239 research outputs found
An adaptive pseudospectral method for discontinuous problems
The accuracy of adaptively chosen, mapped polynomial approximations is studied for functions with steep gradients or discontinuities. It is shown that, for steep gradient functions, one can obtain spectral accuracy in the original coordinate system by using polynomial approximations in a transformed coordinate system with substantially fewer collocation points than are necessary using polynomial expansion directly in the original, physical, coordinate system. It is also shown that one can avoid the usual Gibbs oscillation associated with steep gradient solutions of hyperbolic pde's by approximation in suitably chosen coordinate systems. Continuous, high gradient solutions are computed with spectral accuracy (as measured in the physical coordinate system). Discontinuous solutions associated with nonlinear hyperbolic equations can be accurately computed by using an artificial viscosity chosen to smooth out the solution in the mapped, computational domain. Thus, shocks can be effectively resolved on a scale that is subgrid to the resolution available with collocation only in the physical domain. Examples with Fourier and Chebyshev collocation are given
Introducing PHAEDRA: a new spectral code for simulations of relativistic magnetospheres
We describe a new scheme for evolving the equations of force-free
electrodynamics, the vanishing-inertia limit of magnetohydrodynamics. This
pseudospectral code uses global orthogonal basis function expansions to take
accurate spatial derivatives, allowing the use of an unstaggered mesh and the
complete force-free current density. The method has low numerical dissipation
and diffusion outside of singular current sheets. We present a range of one-
and two-dimensional tests, and demonstrate convergence to both smooth and
discontinuous analytic solutions. As a first application, we revisit the
aligned rotator problem, obtaining a steady solution with resistivity localised
in the equatorial current sheet outside the light cylinder.Comment: 23 pages, 18 figures, accepted for publication in MNRA
Spectral methods in fluid dynamics
Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome
Pseudospectral methods for density functional theory in bounded and unbounded domains
Classical Density Functional Theory (DFT) is a statistical-mechanical
framework to analyze fluids, which accounts for nanoscale fluid inhomogeneities
and non-local intermolecular interactions. DFT can be applied to a wide range
of interfacial phenomena, as well as problems in adsorption, colloidal science
and phase transitions in fluids. Typical DFT equations are highly non-linear,
stiff and contain several convolution terms. We propose a novel, efficient
pseudo-spectral collocation scheme for computing the non-local terms in real
space with the help of a specialized Gauss quadrature. Due to the exponential
accuracy of the quadrature and a convenient choice of collocation points near
interfaces, we can use grids with a significantly lower number of nodes than
most other reported methods. We demonstrate the capabilities of our numerical
methodology by studying equilibrium and dynamic two-dimensional test cases with
single- and multispecies hard-sphere and hard-disc particles modelled with
fundamental measure theory, with and without van der Waals attractive forces,
in bounded and unbounded physical domains. We show that our results satisfy
statistical mechanical sum rules
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