Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction

Aboiyar, T.; Georgoulis, Emmanuil H.; Iske, Armin

doi:10.1137/100792573

Cited by 45 publications

(68 citation statements)

References 31 publications

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“…The seminal work of [2] proposes a new type of WENO method based on nonpolynomial reconstructions. More specifically, in [2] and in related work [1,23], the mesh-free feature of Radial Basis Functions (RBFs) is successfully combined with ENO/WENO ideas for scalar conservation laws in general geometries using unstructured grids.…”

Section: Introductionmentioning

confidence: 99%

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

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We explore the use of radial basis functions (RBF) in the weighted essentially non-oscillatory (WENO) reconstruction process used to solve hyperbolic conservation laws, resulting in a numerical method of arbitrarily high order to solve problems with discontinuous solutions. Thanks to the mesh-less property of the RBFs, the method is suitable for non-uniform grids and mesh adaptation. We focus on multiquadric radial basis functions and propose a simple strategy to choose the shape parameter to control the balance between achievable accuracy and the numerical stability. We also develop an original smoothness indicator which is independent of the RBF for the WENO reconstruction step. Moreover, we introduce type I and type II RBF-WENO methods by computing specific linear weights. The RBF-WENO method is used to solve linear and nonlinear problems for both scalar and systems of conservation laws, including Burgers equation, the Buckley-Leverett equation, and the Euler equations. Numerical results confirm the performance of the proposed method. We finally consider an effective conservative adaptive algorithm that captures moving shocks and rapidly varying solutions well. Numerical results on moving grids are presented for both Burgers equation and the more complex Euler equations.

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Section: Introductionmentioning

confidence: 99%

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Bigoni

Hesthaven

2017

J Sci Comput

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“…We remark that there are efficient algorithms from computational geometry [19] for the construction and maintenance of the Voronoi diagram V Ξ and its dual Delaunay tesselation. The combination between Voronoi diagrams and finite volumes yields through the basic concept of the FVPM a flexible particle method for the numerical solution of (1), (2). We further remark that a more general concept of the FVPM [10,14], allows for overlapping influence areas {V Ξ (ξ )} ξ ∈Ξ in which case, however, the FVPM needs to be combined with a partition of unity method (PUM).…”

Section: Finite Volume Particle Methods (Fvpm)mentioning

confidence: 99%

“…Such discontinuities of the solution u in (1) can easily develop spontaneously even from smooth initial data u 0 in (2).…”

Section: Hyperbolic Conservation Lawsmentioning

confidence: 99%

“…Therefore, nonlinear flow simulation requires more sophisticated and flexible mathematical and computational methods to numerically solve the Cauchy problem (1), (2). For a comprehensive introduction to numerical methods for hyperbolic problems we recommend the textbook [16].…”

Section: Hyperbolic Conservation Lawsmentioning

confidence: 99%

“…We remark that commonly used ENO/WENO schemes work with polynomial reconstruction, which, however, may lead to severe numerical instabilities and other disadvantages, especially for anisotropic distributions of particles, see [1,2]. In Section 6, we propose a numerically stable reconstruction method of arbitrary high order, which essentially avoids (plain) polynomial reconstruction.…”

Section: Weno Reconstructionmentioning

confidence: 99%

See 2 more Smart Citations

On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

Iske

2013

Notes on Numerical Fluid Mechanics and Multidisciplinary Design

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This contribution discusses the construction of kernel-based adaptive particle methods for numerical flow simulation, where the finite volume particle method (FVPM) is used as a prototype. In the FVPM, scattered data approximation algorithms are required in the recovery step of the WENO reconstruction. We first show how kernel-based approximation schemes can be used in the recovery step of particle methods, where we give preference to the radial polyharmonic spline kernel. Then we discuss important aspects concerning the numerical stability and approximation behaviour of polyharmonic splines. Moreover, we propose customized coarsening and refinement rules for the adaptive resampling of the particles. Supporting numerical examples and comparisons with other radial kernels are provided.

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A semi‐implicit scheme for 3D free surface flows with high‐order velocity reconstruction on unstructured Voronoi meshes

Boscheri

Dumbser

Righetti

2012

Numerical Methods in Fluids

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SUMMARYIn this paper, we present a computationally efficient semi‐implicit scheme for the simulation of three‐dimensional hydrostatic free surface flow problems on staggered unstructured Voronoi meshes. For each polygonal control volume, the pressure is defined in the cell center, whereas the discrete velocity field is given by the normal velocity component at the cell faces. A piecewise high‐order polynomial vector velocity field is then reconstructed from the scalar normal velocities at the cell faces by using a new high‐order constrained least‐squares reconstruction operator. The reconstructed high‐order piecewise polynomial velocity field is used for trajectory integration in a semi‐Lagrangian approach to discretize the nonlinear convective terms in the governing PDE. For that purpose, a high‐order Taylor method is used as ODE integrator. The resulting semi‐implicit algorithm is extensively validated on a large set of different academic test problems with exact analytical solution and is finally applied to a real‐world engineering problem consisting of a curved channel upstream of two micro‐turbines of a hydroelectric power plant. For this realistic case, some experimental reference data are available from field measurements. Copyright © 2012 John Wiley & Sons, Ltd.

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Adaptive ADER Methods Using Kernel-Based Polyharmonic Spline WENO Reconstruction

Cited by 45 publications

References 31 publications

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

Adaptive WENO Methods Based on Radial Basis Function Reconstruction

On the Construction of Kernel-Based Adaptive Particle Methods in Numerical Flow Simulation

A semi‐implicit scheme for 3D free surface flows with high‐order velocity reconstruction on unstructured Voronoi meshes

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