QPGreen

FFTCache(N::Integers, grid_size::Integer, params::NamedTuple, T=Float64)

Construct a containers for FFT operations.

Arguments

• N: total number of grid points in one dimension.
• grid_size: Number of half of the grid points in one dimension.
• params: Physical and numerical constants
• T: Floating-point type for allocations. Defaults to Float64.

Returns

An FFTCache object containing:

• j_idx: Vector of integers for the computations of Fourier coefficients [-grid_size, grid_size-1].

• t_j_fft: Spatial grid points in [-c̃, c̃].

• Preallocated complex vectors for FFT operations:

  • eval_int_fft_1D: 1D integration using FFT.
  • shift_sample_eval_int: Shifted samples for FFT.
  • fft_eval: FFT evaluation.
  • shift_fft_1d: Shifted FFT result.
  • fft_eval_flipped: Transposed result of FFT.

struct IntegrationCache{T1<:Real, T2<:Signed} <: QPGreen.AbstractIntegrationCache

Structure storing the normalization factor and the parameters of integration for the cutoff functions.

• normalization::Real: Normalization factor

• params::QPGreen.IntegrationParameters: Parameters of integration

struct IntegrationParameters{T1<:Real, T2<:Signed}

Structure storing the parameters of integration for the cutoff functions.

• a::Real: Lower bound

• b::Real: Upper bound

• order::Signed: Order of the cutoff function

S_even(l, β, k, d, M)

Compute the lattice sum Sₗ for even values of l. Input arguments:

• l: integer value.
• β: parameter β.
• k: parameter k.
• d: parameter d.
• M: number of terms in the sum.

Returns the value of the lattice sum Sₗ for even values of l.

S_odd(l, β, k, d, M)

Compute the lattice sum Sₗ for odd values of l.

Input arguments:

• l: integer value.
• β: parameter β.
• k: parameter k.
• d: parameter d.
• M: number of terms in the sum.

Returns the value of the lattice sum Sₗ for odd values of l.

S₀(β, k, d, M)

Compute the lattice sum S₀.

Input arguments:

• β: parameter β.
• k: parameter k.
• d: parameter d.
• M: number of terms in the sum.

Returns the value of the lattice sum S₀.

Yε(x, cache::IntegrationCache)

Evaluate the cutoff function Yε at the point x (C^∞ function).

Input arguments

• x: point at which the cutoff function is evaluated.
• cache: see IntegrationCache.

Returns

• The value of the cutoff function Yε at x.

Yε_1st_der(x, cache::IntegrationCache)

Evaluate the derivative of the cutoff function Yε at the point x.

Input arguments

• x: point at which the derivative of the cutoff function is evaluated.
• cache: see IntegrationCache.

Returns

• The value of the derivative of the cutoff function Yε at x.

Yε_2nd_der(x, cache::IntegrationCache)

Evaluate the 2nd order derivative of the cutoff function Yε at the point x.

Input arguments

• x: point at which the 2nd order derivative of the cutoff function is evaluated.
• cache: see IntegrationCache.

Returns

• The value of the 2nd order derivative of the cutoff function Yε at x.

asin_ls(x)

Compute the inverse sine function (arcsine function) for values outside of the interval [-1,1]. Input arguments:

• x: real values outside the interval [-1,1].

Returns the inverse sine function (arcsine function) evaluated at x. For values inside the interval [-1,1], use the standard asin function.

bernoulli(n, x)

Compute the Bernoulli polynomial of degree n evaluated at x. Input arguments:

• n: degree of the Bernoulli polynomial.
• x: value at which the polynomial is evaluated.

Returns the nth Bernoulli polynomial evaluated at x.

check_compatibility(alpha, k)

Check if the parameters alpha and k are compatible, i.e., if βₙ is different from zero. For βₙ equal to zero, the algorithm fails.

eigfunc_expansion(z, params::NamedTuple; period=2π, nb_terms=50)

Compute the α-quasi-periodic Green's function using the eigenfunction expansion.

Input arguments

• z: coordinates of the difference between the target point and source point.
• params: named tuple of the physical and numerical parameters for the problem definition.

Returns

• The value of the α-quasi-periodic Green's function for 2D Helmholtz equation at the point z, computed by the basic eigenfunction expansion.

eigfunc_expansion_derivative(z, params::NamedTuple; period=2π, nb_terms=50)

Compute the derivative of the α-quasi-periodic Green's function using the eigenfunction expansion.

