The Defect Correction Approach

Böhmer, Klaus; Hemker, Piet; Stetter, Hans J.

doi:10.1007/978-3-7091-7023-6_1

Cited by 90 publications

(88 citation statements)

References 42 publications

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“…Therefore, one or more DeC iteration steps are sufficient to improve the order of accuracy. This is a well known result which also holds for nonlinear problems [2].…”

Section: N}(w)supporting

confidence: 57%

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Second order accurate upwind solutions of the 2D steady Euler equations by the use of a defect correction method

Spekreijse

1986

Lecture Notes in Mathematics

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In this paper a description is given of first and second order finite volume upwind schemes for the 2D steady Euler equations in generalized coordinates. These discretizations are obtained by projectionevolution stages, as suggested by Van Leer. The first order schemes can be solved efficiently by multigrid methods. Second order approximations are obtained by a defect correction method. In order to maintain monotone solutions, a limiter is introduced for the defect correction method.

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“…Therefore, one or more DeC iteration steps are sufficient to improve the order of accuracy. This is a well known result which also holds for nonlinear problems [2].…”

Section: N}(w)supporting

confidence: 57%

“…Unfortunately however, such an iterative DeC process will converge very slowly or might not even converge at all. On the other hand, it is well known [2] that just one or more DeC iterations are enough to obtain a second order accurate approximation. Therefore we shall apply the DeC iteration steps only a few times.…”

Section: Introductionmentioning

confidence: 99%

Second order accurate upwind solutions of the 2D steady Euler equations by the use of a defect correction method

Spekreijse

1986

Lecture Notes in Mathematics

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“…Further improvement can be obtained by higher-order discretization. This, however, will not influence the rate of convergence of the FAS iteration, if the higher order is obtained by means of the defect correction (1], where only nonlinear systems of the type ( 5 .1) are solved and the second order is obtained by the construction of the proper right-hand side r h.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…In this paper Osher's approximate Riemann-solver is used for the numerical flux f( %' q 1 ) in (2.8). In the remainder of this section we give a short description of this function.…”

Section: Finite Volume Osher Discretization For the 2-d Steady Euler mentioning

confidence: 99%

Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations

Hemker¹,

Spekreijse²

1986

Applied Numerical Mathematics

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An iterative method is developed for the .solution of the steady Euler equations for inviscid flow. The system of hyperbolic conservation laws is discretized by a finite-volume Osher-discretization. The iterative method is a multiple grid (FAS) iteration with symmetric Gauss-Seidel (SGS) as a relaxation method. Initial estimates are obtained by full multigrid (FMG). In the pointwise relaxation the equations are kept in block-coupled form and local linearization of the equations and the boundary conditions is considered. The efficient formulation of Osher's discretization of the 2-D non-isentropic steady Euler equations and its linearization is presented. The efficiency of FAS-SGS iteration is shown for a transonic model problem. It appears that, for the problem considered, the rate of convergence is almost independent of the gridsize and that for all meshsizes the discrete system is solved up to truncation error accuracy in only a few (2 or 3) iteration cycles.

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“…The main idea is as follows: the coarse model optimum approximates efficiently the fine one, the fine model corrects that estimate using the coarse model optimization routine as a workhorse in an iterative procedure. We see a clear analogy with multigrid [1,2] and defect correction [3] algorithms. Only the precise forward problem is solved; inversion is carried out with respect to the coarse model.…”

Section: Introductionmentioning

confidence: 94%

Towards Multi-level Optimization: Space-Mapping and Manifold-Mapping

Echeverria¹,

Tong²

2006

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In this report we study space-mapping and manifold-mapping, two multi-level optimization techniques that aim at accelerating expensive optimization procedures with the aid of simple auxiliary models. Manifoldmapping improves in accuracy the solution given by space-mapping. In this report, the two mentioned techniques are basically described and then applied in the solving of two minimization problems. Several coarse models are tried, both from a two and a three level perspective. The results with these simple tests confirm the speed-up expected for the multi-level approach.

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The Defect Correction Approach

Abstract: This is an introductory survey of the defect correction approach which may serve as a unifying frame of reference for the subsequent papers on special subjects.

Cited by 90 publications

References 42 publications

Second order accurate upwind solutions of the 2D steady Euler equations by the use of a defect correction method

Second order accurate upwind solutions of the 2D steady Euler equations by the use of a defect correction method

Multiple grid and Osher's scheme for the efficient solution of the steady Euler equations

Towards Multi-level Optimization: Space-Mapping and Manifold-Mapping

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