Analysis of local defect correction and high-order compact finite differences

Sizov, M.; Anthonissen, M.J.H.; Mattheij, R.M.M.

doi:10.1002/num.20124

Cited by 4 publications

(4 citation statements)

References 22 publications

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“…The work is going on in several directions including extending this scheme to multiscale cartesian grids using LDC [22][23][24] and to irregular geometries using the cartesian embedded boundary technique [25][26][27].…”

Section: Discussionmentioning

confidence: 99%

A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

Abbas

2013

Numerical Methods Partial

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We present a fourth-order Hermitian box-scheme (HB-scheme) for the Poisson problem in a cube. A singlenonstaggered regular grid is used supporting the discrete unknowns u and ∇u. The scheme is fourth-order accurate for u and ∇u in L ∞ norm. The fast numerical resolution uses a matrix capacitance method, resulting in a computational complexity of O(N 3 log 2 (N)). Numerical results are reported on several examples including nonseparable problems. The present scheme is the extension to the three-dimensional case of the HB-scheme presented in Abbas and Croisille [J Sci Comp 49 (2011), 239-267].

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Section: Discussionmentioning

confidence: 99%

A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

Abbas

2013

Numerical Methods Partial

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“…The default choice for the switch, i.e., s i (v) = 1, selects the central-biased weight matrix given by (15) if n is even or (16) if n is odd. The transition from central to noncentral weights may occur only when a discontinuity is detected in the cells of the reconstruction stencil or in a compressive region of v. Compressive regions of v are nonoverlapping subsets of the integration domain where f (v) is a nonincreasing quantity, i.e., where the characteristic curves tend to intersect thus leading to the shock formation.…”

Section: W σ I (V; X)a σ I and Size (N+1)×(n+1)mentioning

confidence: 99%

“…For example, it has been considered within compact finite differences, c.f. [15] and the recent extension to local defect correction methods [16]. Upwinding is achieved in ENO reconstruction by selecting the stencil that locally provides the smoothest possible interpolation polynomial, and in WENO reconstruction by a careful definition of the weights of a convex linear combination of all the possible ENO polynomials of a given degree.…”

Section: Introductionmentioning

confidence: 99%

An efficient and conservative hybrid method for solving multidimensional conservation laws

Manzini¹

2008

Numerical Methods Partial

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We have developed an efficient hybrid technique for solving nonlinear conservation laws. Previous hybrid techniques have been accurate but lacked the property of conservation, whereas our technique is both accurate and conservative. To achieve this, we superimposed all possible stencils for ENO polynomials and weighted the value from each cells in a way that depends on the numerical solution. Computational efficiency relies on switching from central data where the exact solution is smooth to noncentral data near discontinuities. We prove the theoretical consistency of the technique and discuss the connection with ENO and WENO methods. We introduce time dependency by combining our method with Runge-Kutta schemes that are TVD preserving. We have verified our technique experimentally by solving a suite of test problems with convex and non-convex flux functions taken from the literature.

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“…The method is combined with finite volume discretizations in [4] and with finite elements in [19]. So-called high order compact difference schemes can be used in an LDC technique too [16]. In [9,17] LDC is studied with different grid types.…”

Section: Introductionmentioning

confidence: 99%

Local defect correction in numerical simulation of surface remelting

Anthonissen

2007

International Journal of Computer Mathematics

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We study the efficient numerical simulation of laser surface remelting, a process to improve the surface quality of steel components. To this end we use adaptive grids, which are well-suited for problems with moving heat sources. To account for the local high activity due to the heat source, we introduce local uniform grids and couple the solutions on the global coarse and local fine grids using local defect correction (LDC). LDC is based on simple data structures and simple discretization stencils on uniform or tensor-product grids.

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Analysis of local defect correction and high-order compact finite differences

Cited by 4 publications

References 22 publications

A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

A fourth‐order Hermitian box‐scheme with fast solver for the Poisson problem in a cube

An efficient and conservative hybrid method for solving multidimensional conservation laws

Local defect correction in numerical simulation of surface remelting

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