An efficient class of WENO schemes with adaptive order for unstructured meshes

Balsara, Dinshaw S.; Garain, Sudip K.; Florinski, V.; Boscheri, Walter

doi:10.1016/j.jcp.2019.109062

Cited by 68 publications

(61 citation statements)

References 84 publications

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“…At fourth order of accuracy we have used an adaptive order method to avoid the excessive computational cost of performing high order reconstruction on all thirteen stencils. The WENO-AO method has been described in great detail in Balsara et al (2016), while its implementation on unstructured meshes was presented in Balsara et al (2020Balsara et al ( , 2019. Here we only discuss some specifics of its implementation on the geodesic mesh.…”

Section: Limiting the Reconstructionmentioning

confidence: 99%

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

Florinski

Balsara

Garain

et al. 2020

Comput. Astrophys. Cosmol.

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Many important problems in astrophysics, space physics, and geophysics involve flows of (possibly ionized) gases in the vicinity of a spherical object, such as a star or planet. The geometry of such a system naturally favors numerical schemes based on a spherical mesh. Despite its orthogonality property, the polar (latitude-longitude) mesh is ill suited for computation because of the singularity on the polar axis, leading to a highly non-uniform distribution of zone sizes. The consequences are (a) loss of accuracy due to large variations in zone aspect ratios, and (b) poor computational efficiency from a severe limitations on the time stepping. Geodesic meshes, based on a central projection using a Platonic solid as a template, solve the anisotropy problem, but increase the complexity of the resulting computer code. We describe a new finite volume implementation of Euler and MHD systems of equations on a triangular geodesic mesh (TGM) that is accurate up to fourth order in space and time and conserves the divergence of magnetic field to machine precision. The paper discusses in detail the generation of a TGM, the domain decomposition techniques, three-dimensional conservative reconstruction, and time stepping.

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Section: Limiting the Reconstructionmentioning

confidence: 99%

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

Florinski

Balsara

Garain

et al. 2020

Comput. Astrophys. Cosmol.

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“…Examples of such WENO reconstructions include CWENO [25,26], WENO-ZQ [41], WENO-AO [4], targeted ENO scheme [17,18], hybrid WENO [1,35,40] and their extensions. Especially, there have been systematic studies on WENO-AO [2,3,22]. As in classic WENO-Z [6], smoothness indicators in WENO-ZQ and WENO-AO are properly defined for high order accuracy at critical points.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

Chen¹

2022

CiCP

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We present a new conservative semi-Lagrangian finite difference weighted essentially non-oscillatory scheme with adaptive order. This is an extension of the conservative semi-Lagrangian (SL) finite difference WENO scheme in [Qiu and Shu, JCP, 230 (4) (2011), pp. 863-889], in which linear weights in SL WENO framework were shown to not exist for variable coefficient problems. Hence, the order of accuracy is not optimal from reconstruction stencils. In this paper, we incorporate a recent WENO adaptive order (AO) technique [Balsara et al., JCP, 326 (2016), pp. 780-804] to the SL WENO framework. The new scheme can achieve an optimal high order of accuracy, while maintaining the properties of mass conservation and non-oscillatory capture of solutions from the original SL WENO. The positivity-preserving limiter is further applied to ensure the positivity of solutions. Finally, the scheme is applied to high dimensional problems by a fourth-order dimensional splitting. We demonstrate the effectiveness of the new scheme by extensive numerical tests on linear advection equations, the Vlasov-Poisson system, the guiding center Vlasov model as well as the incompressible Euler equations.

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“…Although the frameworks are significantly different in principle the high-order spatial accuracy is achieved through a natural extension of the representation of the solution within each cell by polynomials. Two of the most popular families of schemes for providing non-oscillatory capabilities to a numerical framework is firstly the family of schemes that can detect when some solution bounds have been violated and switch to a lower-order approximation of the solution such as total-variation bounded (TVB) [34], total-variation diminishing (TVD) [35][36][37], monotone upstream scheme for conservation laws (MUSCL) [38][39][40][41][42][43][44][45][46], and multi-dimensional optimal order detection (MOOD) [3,4,[47][48][49], and secondly by techniques where the spatial-regions with the best quality of information (smooth data) will have the largest influence in the approximation of the solution, such as the weighted essentially non-oscillatory (WENO) [2,5,6,13,[50][51][52][53][54][55] schemes. It needs to be noted that high order WENO reconstruction in space is also the key ingredient of the ADER class of finite volume schemes of Toro and Titarev, see e.g.…”

Section: Introductionmentioning

confidence: 99%

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

Tsoutsanis

Dumbser

2021

Computers & Fluids

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An efficient class of WENO schemes with adaptive order for unstructured meshes

Cited by 68 publications

References 84 publications

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

Technologies for supporting high-order geodesic mesh frameworks for computational astrophysics and space sciences

Adaptive Order WENO Reconstructions for the Semi-Lagrangian Finite Difference Scheme for Advection Problem

Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

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