The method of iterated defect-correction and its application to two-point boundary value problems

Applied Mathematics and Computation

Weinmüller

2004

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.

Section: The Numerical Methodsmentioning

confidence: 99%

“…. , N. Now p [1] (t) is used to construct a new neighboring problem analogous to (8), where again the exact solution is known, and the numerical solution of this neighboring problem serves to obtain the second improved solution v [2] h :…”

Section: The Numerical Methodsmentioning

confidence: 99%

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

Applied Mathematics and Computation

Weinmüller

2004

“…First, we describe the classical version of Iterated Defect Correction (IDeC) [3]. Consider an initial value problem in n dimensions y 0 ðtÞ ¼ f ðt; yðtÞÞ; yðt 0 Þ ¼ y 0 ; ð1Þ to be solved on the interval ½t 0 ; t end .…”

Section: Iterated Splitting Defect Correctionmentioning

confidence: 99%

“…The idea can also be used to successively improve the accuracy of the numerical solution ( [1,3], and the references therein). 1 In this acceleration technique, a number of neighboring problems have to be solved, which are not necessarily of the same type as the original problem.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a defect correction method for geometric integrators

Hofstätter

2006

Numer Algor

We discuss a new variant of Iterated Defect Correction (IDeC), which increases the range of applicability of the method. Splitting methods are utilized in conjunction with special integration methods for Hamiltonian systems, or other initial value problems for ordinary differential equations with a particular structure, to solve the neighboring problems occurring in the course of the IDeC iteration. We demonstrate that this acceleration technique serves to rapidly increase the convergence order of the resulting numerical approximations, up to the theoretical limit given by the order of certain superconvergent collocation methods.

“…Therefore, we use Iterated Defect Correction (IDeC) based on the implicit Euler method for the solution of the involved initial value problems. This acceleration technique was investigated for regular problems in [9], [10] and [11]. When applied to singular initial value problems, this method shows its classical convergence behavior, which means that any convergence order O(h p ) can be obtained comparatively cheaply for sufficiently smooth data; see [22] for the analysis and [2], [21] for numerical evidence.…”

Section: Y (T) = M (T) T Y(t) + F (T Y(t)) T ∈ (0 1] (11a)mentioning

confidence: 99%

The convergence of shooting methods for singular boundary value problems

Weinmüller²

2001

Math. Comp.

Abstract. We investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems. Particular attention is paid to the well-posedness of the process. It is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available. Moreover, convergence results for a perturbed Newton iteration are discussed. PreliminariesWe discuss the numerical solution of the following nonlinear singular boundary value problems of the first order:where y and f are vector-valued functions of dimension n, M is an n × n matrix and g : R n × R n → R r , r ≤ n, is a smooth function.The search for a method to solve problems (1.1) is strongly motivated by numerous applications from physics (see [3] Our aim is to investigate the convergence of shooting procedures (see [1] or [18]) for the approximate solution of (1.1), based on an efficient numerical solution of the associated singular initial value problems.The unsatisfactory convergence properties of direct higher order methods are the main motivation to use shooting methods for the solution of singular boundary value problems. For collocation schemes only the stage order can be observed; the superconvergence does not hold in general, see [16]. In the presence of a singularity, other direct higher order methods (finite differences) show order reductions and become inefficient. Moreover, the standard acceleration techniques based on low-order methods do not work efficiently either, because in general, a proper asymptotic error expansion for the basic scheme does not exist, cf. [12]. Moreover, multiple shooting seems to be a particularly attractive alternative, because within its framework one can use different controlling mechanisms close to and away from