The automatic solution of partial differential equations using a global spectral method

Townsend, Alex; Olver, Sheehan

doi:10.1016/j.jcp.2015.06.031

Cited by 65 publications

(72 citation statements)

References 46 publications

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“…(31) and (32) The trick of writing f (r) = (1 − r)g(r) has to be used when there are two boundary conditions at r = 1. If there would be just one, the value of f n r can be obtained from those at the inner points directly from the approximate boundary condition.…”

Section: Discussionmentioning

confidence: 99%

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Radial collocation methods for the onset of convection in rotating spheres

Sánchez

García

Net

2016

Journal of Computational Physics

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The viability of using collocation methods in radius and spherical harmonics in the angular variables to calculate convective flows in full spherical geometry is examined. As a test problem the stability of the conductive state of a self-gravitating fluid sphere subject to rotation and internal heating is considered. A study of the behavior of different radial meshes previously used by several authors in polar coordinates, including or not the origin, is first performed. The presence of spurious modes due to the treatment of the singularity at the origin, to the spherical harmonics truncation, and to the initialization of the eigenvalue solver is shown, and ways to eliminate them are presented. Finally, to show the usefulness of the method, the neutral stability curves at very high Taylor and moderate and small Prandtl numbers are calculated and shown.

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Section: Discussionmentioning

confidence: 99%

“…Moreover the linear systems are well-conditioned. These methods have been recently extended to two-dimensional partial differential equations in rectangular domains in [31], and used in fluid flow applications [32].…”

Section: Introductionmentioning

confidence: 99%

Radial collocation methods for the onset of convection in rotating spheres

Sánchez

García

Net

2016

Journal of Computational Physics

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“…For the spatial discretization, we adopt a spectral method, expanding functions in a basis composed of Fourier modes and Chebyshev polynomials of the first kind. The implicit inversion is discretized using the well-conditioned scheme introduced by Olver and Townsend [49,50]. Since the system is periodic in the horizontal direction, the linear operator separates into one-dimensional operators, one for each Fourier mode.…”

Section: Appendix A: Numerical Methodsmentioning

confidence: 99%

“…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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“…Examples of non-adaptive discretization are: standard finite differences [63,77], standard continuous or discontinuous finite elements [34,52,53,83,98], and spectral methods [79,100,101], among many others. They are very general in the sense that they can be used for a variety of different problems.…”

Section: Introductionmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

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The automatic solution of partial differential equations using a global spectral method

Cited by 65 publications

References 46 publications

Radial collocation methods for the onset of convection in rotating spheres

Radial collocation methods for the onset of convection in rotating spheres

Geometry-dependent viscosity reduction in sheared active fluids

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

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