chebyshev

Chebyshev pseudospectral discretization

elementwise_derivative!(coord, ff, adv_fac, spectral::chebyshev_info)

Chebyshev transform f to get Chebyshev spectral coefficients and use them to calculate f'.

Note: Chebyshev derivative does not make use of upwinding information within each element.

elementwise_derivative!(coord, ff, chebyshev::chebyshev_info)

Chebyshev transform f to get Chebyshev spectral coefficients and use them to calculate f'.

derivative matrix for Gauss-Lobatto points using the analytical specification from Chapter 8.2 from Trefethen 1994 https://people.maths.ox.ac.uk/trefethen/8all.pdf full list of Chapters may be obtained here https://people.maths.ox.ac.uk/trefethen/pdetext.html

Derivative matrix for Chebyshev-Radau grid using the FFT. Note that a similar function could be constructed for the Chebyshev-Lobatto grid, if desired.

takes the real function ff on a Chebyshev grid in z (domain [-1, 1]), which corresponds to the domain [π, 2π] in variable theta = ArcCos(z). interested in functions of form f(z) = sumn cn Tn(z) using Tn(cos(theta)) = cos(ntheta) and z = cos(theta) gives f(z) = sumn cn cos(ntheta) thus a Chebyshev transform is equivalent to a discrete cosine transform doing this directly turns out to be slower than extending the domain from [0, 2pi] and using the fact that f(z) must be even (as cosines are all even) on this extended domain, can do a standard complex-to-complex fft fext is an array used to store f(theta) on the extended grid theta ∈ [0,2π) ff is f(theta) on the grid [π,2π] the Chebyshev coefficients of ff are calculated and stored in chebyf n is the number of grid points on the Chebyshev-Gauss-Lobatto grid transform is the plan for the complex-to-complex, in-place fft

use Chebyshev basis to compute the first derivative of f

compute and return modified Chebyshev moments of the first kind: ∫dx Tᵢ(x) over range [-1,1]

returns the Chebyshev-Gauss-Lobatto grid points on an n point grid

returns wgts array containing the integration weights associated with all grid points for Clenshaw-Curtis quadrature

initialize chebyshev grid scaled to interval [-boxlength/2, boxlength/2] we no longer pass the boxlength to this function, but instead pass precomputed arrays elementscale and element_shift that are needed to compute the grid.

ngrid – number of points per element (including boundary points) nelementlocal – number of elements in the local (distributed memory MPI) grid n – total number of points in the local grid (excluding duplicate points) elementscale – the scale factor in the transform from the coordinates where the element limits are -1, 1 to the coordinate where the limits are Aj = coord.grid[imin[j]-1] and Bj = coord.grid[imax[j]] elementscale = 0.5*(Bj - Aj) elementshift – the centre of the element in the extended grid coordinate element_shift = 0.5*(Aj + Bj) imin – the array of minimum indices of each element on the extended grid. By convention, the duplicated points are not included, so for element index j > 1 the lower boundary point is actually imin[j] - 1 imax – the array of maximum indices of each element on the extended grid.

create arrays needed for explicit Chebyshev pseudospectral treatment and create the plans for the forward and backward fast Fourier transforms

compute the Chebyshev spectral coefficients of the spatial derivative of f

Chebyshev transform f to get Chebyshev spectral coefficients