an alternative to the standard use of global error control, as the basis for termination and mesh redistribution criteria.
Research Interests:
There has been considerable recent progress in the analysis and development of interpolation schemes that can be associated with discrete Runge-Kutta methods. With the availability of these schemes it can now be asked that a numerical... more
There has been considerable recent progress in the analysis and development of interpolation schemes that can be associated with discrete Runge-Kutta methods. With the availability of these schemes it can now be asked that a numerical method provide a ...
Research Interests:
Research Interests: Applied Mathematics, Numerical Method, Problem Solving, Error Control Coding, Implementation, and 10 moreInterpolation, Numerical Analysis and Computational Mathematics, Differential equation, Numerical Solution, Boundary Value Problem, Runge Kutta, Formulas de interpolación, Ordinary Differential Equation, Runge Kutta Methods, and Interpolation Formula
ABSTRACT The standard method of multiple shooting for a system of n first-order differential equations with k unknown initial conditions requires the integration of k sets of variational equations on the first shot and n sets of... more
ABSTRACT The standard method of multiple shooting for a system of n first-order differential equations with k unknown initial conditions requires the integration of k sets of variational equations on the first shot and n sets of variational equations on every shot thereafter. This paper describes a variant of multiple shooting that requires the solution of k sets of variational equations on every shot. The technique applies to both linear and nonlinear boundary-value problems. Techniques to deal with difficulties unique to the solution of nonlinear problems are suggested.