A hybrid framework for coupling arbitrary summation-by-parts schemes on general meshes

Lundquist, Tomas; Malan, Arnaud G.; Nordström, Jan

doi:10.1016/j.jcp.2018.02.018

Cited by 28 publications

(66 citation statements)

References 33 publications

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“…It can easily be shown that the successive application of several IPP operators is always an IPP operation in itself, see Lemma 1 in [18]. Thus, since each step above involves the use of an IPP operator, it follows that the whole sequence is IPP with respect to the reverse operation.…”

Section: Operators In Two Dimensionsmentioning

confidence: 95%

“…for all polynomials ϕ up to (and including) a certain order p, where ϕ(X n ), ϕ(X n+1 ) are vectors with the value of ϕ in each node. For the reader's convenience we restate below the central accuracy result for IPP operators originally proven in [18].…”

Section: Inner Product Preserving Interpolationmentioning

confidence: 99%

“…Inner product preserving (IPP) interpolation has recently emerged as a powerful framework for locally adapting the refinement level of a computational mesh with the use of hanging nodes. Developed for numerical methods ranging from finite difference (FD) [20], masslumped discontinuous Galerkin (dG) [7] as well as finite volume and various hybrid methods [14,18], the framework of IPP is ideally suited to be used in conjunction with the energy method in order to prove stability of interpolation based interface coupling conditions between general blocks or elements in space. The successful use of IPP operators in the references cited above rely heavily on the interplay between the IPP concept and an underlying formal SBP framework for numerical differentiation on each of the individual blocks or element.…”

Section: Introductionmentioning

confidence: 99%

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Stable Dynamical Adaptive Mesh Refinement

2021

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We consider accurate and stable interpolation procedures for numerical simulations utilizing time dependent adaptive meshes. The interpolation of numerical solution values between meshes is considered as a transmission problem with respect to the underlying semi-discretized equations, and a theoretical framework using inner product preserving operators is developed, which allows for both explicit and implicit implementations. The theory is supplemented with numerical experiments demonstrating practical benefits of the new stable framework. For this purpose, new interpolation operators have been designed to be used with multi-block finite difference schemes involving non-collocated, moving interfaces.

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Section: Operators In Two Dimensionsmentioning

confidence: 95%

Section: Inner Product Preserving Interpolationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Stable Dynamical Adaptive Mesh Refinement

2021

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“…The interpolation operator P 1c interpolates from the cell-centered grid to the edge-1 grid, and so forth. By applying the SBP interpolation property (14) to (15), we find…”

Section: Two-dimensional Sbp Operatorsmentioning

confidence: 99%

Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

OReilly

Petersson

2020

Journal of Computational Physics

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We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to unity.

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“…The stability requirements on the interpolation operators limit the accuracies according to (see [12]) q u2v + q v2u ≤ 2p + 1.…”

Section: Local Truncation Errors and Convergence Ratesmentioning

confidence: 99%

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

Almquist¹,

Wang²,

Werpers³

2019

SIAM J. Sci. Comput.

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We study non-conforming grid interfaces for summation-by-parts finite difference methods applied to partial differential equations with second derivatives in space. To maintain energy stability, previous efforts have been forced to accept a reduction of the global convergence rate by one order, due to large truncation errors at the non-conforming interface. We avoid the order reduction by generalizing the interface treatment and introducing order preserving interpolation operators. We prove that, given two diagonal-norm summation-by-parts schemes, order preserving interpolation operators with the necessary properties are guaranteed to exist, regardless of the grid-point distributions along the interface. The new methods retain the stability and global accuracy properties of the underlying schemes for conforming interfaces.

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A hybrid framework for coupling arbitrary summation-by-parts schemes on general meshes

Cited by 28 publications

References 33 publications

Stable Dynamical Adaptive Mesh Refinement

Stable Dynamical Adaptive Mesh Refinement

Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids

Order-Preserving Interpolation for Summation-by-Parts Operators at Nonconforming Grid Interfaces

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