Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations

Frank, Reinhard; Ueberhuber, Christoph W.

doi:10.1007/bf01932286

Cited by 79 publications

(53 citation statements)

References 9 publications

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“…Instead, a piecewise polynomial function composed of polynomials of (moderate) degree m is defined to specify the neighboring problem. There are two different ways in which we can extend the solution from a subinterval where one Zadunaisky polynomial is used to the next interval, see for example [9]. For the global connection strategy, we apply the implicit Euler method on the whole integration interval, and interpolate the values with a continuous, piecewise function p [0] (t).…”

Section: The Numerical Methodsmentioning

confidence: 99%

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Analytical and numerical treatment of a singular initial value problem in avalanche modeling

Koch

Weinmüller

2004

Applied Mathematics and Computation

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We discuss a leading-edge model used in the computation of the run-out length of dry-flowing avalanches. The model has the form of a singular initial value problem for a scalar ordinary differential equation describing the avalanche dynamics. Existence, uniqueness and smoothness properties of the analytical solution are shown. We also prove the existence of a unique root of the solution. Moreover, we present a FORTRAN 90 code for the numerical computation of the run-out length. The code is based on a solver for singular initial value problems which is an implementation of the acceleration technique known as Iterated Defect Correction based on the implicit Euler method.

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Section: The Numerical Methodsmentioning

confidence: 99%

“…Apparently, this variant suffers from similar drawbacks as many common high-order one-step methods for singular problems (see for example [12]), because restarting the procedure on every subinterval amounts to the same as applying a one-step method of order m, see [9].…”

Section: Further Iteration Does Not Increase the Asymptotic Order Of mentioning

confidence: 99%

Analytical and numerical treatment of a singular initial value problem in avalanche modeling

Koch

Weinmüller

2004

Applied Mathematics and Computation

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“…Frank and Ueberhuber [13] describe the use of iterated defect correction and a variety of schemes have been discussed by Butcher [3]. He suggested how higher order methods could be used in combination with diagonally implicit methods through an iterated defect correction process.…”

Section: Introductionmentioning

confidence: 99%

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

Vigneswaran¹,

Kajanthan²

2017

Int. J. of Pure and Appl. Math.

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The non-linear equations, when implementing implicit Runge-Kutta methods, may be solved by a modified Newton scheme and by several linear iteration schemes which sacrificed superlinear convergence for reduced linear algebra costs. A linear scheme of this type was proposed, which requires some additional computation in each iteration step. The rate of convergence of this scheme is examined when it is applied to the scalar test problem

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“…Frank and Ueberhuber [13] describe the use of iterated defect corrections. A variety of other iteration schemes have been discussed by Butcher [6].…”

Section: Introductionmentioning

confidence: 99%

On the Implementation of Singly Implicit Runge-Kutta Methods

Cooper¹

1991

Mathematics of Computation

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Abstract. A modified Newton method is often used to solve the algebraic equations that arise in the application of implicit Runge-Kutta methods. When the Runge-Kutta method has a coefficient matrix A with a single point spectrum (with eigenvalue X), the efficiency of the modified Newton method is much improved by using a similarity transformation of A . Each iteration involves vector transformations. In this article an alternative iteration scheme is obtained which does not require vector transformations and which is simpler in other respects also. Both schemes converge in a finite number of iterations when applied to linear systems of differential equations, but the new scheme uses the nilpotency of A -XI to achieve this. Numerical results confirm the predicted convergence for nonlinear systems and indicate that the scheme may be a useful alternative to the modified Newton method for low-dimensional systems. The scheme seems to become less effective as the dimension increases. However, it has clear advantages for parallel computation, making it competitive for high-dimensional systems.

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Iterated defect correction for the efficient solution of stiff systems of ordinary differential equations

Cited by 79 publications

References 9 publications

Analytical and numerical treatment of a singular initial value problem in avalanche modeling

Analytical and numerical treatment of a singular initial value problem in avalanche modeling

Improving Rate of Convergence of an Iterative Scheme With Extra Sub-Steps for Two Stage Gauss Method

On the Implementation of Singly Implicit Runge-Kutta Methods

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