Input arguments

• z: coordinates of the difference between the target point and source point.
• params: named tuple of the physical and numerical parameters for the problem definition.

Returns

• The value of the derivative of the α-quasi-periodic Green's function for 2D Helmholtz equation at the point z, computed via the basic eigenfunction expansion.

eval_qp_green(x, params::NamedTuple, interpolator, Yε_cache::IntegrationCache; nb_terms=50)

Compute the quasiperiodic Green's function $G(x)$ using the FFT-based method [1] with series expansion fallback.

Input arguments

• x: 2D point at which to evaluate the Green's function.

• params: Physical and numerical parameters containing:

  • alpha: Quasiperiodicity coefficient
  • k: Wavenumber
  • c: Lower cutoff parameter for function χ.
  • c_tilde: Upper cutoff parameter for function χ.
  • epsilon: cutoff parameter for function Yε (recommended: 0.4341).
  • order: Quadrature order.

• value_interpolator: Bicubic spline interpolator of Ln.

• Yε_cache: Precomputed cache for cutoff function Yε evaluations.

Keyword Arguments

• nb_terms: Number of terms to use in the eigenfunction expansion fallback (when x₂ ∉ [-c, c])

Returns

• G: The approximate value of the quasiperiodic Green's function at point x

eval_smooth_qp_green(x, params::NamedTuple, value_interpolator, Yε_cache::IntegrationCache; nb_terms=50)

Compute the smooth α-quasi-periodic Green's function $G_0(x)$ (i.e. without the term $H_0^{(1)(k|x|)}$ using the FFT-based method [1] with series expansion fallback.

Input arguments

• x: 2D point at which to evaluate the Green's function.

• params: Physical and numerical parameters containing:

  • alpha: Quasiperiodicity coefficient
  • k: Wavenumber
  • c: Lower cutoff parameter for function χ.
  • c_tilde: Upper cutoff parameter for function χ.
  • epsilon: cutoff parameter for function Yε (recommended: 0.4341).
  • order: Quadrature order

• value_interpolator: Bicubic spline interpolator for Ln.

• Yε_cache: Precomputed cache for cutoff function Yε evaluations.

Keyword Arguments

• nb_terms: Number of terms to use in the eigenfunction expansion fallback (when x₂ ∉ [-c, c])

Returns

• G_0: The approximate value of the quasiperiodic Green's function at point x

ewald(z, csts)

Evaluation of the Green's function using Ewald's method. Input arguments:

• z: coordinates.
• csts: tuple of constants (a, M₁, M₂, N, β, k, d).

Returns the value of the Green's function at the point z.

f₁(x, cache::IntegrationCache)

Calculate the function f₁.

Input arguments

• x: point at which the function is evaluated
• cache: cache for the cut-off functionYε`

Returns

• The value of the function f₁ at the point x.

f₂(x, cache::IntegrationCache)

Calculate the function f₂.

Input arguments

• x: point at which the function is evaluated
• cache: cache for the cut-off function Yε

Returns

• The value of the function f₂.

get_F̂ⱼ(j₁, j₂, c̃, Φ̂₁ⱼ, Φ̂₂ⱼ, F̂₁ⱼ₀::Complex{T}, ::Type{T}) where {T}

Calculate the Fourier coefficients F̂ⱼ.

Input arguments

• j₁: first index
• j₂: second index
• c̃: parameter of the periodicity
• Φ̂₁ⱼ: Fourier coefficients of the function Φ₁
• Φ̂₂ⱼ: Fourier coefficients of the function Φ₂
• F̂₁ⱼ₀: Fourier coefficient of the function F₁ at |j| = 0
• T: type of the parameter α

Returns

• The Fourier coefficients F̂ⱼ.

get_K̂ⱼ!(K̂ⱼ, params::NamedTuple, N, i, fft_cache::FFTCache{T}, cache::IntegrationCache, p) where {T}

Mutating function that computes the Fourier coefficients K̂ⱼ.

Input arguments

• K̂ⱼ: matrix to store the Fourier coefficients
• params: Named tuple containing physical and numerical constants
• N: size of the grid
• i: index of the grid points
• fft_cache: cache for the FFT
• cache: cache for the cut-off function Yε
• p: plan for the FFT

Returns

• The Fourier coefficients K̂ⱼ.

get_t(x)

Find the value of t in the interval [-π, π[ such that x = 2nπ + t.

