On the line and its tensor products, Fekete points are known to be Gauss--Lobatto quadrature points. But unlike high-order quadrature, generalize non-tensor-product domains such as triangle. Thus might serve an alternative for certain applications. In this work we present a new algorithm compute give results up degree 19 For d > 10 these have smallest Lebesgue constant currently known. The computations validate conjecture of Bos [ J. Approx. Theory, 64 (1991), pp. 271--280] that along...

We apply a spectral element method based upon conforming mesh of quadrangles and triangles to the problem 2-D elastic wave propagation. The retains advantages classical methods only. It makes use Gauss–Lobatto–Legendre formulation on quadrangles, while discretization is interpolation at Fekete points. obtain global diagonal mass matrix which allows us keep explicit structure solvers. demonstrate accuracy efficiency by comparing results obtained for pure quadrangle meshes with those using...

We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time of highly oscillatory nature. The combines the parareal method---a parallel-in-time scheme introduced [J.-L. Lions, Y. Maday, and G. Turinici, C. R. Acad. Sci. Paris Ser. I Math., 332 (2001), pp. 661--668]---with techniques from heterogeneous multiscale method (cf. [W. E B. Engquist, Notices Amer. Math. Soc., 50 (2003), 1062--1070]), which make use slow asymptotic structure equations [A. J....

We present a new algorithm for numerically computing quadrature formulas arbitrary domains which exactly integrate given polynomial space. An effective method constructing has been to solve nonlinear set of equations the points and their associated weights. Symmetry conditions are often used reduce number unknowns. Our instead relies on construction cardinal functions thus requires that N be equal dimension prescribed lower dimensional The allow us treat weights as dependent variables remove...

Tensor products of Gauss-Lobatto quadrature points are frequently used as collocation in spectral element methods. Unfortunately, it is not known if exist non-tensor-product domains like the simplex. In this work, we show that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math>...

The manuscript presents a technique for efficiently solving the classical wave equation, shallow water equations, and, more generally, equations of form |$\partial u /\partial t = \mathcal {L}u$|⁠, where |$\mathcal {L}$| is skew-Hermitian differential operator. idea to explicitly construct an approximation time-evolution operator |$\exp (\tau {L})$| relatively large time-step |$\tau $|⁠. Recently developed techniques approximating oscillatory scalar functions by rational functions, and...

In this paper we explore the fast rotation, nonhydrostatic limit of rotating and stratified Boussinesq equations. We derive new reduced equations for slow dynamics that describe Taylor-Proudman flows. One aspect is a decoupling horizontal kinetic energy, described by 2D Navier-Stokes, from coupling vertical energy buoyancy. support theory with high resolution numerical simulations full that, in limit, reveal spontaneous formation columns their dynamics.

This paper presents, discusses and analyses a massively parallel-in-time solver for linear oscillatory partial differential equations, which is key numerical component evolving weather, ocean, climate seismic models. The time parallelization in this allows us to significantly exceed the computing resources used by parallelization-in-space methods results correspondingly reduced wall-clock time. One of major difficulties achieving Exascale performance weather prediction that strong scaling...

We present several new quadrature formulas in the triangle for exact integration of polynomials. The points were computed numerically with a cardinal function algorithm which imposes that number $N$ be equal to dimension lower dimensional polynomial space. Quadrature forumulas are presented up degree $d=25$, all have positive weights and contain no outside triangle. Seven these improve on previously known results.

Ocean modeling presents several unique technical challenges: there is a tremendous range of spatial scales; the kinetic energy forcing scale occurs at Rossby radius deformation (20–100 km), which often or below grid resolution; and mixing strongly anisotropic, occurring primarily along nearly horizontal isopycnal surfaces. We present analysis numerical results to show that Lagrangian-averaged Navier–Stokes alpha (LANS-α) turbulence parameterization and, lesser extent, Leray are well suited...

It is shown that for Boussinesq flows in which rotation and stratification are equally strong, the forward cascade of potential enstrophy constrains spectral distribution horizontal kinetic energy energy. Horizontal suppressed small–aspect-ratio wave modes, large–aspect-ratio modes. Scaling estimates based on phenomenological arguments yield scaling kh- 3 kz- 3, respectively, two spectra. High-resolution numerical simulations equations relevant parameter regimes show exponent closer to - 4,...

Abstract The class of alpha models for turbulence may be derived by applying Lagrangian averaging to the exact fluid equations and then making a closure approximation based on Taylor’s hypothesis frozen-in fluctuations. This derivation provides closed expression unknown pseudomomentum in generalized mean theory Andrews McIntyre. In current study, effects baroclinic instability are explored, as determined two-layer quasigeostrophic-alpha model quasigeostrophic (QG) balance. QG-alpha is found...

In [SIAM J. Sci. Comput., 36 (2014), pp. A693--A713] the authors present a new coarse propagator for parareal method applied to oscillatory PDEs that exhibit time-scale separation and show, under certain regularity constraints, superlinear convergence which leads significant parallel speedups over standard methods. The error bound depends on degree of separation, $\epsilon$, time step, $\Delta T$, relies holds only in limit small $\epsilon$. main result paper is generalization this also...

Abstract The effect of non-slow (typically fast) components a rotating stratified Boussinesq flow on the dynamics slow manifold is quantified using decomposition that isolates part living manifold. In this system, there are three distinct asymptotic limits with corresponding reduced equations, each defining All these limits, namely rapid rotation, strong stratification, and simultaneous stratification rotation (quasi-geostrophy), considered. Numerical simulations indicate that, for geometry...

A framework is developed to describe the two-point statistics of potential vorticity in rotating and stratified turbulence as described by Boussinesq equations. The Karman-Howarth equation for dynamics correlation function reveals possibility inertial-range certain regimes Rossby, Froude, Prandtl Reynolds number parameters. For case large Rossby Froude numbers, quasi-geostrophic dynamics, a linear scaling law with 2/3 prefactor derived third-order mixed between velocity, result that...

Abstract A linear stability analysis of the inviscid stratified Boussinesq equations is presented given a steady zonal flow with constant vertical shear in tilted f plane. Full nonhydrostatic terms are included: 1) acceleration velocity and 2) Coriolis force arising from meridional component earth’s rotation vector. Calculations growth rates, critical wavenumbers, dominance regimes for baroclinic symmetric instabilities compared results traditional equations, which include strictly vector,...

This work investigates the numerical time stability of Lagrangian-averaged shallow water α model (SW-α). The main result is an analytical estimate for maximum allowable step. shows that as grid refined step becomes independent mesh spacing and instead depends on length scale, α, a parameter model. achieves this through changes in equations motion reduce frequency linear waves at high wavenumbers. type reduction high-wavenumber also characteristic time-implicit methods. Consequently, analogy...