Strong and Auxiliary Forms of the Semi-Lagrangian Method for Incompressible Flows

Xiu, Dongbin; Sherwin, Spencer J.; Dong, Shen; Karniadakis, George Em

doi:10.1007/s10915-004-4647-1

Cited by 21 publications

(8 citation statements)

References 31 publications

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“…A possible remedy is to add an additional stabilization term that includes artificial diffusion [10]. Other semi-Lagrangian methods for incompressible Navier-Stokes equations directly transport point values with the nodes of a mesh instead of evaluating integrals of transported quantities, see [34,29,47,46,13] and [16], where the last work makes use of exponential integrators [17]. Most authors employ spectral elements for the discretization in space [34,29], but any type of finite-element space with degrees-offreedom relying on point evaluations can be used.…”

Section: Outline and Related Workmentioning

confidence: 99%

Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

Tonnon¹,

Hiptmair²

2023

Preprint

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Section: Outline and Related Workmentioning

confidence: 99%

Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

Tonnon¹,

Hiptmair²

2023

Preprint

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“…If we wanted to solve the incompressible Navier-Stokes equations, we could use a splitting scheme. Applying a first-order strong semi-Lagrangian splitting scheme [20] to the Navier-Stokes equations gives a solution algorithm consisting of three steps: convection…”

Section: Synthetic Turbulence Simulationmentioning

confidence: 99%

“…Caustics formation is characterized by the existence of relatively large particle velocity differences over short distances, and we can use structure functions to identify this. We use the longitudinal particle velocity structure function of order p, which is defined as (20) Plots of S p (r) (not shown) reveal power laws S p (r) ∼ r ξ p in the dissipative range, i.e., for low values of r. For a smooth velocity field we expect to find ξ p = p. The first-order exponent ξ 1 is plotted in Fig. 9 for the full range of Stokes numbers.…”

Section: High Stokes Number Clusteringmentioning

confidence: 99%

Mechanisms of particle clustering in Gaussian and non-Gaussian synthetic turbulence

Nilsen

Andersson

2014

Phys. Rev. E

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We use synthetic turbulence simulations to study how inertial particles cluster in a turbulent flow, for a wide range of Stokes numbers. Two different types of synthetic turbulence are used: one Gaussian, where the time evolution of the velocity field is a simple phase shift, and one non-Gaussian, where convection is used to evolve the velocity field in time. In both flow types we observe significant particle clustering over a wide range of scales and Stokes numbers. The clustering found at low Stokes numbers can be attributed to the vortex centrifuge effect, where heavy particles are expelled from regions dominated by vorticity. This mechanism is much more effective in the non-Gaussian turbulence, because local flow structures are convected with the particles. The preferential sampling of regions with low vorticity is almost negligible in the Gaussian turbulence. At higher Stokes numbers, caustics are formed in a very similar manner in both Gaussian and non-Gaussian synthetic turbulence. In non-Gaussian turbulence, heavy particles cluster in regions of low fluid kinetic energy, while the opposite is true in Gaussian turbulence. Our results show that synthetic simulations cannot correctly predict how the particle clustering correlates with local fluid flow properties, without including convection.

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“…In particular, the backward semi‐Lagrangian method (BSLM) has a good stability in implicitly solving problems the characteristic curves of particles in the opposite direction involving a large time‐step size. Because of these advantages, BSLM has been widely used to deal with advection–diffusion type problems as much as the pure advection problem which also includes an advection problem induced by the splitting method.…”

Section: Introductionmentioning

confidence: 99%

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Bak

2019

Numerical Methods Partial

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In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.

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Strong and Auxiliary Forms of the Semi-Lagrangian Method for Incompressible Flows

Cited by 21 publications

References 31 publications

Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

Semi-Lagrangian Finite-Element Exterior Calculus for Incompressible Flows

Mechanisms of particle clustering in Gaussian and non-Gaussian synthetic turbulence

High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

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