Two-dimensional parallel solver for the solution of Navier–Stokes equations with constant and variable coefficients using ADI on cells

High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements

“…[11] and references therein), including methods for solution of a variety of linear and nonlinear PDEs [12][13][14][15][16][17][18][19][20][21], as well as methods of high-order of spatial and temporal accuracy [22][23][24][25][26]. As suggested above, previous unconditionally stable alternating-direction methods can only achieve high-order accuracy in presence of a formulation of the given PDE on domains given by the union of a finite number of rectangular regions.…”

“…In particular, in [16,[48][49][50] an alternatingdirection approach was proposed that uses a Fourier basis for differentiation. Applications of previous Fourier-based techniques have thus far been restricted to rectangular geometries in spite of efforts seeking generalization to general domains [16,48]. (In fact, the previous Fourier approaches exhibit even less flexibility than their finite-difference counterparts: the latter are applicable to general domains, albeit with first-order accuracy.)…”

High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations

“…(A number of attempts have been made to combine the unconditional stability of the alternating direction type schemes with the spectral character of Fourier bases [7][8][9][10]. We expect that, like our FC-AD method, these Fourier-based approaches do not suffer from pollution errors.…”

Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

“…The smoothing replaces these discontinuous functions by other functions that are similar, but infinitely differentiable. Smoothed ramps and bells are useful to construct local Fourier bases (Averbuch et al [1][2][3][4][5], Israeli et al [32,33], Vozovoi et al [42,43], Coifman and Meyer [21], Jawerth and Sweldens [34], Bittner and Chui [6], Matviyenko [35]). Ramps and bells are also useful to solve the Fourier extension problem, in which a non-periodic f (x) defined on a certain interval is transformed into a functionf which is periodic on a larger interval (Boyd [17][18][19], Elghaoui and Pasquetti [23,22], Nordström et al [38], Högberg and Henningson [31], Garbey and Tromeur Dervout [29], Garbey [28], Haugen and Machenhauer [30]).…”