Input arguments

• x: point at which the function is evaluated

Returns

• The value of t.

get_Ĥⱼ(j₁, j₂, params::NamedTuple, F̂₁ⱼ, F̂₂ⱼ, ρ̂ₓⱼ, Φ̂₃ⱼ, ĥ₁ⱼ, ĥ₂ⱼ, Ĥ₁ⱼ₀, T)

Calculate the Fourier coefficients Ĥⱼ.

Input arguments

- `j₁`: first index
- `j₂`: second index
- `params`: named tuple containing physical and numerical constants
- `F̂₁ⱼ`: Fourier coefficients of the function `F₁`
- `F̂₂ⱼ`: Fourier coefficients of the function `F₂`
- `ρ̂ₓⱼ`: Fourier coefficients of the function `ρₓ`
- `Φ̂₃ⱼ`: Fourier coefficients of the function `Φ₃`
- `ĥ₁ⱼ`: Fourier coefficients of the function `h₁_reduced`
- `ĥ₂ⱼ`: Fourier coefficients of the function `h₂_reduced`
- `Ĥ₁ⱼ₀`: Fourier coefficient of the function `H₁` at `|j| = 0`
- `T`: type of the parameter `α`

Returns

• The Fourier coefficients Ĥⱼ.

grad_qp_green(x, params::NamedTuple, grad::NamedTuple, Yε_cache::IntegrationCache; nb_terms=50)

Compute the first order derivative of the α-quasi-periodic Green's function $G(x)$ using the FFT-based method [1] with series expansion fallback.

Input arguments

• x: 2D point at which to evaluate the Green's function.

• params: Physical and numerical parameters containing:

  • alpha: Quasiperiodicity coefficient
  • k: Wavenumber
  • c: Lower cutoff parameter for function χ.
  • c_tilde: Upper cutoff parameter for function χ.
  • epsilon: cutoff parameter for function Yε (recommended: 0.4341).
  • order: Quadrature order

• grad: Bicubic spline interpolator for the gradient of Ln.

• Yε_cache: Precomputed cache for cutoff function Yε evaluations.

Keyword Arguments

• nb_terms: Number of terms to use in the eigenfunction expansion fallback (when x₂ ∉ [-c, c])

Returns

• ∇G: The approximate value of the fist order derivative of the quasiperiodic Green's function at point x

grad_smooth_qp_green(x, params::NamedTuple, grad::NamedTuple, Yε_cache::IntegrationCache; nb_terms=50)

Compute the first order derivative of the smooth α-quasi-periodic Green's function using the FFT-based method [1] with series expansion fallback.

Input arguments

• x: 2D point at which to evaluate the Green's function.

• params: Physical and numerical parameters containing:

  • alpha: Quasiperiodicity coefficient
  • k: Wavenumber
  • c: Lower cutoff parameter for function χ.
  • c_tilde: Upper cutoff parameter for function χ.
  • epsilon: cutoff parameter for function Yε (recommended: 0.4341).
  • order: Quadrature order

• grad: Bicubic spline interpolator for the gradient of Ln.

• Yε_cache: Precomputed cache for cutoff function Yε evaluations.

Keyword Arguments

• nb_terms: Number of terms to use in the eigenfunction expansion fallback (when x₂ ∉ [-c, c])

Returns

• ∇G_0: The approximate value of the fist order derivative of the smooth quasiperiodic Green's function at point x

h₁(x, params::NamedTuple, cache::IntegrationCache)

Calculate the function h₁.

Input arguments

• x: point at which the function is evaluated
• params: Named tuple containing physical and numerical constants

Returns

• The value of the function h₁.

h₁_reduced(x, x_norm, α, cache::IntegrationCache)

Calculate the function h₁_reduced (removal of the singularity in the Fourier space for the 1st order derivative).

h₂(x, params::NamedTuple, cache)

Calculate the function h₂.

Input arguments

• x: point at which the function is evaluated
• params: Named tuple containing physical and numerical constants

Returns

• The value of the function h₂.

h₂_reduced(x, x_norm, α, cache::IntegrationCache)

Calculate the function h₂_reduced (removal of the singularity in the Fourier space for the 1st order derivative).

image_expansion(z, params::NamedTuple; period=2π, nb_terms=50)

Compute the α-quasi-periodic Green's function using the image expansion.

Input arguments

• z: coordinates of the difference between the target point and source point.
• params: named tuple of the physical and numerical parameters for the problem definition.

Returns

• The value of the α-quasi-periodic Green's function for the 2D Helmholtz equation at the point z, computed via the basic image expansion.

image_expansion_derivative(z, params::NamedTuple; period=2π, nb_terms=50)

Compute the derivative of the α-quasi-periodic Green's function using the image expansion.

