A Fast and Well-Conditioned Spectral Method

Olver, Sheehan; Townsend, Alex

doi:10.1137/120865458

Cited by 205 publications

(326 citation statements)

References 44 publications

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“…Numerical solution of the coupled nonlinear sixth-order PDEs (3) with non-periodic boundary conditions for experimentally relevant domain sizes [5][6][7]10] is computationally challenging. We developed an algorithm that achieves the required numerical accuracy by combining a well-conditioned Chebyshev-Fourier spectral method [49,50] with a third-order semi-implicit time-stepping scheme [51] and integral conditions for the vorticity field [48] (App. A).…”

Section: Simulationsmentioning

confidence: 99%

Geometry-dependent viscosity reduction in sheared active fluids

Słomka

Dunkel

2017

Phys. Rev. Fluids

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We investigate flow pattern formation and viscosity reduction mechanisms in active fluids by studying a generalized Navier-Stokes model that captures the experimentally observed bulk vortex dynamics in microbial suspensions. We present exact analytical solutions including stress-free vortex lattices and introduce a computational framework that allows the efficient treatment of previously intractable higher-order shear boundary conditions. Large-scale parameter scans identify the conditions for spontaneous flow symmetry breaking, geometry-dependent viscosity reduction and negative-viscosity states amenable to energy harvesting in confined suspensions. The theory uses only generic assumptions about the symmetries and long-wavelength structure of active stress tensors, suggesting that inviscid phases may be achievable in a broad class of non-equilibrium fluids by tuning confinement geometry and pattern scale selection.Self-driven vortical flows in microbial [1] and synthesized active liquids [2-4] often exhibit a dominant length scale [5][6][7][8], distinctly different from the scale-free spectra of conventional turbulence [9]. Experimentally observed vortices in dense bacterial suspensions typically have diameters Λ ∼ 50 − 100 µm [5,8,10] and decay within a few seconds in a bulk fluid [10]. However, when the suspension is enclosed by a small container of dimensions comparable to Λ, individual vortices become stabilized for several minutes [11,12] and can be coupled together to form magnetically ordered vortex lattices [13]. Another form of confinement-induced symmetry breaking was observed recently in a microfluidic realization of bacterial 'racetracks' [14]. For sufficiently narrow tracks of diameter Λ, bacteria spontaneously aligned their swimming directions to form persistent unidirectional currents. These examples illustrate the importance of confinement geometry for flow-pattern formation in non-equilibrium liquids. Conversely, biologically or chemically powered fluids may profoundly affect the dynamics of moving boundaries as active components can significantly alter the effective viscosity of the surrounding solvent fluid [15][16][17]. In particular, recent shear experiments suggest that Escherichia coli bacteria can create effectively inviscid flow if their concentration and activity are sufficiently large to support coherent collective swimming [18]. From a theory perspective, it is desirable to formulate a minimal hydrodynamic model that is analytically tractable and can account for all the aforementioned experimental observations without overfitting.Previous theoretical work [19][20][21][22][23][24] identified potential viscosity reduction mechanisms [15,18] in certain classes of active suspensions, but the complexity and specific nature of the underlying multi-field models have made analytical insight, time-resolved dynamical studies and comparison with experiment challenging. To better understand the general conditions under which active fluids can develop spontaneous symmetry-breaking and quasi-invis...

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Section: Simulationsmentioning

confidence: 99%

Geometry-dependent viscosity reduction in sheared active fluids

Słomka

Dunkel

2017

Phys. Rev. Fluids

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“…The only steps that require an N log N transform to physical space are (i) the multiplications with variable coefficients and (ii) the computation of Q r (x) via the sine transform. A banded-operator approach [67] could be used to perform the variable-coefficient multiplications (to a specified tolerance) while remaining in spectral space. Such a method would therefore avoid several of the N log N transforms, but not the sine transform associated with Q r (x), and thus the overall computational cost would remain O (N log N ).…”

Section: Algorithm 3 Main Algorithm To Solve For the Wing Kinematicsmentioning

confidence: 99%

A fast Chebyshev method for simulating flexible-wing propulsion

Moore¹

2017

Journal of Computational Physics

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We develop a highly efficient numerical method to simulate small-amplitude flapping propulsion by a flexible wing in a nearly inviscid fluid. We allow the wing's elastic modulus and mass density to vary arbitrarily, with an eye towards optimizing these distributions for propulsive performance. The method to determine the wing kinematics is based on Chebyshev collocation of the 1D beam equation as coupled to the surrounding 2D fluid flow. Through small-amplitude analysis of the Euler equations (with trailing-edge vortex shedding), the complete hydrodynamics can be represented by a nonlocal operator that acts on the 1D wing kinematics. A class of semi-analytical solutions permits fast evaluation of this operator with O (N log N ) operations, where N is the number of collocation points on the wing. This is in contrast to the minimum O N 2 cost of a direct 2D fluid solver. The coupled wing-fluid problem is thus recast as a PDE with nonlocal operator, which we solve using a preconditioned iterative method. These techniques yield a solver of nearoptimal complexity, O (N log N ), allowing one to rapidly search the infinite-dimensional parameter space of all possible material distributions and even perform optimization over this space.

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“…While this approach is well understood, it produces dense matrices that often are ill-conditioned for variable coefficient boundary value problem. Recently, Olver and Townsend [8] developed a spectral method for linear ordinary differential equations which results in almost banded, sparse matrices that can solve linear differential equations in O(m 2 n) operations, where m and n are respectively the numbers of Chebyshev points to resolve the differential operators and the solution to the differential equation.…”

Section: Linear Solvermentioning

confidence: 99%

“…The differential operators D λ act on Chebyshev polynomials and differentiate in Gegenbauer space, and S λ are conversion operators from Chebyshev to Gegenbauer space. Full details on the calculation of these matrices can be found in the paper by Olver and Townsend [8]. Numerically implementing the homotopy analysis method using the aforementioned linear discretization gives the Gegenbauer homotopy analysis method (gham) …”

Section: C62mentioning

confidence: 99%

A fast, spectrally accurate solver for the Falkner--Skan equation

Cullen

Clarke²

2017

ANZIAMJ

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We present a new numerical technique, the Gegenbauer homotopy analysis method, which allows for the construction of iterative solutions to nonlinear differential equations. This technique is a numerical extension of the semi-analytic homotopy analysis method that exhibits spectral convergence while performing sparse matrix operations in Gegenbauer space. This technique is used to present solutions to the Falkner-Skan equation, a well known problem in boundary layer fluid dynamics. These solutions are compared to previously published works, and the convergence properties exhibited by this new technique are considered.

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A Fast and Well-Conditioned Spectral Method

Cited by 205 publications

References 44 publications

Geometry-dependent viscosity reduction in sheared active fluids

Geometry-dependent viscosity reduction in sheared active fluids

A fast Chebyshev method for simulating flexible-wing propulsion

A fast, spectrally accurate solver for the Falkner--Skan equation

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