High-Order Numerical Integration over Discrete Surfaces

Ray, Navamita; Wang, Duo; Jiao, Xiangmin; Glimm, James

doi:10.1137/110857404

Cited by 15 publications

(13 citation statements)

References 8 publications

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“…We addressed the fundamental question of whether one can achieve high-order accuracy given a piecewise linear approximation to the surface. We proposed a novel method, which can compute the surface integrals to high-order of accuracy over piecewise-linear meshes [29,9]. We proved theoretically that high-order of accuracy can be achieved for surface integrals using a least-squares based approximation over linear meshes and demonstrated numerical experiments achieving the predicted order of accuracy.…”

Section: Numerical Tools For Complex Dynamic Geometrymentioning

confidence: 91%

Mixing, Combustion, and Other Interface Dominated Flows; Paragraphs 3.2.1 A, B, C and 3.2.2 A

Glimm¹,

Li²,

Jiao³

et al. 2014

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Our progress has focused on advanced numerical algorithms and supporting mathematical theory, with a concentration on higher order geometry in the description of sharp interfaces, stochastic issues in modeling and simulation, and nonlinearities caused by high levels of deformation in the coupling of fluid to solid codes. These general themes are addressed within the context of ARO interests, including parachute drop, brittle fracture, turbulent modeling and turbulent combustion. We developed a new formulation and interpretation of convergence for turbulent combustion simulations with a new rigorous mathematical analysis of turbulent flow. We modeled the

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Section: Numerical Tools For Complex Dynamic Geometrymentioning

confidence: 91%

Mixing, Combustion, and Other Interface Dominated Flows; Paragraphs 3.2.1 A, B, C and 3.2.2 A

Glimm¹,

Li²,

Jiao³

et al. 2014

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“…The remap method in this work utilizes two high-order reconstruction techniques, known as CMF (Continuous Moving Frames) and WALF (Weighted Average of Least-squares Fittings) [52], both of which are based on weighted least squares (WLS ). These techniques were first proposed for reconstructing surfaces and later adapted to reconstruct functions on surfaces [81]. CMF shares some similarities with some variant of MMLS (such as that in [94]), in that CMF constructs a least-squares fitting at each reconstruction point and it achieves accuracy and stability through some local adaptivity (instead of the global construction of the original MLS for smoothness [65]).…”

Section: High-order Reconstructionsmentioning

confidence: 99%

“…As shown in [53] and [81], if the condition number of the rescaled Vandermonde system Ã is bounded, then the function values of a smooth function f : Γ → R can be reconstructed to O(h p+1 ) on a discrete surface Γ, where h is proportional to the "radius" of the stencil. Furthermore, if the stencils are (nearly) symmetric about the origin of the local coordinate system, the reconstruction can superconverge at O(h p+2 ) for even-degree p due to error cancellation [53,52], analogous to the error cancellation in centered differences.…”

Section: Optimizing Weights For Remapmentioning

confidence: 99%

WLS-ENO Remap: Superconvergent and Non-Oscillatory Weighted Least Squares Data Transfer on Surfaces

Li,

Chen,

Wang

et al. 2019

Preprint

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Data remap between non-matching meshes is a critical step in multiphysics coupling using a partitioned approach. The data fields being transferred often have jumps in function values or derivatives. It is important but very challenging to avoid spurious oscillations (a.k.a. the Gibbs Phenomenon) near discontinuities and at the same time to achieve high-order accuracy away from discontinuities. In this work, we introduce a new approach, called WLS-ENOR, or Weighted-Least-Squares-based Essentially Non-Oscillatory Remap, to address this challenge. Based on the WLS-ENO reconstruction technique proposed by Liu and Jiao (J. Comput. Phys. vol 314, pp 749--773, 2016), WLS-ENOR differs from WLS-ENO and other WENO schemes in that it resolves not only the O(1) oscillations due to C 0 discontinuities, but also the accumulated effect of O(h) oscillations due to C 1 discontinuities. To this end, WLS-ENOR introduces a robust detector of discontinuities and a new weighting scheme for WLS-ENO near discontinuities. We also optimize the weights at smooth regions to achieve superconvergence. As a result, WLS-ENOR is more than fifth-order accurate and highly conservative in smooth regions, while being non-oscillatory and minimally diffusive near discontinuities. We also compare WLS-ENOR with some commonly used methods based on L 2 projection, moving least squares, and radial basis functions.

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“…Here, we address three issues: the application programming interface (API), a conservative version of the API and a graceful end to the tracking of the interface in overly complex frontal topologies. Before going into the details, we describe the front or interface software, a product of Jiao and students [21][22][23], and replacing our older interface software. Conceptually the ideas are simple.…”

Section: An Application Programming Interface For Front Trackingmentioning

confidence: 99%

Euler equation existence, non-uniqueness and mesh converged statistics

Glimm

Sharp

Lim

et al. 2015

Phil. Trans. R. Soc. A.

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One contribution of 15 to a theme issue 'Free boundary problems and related topics' . We review existence and non-uniqueness results for the Euler equation of fluid flow. These results are placed in the context of physical models and their solutions. Non-uniqueness is in direct conflict with the purpose of practical simulations, so that a mitigating strategy, outlined here, is important. We illustrate these issues in an examination of mesh converged turbulent statistics, with comparison to laboratory experiments.

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High-Order Numerical Integration over Discrete Surfaces

Cited by 15 publications

References 8 publications

Mixing, Combustion, and Other Interface Dominated Flows; Paragraphs 3.2.1 A, B, C and 3.2.2 A

Mixing, Combustion, and Other Interface Dominated Flows; Paragraphs 3.2.1 A, B, C and 3.2.2 A

WLS-ENO Remap: Superconvergent and Non-Oscillatory Weighted Least Squares Data Transfer on Surfaces

Euler equation existence, non-uniqueness and mesh converged statistics

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