Input arguments

• z: coordinates of the difference between the target point and source point.
• params: named tuple of the physical and numerical parameters for the problem definition.

Returns

• The value of the derivative of the α-quasi-periodic Green's function for the 2D Helmholtz equation at the point z, computed via the basic image expansion.

image_expansion_derivative_smooth(z, params:NamedTuple; period=2π, nb_terms=50)

Compute the derivative of the smooth α-quasi-periodic Green's function using the image expansion.

Input arguments

• z: coordinates of the difference between the target point and source point.
• params: named tuple of the physical and numerical parameters for the problem definition.

Returns

• The value of the derivative of the smooth α-quasi-periodic Green's function for 2D Helmholtz equation at the point z, computed via the basic image expansion.

image_expansion_smooth(z, params::NamedTuple; period=2π, nb_terms=50)

Compute the smooth α-quasi-periodic Green's function (i.e. without the term $H_0^{(1)(k|x|)}$ using the image expansion.

Input arguments

• z: coordinates of the difference between the target point and source point.
• params: named tuple of the physical and numerical parameters for the problem definition.

Returns

• The value of the smooth α-quasi-periodic Green's function for the 2D Helmholtz equation at the point z, computed via the basic image expansion.

init_qp_green_fft(params::NamedTuple, grid_size::Integer; derivative=false)

Preparation step of the FFT-based algorithm.

Input arguments

• params: Physical and numerical parameters, containing:

  • alpha: Quasiperiodicity coefficient.
  • k: Wave number.
  • c: Lower cutoff parameter for function χ.
  • c_tilde: Upper cutoff parameter for function χ.
  • epsilon: cutoff parameter for function Yε (recommended: 0.4341).
  • order: Quadrature order for integration.

• grid_size: Number of grid points per dimension (grid is 2 * grid_size × 2 * grid_size).

Keyword Arguments

• derivative: if true, computes additionally the first-order derivative (∇G) of the quasi periodic Green's function.

Returns

- If `derivative=false`: a NamedTuple with fields
        + `value`: Spline interpolator for the function `Ln`.
        + `cache`: Precomputed integration cache for reuse in later computations.
- If `derivative=true`: a NamedTuple with fields
        + `value`: Spline interpolator for the function `Ln`.
        + `grad`: Tuple of spline interpolators for the first derivatives of `Ln` (`∂/∂x₁`, `∂/∂x₂`).
        + `cache`: Precomputed integration cache for reuse in later computations.

lattice_sums_calculation(x, csts, Sₗ; c=0.6, nb_terms=100)

Compute the Green's function using lattice sums. Input arguments:

• x: evaluation point
• csts: tuple of constants (β, k, d, M, L)
• Sₗ: lattice sum coefficients
• c: cutoff value
• nb_terms: number of terms in the sum

Keyword arguments:

• c: cutoff value
• nb_terms: number of terms in the sum

Returns the value of the Green's function at the point x.

lattice_sums_preparation(csts)

Perform the preparation step for the lattice sums algorithm. Input arguments:

• csts: tuple of constants (β, k, d, M, L) representing quasi-periodicty parameter, wave number , period, number of terms, and number of terms in the sum respectively.

Returns the lattice sum coefficients.

process_frequency_component!(i, N, params, fft_cache, χ_cache, fft_plan, K̂ⱼ)

Internal helper for processing a single frequency component during quasi-periodic Green's function computation. Handles coefficient calculations and frequency index mapping for position i in the FFT grid.

rfftshift_normalization!(Φ̂₁ⱼ, fft_Φ₁_eval, N, c̃)

Shift the Fourier coefficients obtained by rfft and normalize them.

Φ₁(x, cache::IntegrationCache)

Calculate the function Φ₁ (removal of the singularity in the Fourier space).

ρₓ(x, x_norm, cache::IntegrationCache)

Calculate the function ρₓ (removal of the singularity in the Fourier space for the 1st order derivative).

χ(x, cache::IntegrationCache)

Evaluate the cutoff function χ at the point x (C^∞ function).

Input arguments

• x: point at which the cutoff function is evaluated.
• cache: see IntegrationCache.

Returns

• The value of the cutoff function χ at x.

χ_der(x, cache::IntegrationCache)

Evaluate the derivative of the cutoff function χ at the point x.

Input arguments

• x: point at which the derivative of the cutoff function is evaluated.
• cache: see IntegrationCache.

Returns

• The value of the derivative of the cutoff function χ at x.