Efficient cubature rules

Zandt, James R. Van

doi:10.1553/etna_vol51s219

Cited by 4 publications

(3 citation statements)

References 37 publications

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“…• Last, for three-dimensional compressible flows, a recently introduced D3Q45 velocity set was applied [48], based on a cubature rule by Konyaev [49,50]. This off-lattice velocity set was compared in an off-lattice Boltzmann simulation to the state of the art on-lattice counterparts D3Q103 and D3V107 applied to compressible on-lattice Boltzmann simulations.…”

Section: Cubature In Lattice Boltzmann Methodsmentioning

confidence: 99%

Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

Wilde,

Krämer,

Bedrunka

et al. 2021

Preprint

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Off-lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretizations of time, space, and particle velocities. However, the velocity sets that are mostly used in off-lattice Boltzmann simulations were originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and efficiency of weakly and fully compressible semi-Lagrangian off-lattice Boltzmann simulations is increased by velocity sets derived from cubature rules, i.e. multivariate quadratures, which have not been produced by the Gauß-product rule. In particular, simulations of 2D shock-vortex interactions indicate that the cubature-derived degree-nine D2Q19 velocity set is capable to replace the Gauß-product rule-derived D2Q25. Likewise, the degree-five velocity sets D3Q13 and D3Q21, as well as a degree-seven D3V27 velocity set were successfully tested for 3D Taylor-Green vortex flows to challenge and surpass the quality of the customary D3Q27 velocity set. In compressible 3D Taylor-Green vortex flows with Mach numbers Ma = {0.5; 1.0; 1.5; 2.0} on-lattice simulations with velocity sets D3Q103 and D3V107 showed only limited stability, while the off-lattice degree-nine D3Q45 velocity set accurately reproduced the kinetic energy provided by literature.

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Section: Cubature In Lattice Boltzmann Methodsmentioning

confidence: 99%

Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

Wilde,

Krämer,

Bedrunka

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Therefore, we successfully identified and implemented a degree-nine D3Q45 velocity set with only 45 components, which fulfills all requirements in terms of symmetry. The D3Q45 was derived by a cubature rule after Konyaev [89] and has recently been listed in the supplemental material of a cubature article by van Zandt [90]. The resulting discrete velocities are shown in Fig.…”

Section: B Cubature-based Velocity Setsmentioning

confidence: 99%

High-order semi-Lagrangian kinetic scheme for compressible turbulence

et al. 2021

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Turbulent compressible flows are traditionally simulated using explicit time integrators applied to discretized versions of the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time-step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which circumvents this restriction. While many lattice Boltzmann methods for compressible flows were restricted to two dimensions due to the enormous number of discrete velocities in three dimensions, the SLLBM uses only 45 discrete velocities. Based on compressible Taylor-Green vortex simulations we show that the new method accurately captures shocks or shocklets as well as turbulence in 3D without utilizing additional filtering or stabilizing techniques other than the filtering introduced by the interpolation, even when the time-step sizes are up to two orders of magnitude larger compared to simulations in the literature. Our new method therefore enables researchers to study compressible turbulent flows by a fully explicit scheme, whose range of admissible time-step sizes is dictated by physics rather than spatial discretization.

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“…Therefore, we successfully identified and implemented a degree-nine D3Q45 velocity set with only 45 components, which fulfills all requirements in terms of symmetry. The D3Q45 was derived by a cubature rule after Konyaev [80] and has recently been picked up again by van Zandt in the cubature literature [81]. The resulting discrete velocities are shown in Fig.…”

mentioning

confidence: 99%

High-order semi-Lagrangian kinetic scheme for compressible turbulence

Wilde,

Krämer,

Reith

et al. 2020

Preprint

View full text Add to dashboard Cite

Turbulent compressible flows are traditionally simulated using explicit Eulerian time integration applied to the Navier-Stokes equations. However, the associated Courant-Friedrichs-Lewy condition severely restricts the maximum time step size. Exploiting the Lagrangian nature of the Boltzmann equation's material derivative, we now introduce a feasible three-dimensional semi-Lagrangian lattice Boltzmann method (SLLBM), which elegantly circumvents this restriction. Previous lattice Boltzmann methods for compressible flows were mostly restricted to two dimensions due to the enormous number of discrete velocities needed in three dimensions. In contrast, this Rapid Communication demonstrates how cubature rules enhance the SLLBM to yield a three-dimensional velocity set with only 45 discrete velocities. Based on simulations of a compressible Taylor-Green vortex we show that the new method accurately captures shocks or shocklets as well as turbulence in 3D without utilizing additional filtering or stabilizing techniques, even when the time step sizes are up to two orders of magnitude larger compared to simulations in the literature. Our new method therefore enables researchers for the first time to study compressible turbulent flows by a fully explicit scheme, whose range of admissible time step sizes is only dictated by physics, while being decoupled from the spatial discretization.

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Efficient cubature rules

Cited by 4 publications

References 37 publications

Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

Cubature rules for weakly and fully compressible off-lattice Boltzmann methods

High-order semi-Lagrangian kinetic scheme for compressible turbulence

High-order semi-Lagrangian kinetic scheme for compressible turbulence

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