Examples

We show here some examples on how to compute efficiently Quasi-periodic Green Functions and its first derivatives for the 2D Helmholtz equation. The main focus is on the FFT-based algorithm, but we also provide examples for the lattice sums and Ewald summation algorithms.

Example 1: FFT-based algorithm

The first example is concerned with computing the quasi-periodic Green function for the 2D Helmholtz equation using the FFT-based algorithm (based on the article [1]).

First, load the package, using the following command:

using QPGreen

Then initialize the different parameters, as a NamedTuple, that define your problems. We have

# Parameters
params = (alpha=0.3, k=1.0, c=0.6, c_tilde=1.0, epsilon=0.45, order=8)

The parameters are defined as follows:

• alpha is the quasiperiod,
• k is the wavenumber,
• c and c_tilde are the coefficients introduced in [1] that define the vertical bounds of the computational domains used in the FFT-based algorithm, determining the regions where interpolation and Fourier-based approximations are valid (consider $x=(x_1, x_2) \in \mathbb{R}^2$, then if $|x_2| > c$ the we use the eigenfunction expansion, else the FFT-based algo),

• epsilon controls the support of smooth cutoff functions used to isolate and remove singularities,
• order denotes the regularity from the cut-off functions.

We set the grid size used to generate the 2D tensor product mesh (adjust for desired accuracy).

# Size of the Grid
grid_size = 128

This method relies on two steps: a preparation step and an evaluation step.

Preparation step

The preparation step, independent of evaluating the function at specific points, computes the necessary Fourier coefficients and produces an interpolation object for function and gradient evaluation (if the flag is true), along with a cache of integral approximations. This step needs to be performed only once for a fixed set of parameters and grid resolution.

# Preparation Step
value_interpolator, grad_interpolator, cache = init_qp_green_fft(params, grid_size; derivative=true)

Evaluation step

The evaluation step efficiently computes the Green’s function and its first derivative at any specified point within the periodic domain.

# Evaluation Step for any points
Z = (0.002π, 0.01)
eval_Green_function = eval_qp_green(Z, params, value_interpolator, cache; nb_terms=50)
grad_Green_function = grad_qp_green(Z, params, grad_interpolator, cache; nb_terms=50)
@show "The value of the Green function at the point $(Z) is: $(eval_Green_function)"
@show "The value of the gradient of the Green function at the point $(Z) is: $(grad_Green_function)"

"The value of the gradient of the Green function at the point (0.006283185307179587, 0.01) is: ComplexF64[-7.1188161590804295 + 0.06215371863722293im, -11.4148584648968 - 0.0013267471845958868im]"

Note here that the keyword argument nb_terms is used to control the number of terms in the eigenfunction expansion (in the case where the FFT-based algorithm is not used).

Example 2: Lattice sums algorithm

The second algorithm implemented is the lattice sums algorithm (implementation was done based on the article [2]).

Once again, load the package, using the following command:

using QPGreen

Then initialize the different parameters, as a NamedTuple, that define your problems. We have

X, Y = (0.0, 0.01 * 2π)

β, k, d, M, L = (√2 / (2 * π), 1 / π, 2 * π, 80, 4)
csts = (β, k, d, M, L)

Sl = QPGreen.lattice_sums_preparation(csts);
eval_ls = QPGreen.lattice_sums_calculation((X, Y), csts, Sl; nb_terms=100)

-0.4595073481223693 - 0.35087920819684576im

Example 3: Ewald's method

The last algorithm implemented is the Ewald summation algorithm (implementation was done based on the article [2]).

Load the package, using the following command:

using QPGreen

Then initialize the different parameters, as a NamedTuple, that define your problems. We have

(X, Y) = (0.0, 0.01 * 2π)

a, M₁, M₂, N, β, k, d = (2, 3, 2, 7, √2 / (2 * π), 1 / π, 2 * π)
csts = (a, M₁, M₂, N, β, k, d)

res_ewald = QPGreen.ewald([X, Y], csts)

-0.4595298794735017 - 0.3509130869382092im