Methods For Stiff Differential Equations Research Articles

On the Stability of Predictor-Corrector Methods for Parabolic Equations with Delay

Diffusion problems where the current state depends upon an earlier one give rise to parabolic equations with delay. The efficient numerical solution of classical parabolic equations can be accomplished via methods for stiff differential equations; one such class is predictor-corrcctor-type methods with extended real stability intervals and with reduced storage requirements. Analogous methods for equations with delay are proposed and analysed here. Our analysis will be based on the test equation y(t)=q 1 y(t)+q 2 y(t-ω), where, in view of the class of parabolic delay equations we want to consider, our main interest will be in the case |q 1 |≫|q 2 |. Implementational details of the methods developed are given and numerical results are presented.

A minimum configuration L‐stable fourth‐order non‐autonomous rosenbrock method for stiff differential equations

AbstractOne class of methods for solving stiff systems of ordinary differential equations is the so‐called Rosenbrock method. A fourth‐order method which requires the minimum computational work per step is described.

Norm bounds for rational matrix functions

If the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then ?f(A)?2?1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Pade approximations which are notA-stable.

Comparison of Numerical Methods for Stiff Differential Equations in Biology and Chemistry

Comparison of Numerical Methods for Stiff Differential Equations in Biology and Chemistry

Introduction to Numerical Analysis.

This well written book is enlarged by the following topics: $B$-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for the $LR$ and $QR$ algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations, preconditioning techniques and convergence rate of the conjugate gradient algorithm and multigrid methods for boundary value problems. Cf. also the reviews of the German original editions.

SomeL-stable methods for stiff differential equations

A recent paper by Steihaug and Wolfbrandt [6] gives a second-order method for stiff differential equations, which includes an error estimate. We show how, in most practical circumstances, the error estimate can be obtained more simply, and derive two third-order methods along similar lines. All the methods described are single-step and linearly implicit, and they are designed to avoid frequent evaluations of the Jacobian and inversion of the associated matrix.

Comments on: T.D. Bui, “on an L-stable method for stiff differential equations”

Comments on: T.D. Bui, “on an L-stable method for stiff differential equations”

Unconditionally stable explicit methods for parabolic equations

This paper discussesrational Runge-Kutta methods for stiff differential equations of high dimensions. These methods are explicit and in addition do not require the computation or storage of the Jacobian. A stability analysis (based onn-dimensional linear equations) is given. A second orderA 0-stable method with embedded error control is constructed and numerical results of stiff problems originating from linear and nonlinear parabolic equations are presented.

Implementation of defect correction methods for stiff differential equations

There is a large gap between the theoretical results about iterated defect corrections (IDeC) and practical implementations of IDeC methods for stiff systems. This paper tries to close this gap by providing general principles which are essential for the construction of efficient IDeC codes. Numerical results gathered with one particular IDeC based experimental code furnish evidence of the inherent power of the defect correction concept in the context of stiff systems of ordinary differential equations.

Optimal stimuli for cochlear outer dendrites

To distinguish the functional difference of the two types of cochlear dendrites, simulation studies were done to discern the AP generating site response for different sound input signals. An outer dendrite receives synaptic input from hair cells that cover 200 microns on the cochlear partition, which is associated with optimal sound input frequency per location on the partition. The synaptic boundary condition is V(0,t) = g(t) * (V(0,t) + K) with synaptic conductance g(t), intra‐extracellular equilibrium potential difference K, and V(x,t) denotes dendrite response potential at location x, time t. The nonlinear boundary condition and nonsimple geometry rule out closed form solutions for the dendrite partial differential equation. Simulations were done to examine AP generating site responses. Preserving the geometry,compartmentalization yielded a set of ordinary differential equations with rate constants differing by several orders of magnitude, which have instabilities when solved by usual numerical integration techniques. The set becomes integrable when using methods for stiff differential equations. In descending order of response effectiveness, best inputs are: all synapses active (frequency band sound input), basal to apical synapses active (frequency downsweeps), and apical to basal synapses active (upsweeps).

Some new methods for stiff differential equations

A numerical method applied to the differential equation y′=λy can lead to a solution of the form , where z and w = λh satisfy an equation P(w, z) = 0. In general P(w,z) is a polynomial in both w and z, of degree M in w and N in z. Existing multistep and Runge-Kuta methods correspond to the cases M = 1 and N = 1 respectively. New methods are fottd by taking MS≧2N≧2. The approach here is first to find a suitable polynomial P(w,z) with the desired stability properties, and then to find a process which leads to this polynomial. Third- and fourth-order A- and L-stable processes are given of the semi-explicit arid linearly implicit types.

On an L-stable method for stiff differential equations

On an L-stable method for stiff differential equations

Characterization of Optimal Stepsize Sequences for Methods for Stiff Differential Equations

The amount of work needed for a numerical method to solve a given problem depends mainly on the stepsize sequence that is used. For a given method and a given problem consider all stepsize sequences that give a global error whose norm is everywhere less than a prescribed tolerance. By an optimal stepsize sequence we mean a sequence that requires fewer steps than any other sequence. A large part of this study contains characterizations of optimal stepsize sequences for some classes of problems. From the characterization of the optimal stepsize sequences for stiff systems one can conclude that the stepsize sequence obtained with a fixed bound on the local error per unit step is far from optimal. Existing numerical methods that use a variable stepsize choose their stepsizes on the basis of either a bound on the local error per step or on a bound on the local error per unit step. For systems of stiff differential equations the stepsize sequence obtained with the first criterion, in general, requires less work than the stepsize sequence obtained with the second. A convenient estimate of the global error for a class of methods for solution of systems of stiff differential equations is described. Further a relation (valid for sufficiently smooth stepsize functions) between a bound on the local error and the global error for fixed and variable order methods is established.

On a dangerous property of methods for stiff differential equations

Two mathematically unstable problems are proposed as tests for numerical methods for stiff differential equations. Several methods failed to detect the instability of the problems and produced invalid solutions that for an unsuspecting user could appear to be quite reasonable.

Methods for stiff differential equations

Under supervision of professor G. Dahlquist different approaches to the numerical solution of stiff differential equations have been studied at our institute. As an introduction to the subject a survey of methods and applications up to 1970 (1) was made.

A-stable, Accurate Averaging of Multistep Methods for Stiff Differential Equations

Several low-order numerical solutions of stiff systems of ordinary differential equations are computed by repeated integration, using a multistep formula with parameters. By forming suitable linear combinations of such solutions, higher-order solutions are obtained. If the parameters are properly chosen the underlying solutions, and thus the higher-order one, can be madAe A-stable and strongly damping with respect to the stiff components of the system. A detailed description is given of an algorithmic implementation of the method, which is computationally efficient. Numerical experiments are carried out on some test problems, confirming the validity of the method.

Introduction to Numerical Analysis

This well written book is enlarged by the following topics: $B$-splines and their computation, elimination methods for large sparse systems of linear equations, Lanczos algorithm for eigenvalue problems, implicit shift techniques for the $LR$ and $QR$ algorithm, implicit differential equations, differential algebraic systems, new methods for stiff differential equations, preconditioning techniques and convergence rate of the conjugate gradient algorithm and multigrid methods for boundary value problems. Cf. also the reviews of the German original editions.