FourierCollocation

Package: kwave.toolbox

Derivative calculation using the Fourier collocation spectral method.

Description

Class used to calculate differential operators using the Fourier collocation spectral method. The class constructor takes a kwave.toolbox.Grid object which defines the discrete locations where the input field is defined. Calculations are always performed on the padded grid. Methods are provided for calculating the gradient of a scalar field, and the divergence of a vector field.

Vector fields are stored by concatenating the Cartesian components in the fourth dimension. For example, in 2D, a vector field has size (Nx, Ny, 1, 2), where (:, :, 1, 1) is the x-component, and (:, :, 1, 2) is the y-component. The divergence method acts on each vector component individually using divergenceSplit, and then sums the results. The calculations allow shifting to and from the staggered grid by setting the Staggering option.

To include k-space dispersion correction in the derivative calculations, the property kappa should be defined. As this is a computational parameter, kappa must be defined on the padded grid, or be scalar. If kappa is scalar or not defined, derivative calculations use 1D FFTs, otherwise, ND FFTs are used.

Examples

Calculate the gradient of a scalar field in 2D, and compare with the MATLAB Gradient function.

% Define input field.
[x, y] = meshgrid(-2:.2:2, -2:.2:2);
z = x .* exp(-x.^2 - y.^2);

% Define input field.
kgrid = kwave.toolbox.Grid([40, 40], 0.2);
f = kgrid.x .* exp(-kgrid.x.^2 - kgrid.y.^2);

% Compute gradient using k-Wave.
fourierDiffOps = kwave.toolbox.FourierCollocation(kgrid);
gradf = fourierDiffOps.gradient(f);

% Compute gradient using finite differences with MATLAB gradient function.
[fy, fx] = gradient(f, kgrid.dx);

% Plot.
figure;
subplot(2, 3, 1);
imagesc(gradf(:, :, :, 1));
colorbar;
axis image;
title('k-Wave \partialf/\partialx');

subplot(2, 3, 2);
imagesc(fx);
colorbar;
axis image;
title('Finite Difference \partialf/\partialx');

subplot(2, 3, 3);
imagesc(abs(gradf(:, :, :, 1) - fx));
colorbar;
axis image;
title('Difference');

subplot(2, 3, 4);
imagesc(gradf(:, :, :, 2));
colorbar;
axis image;
title('k-Wave \partialf/\partialy');

subplot(2, 3, 5);
imagesc(fy);
colorbar;
axis image;
title('Finite Difference \partialf/\partialy');

subplot(2, 3, 6);
imagesc(abs(gradf(:, :, :, 2) - fy));
colorbar;
axis image;
title('Difference');

Input Arguments

  • kgrid - (kwave.toolbox.Grid) Object which defines the simulation grid size.

Properties

Input objects:

  • kgrid - (kwave.toolbox.Grid) Handle for grid object.

Other properties:

  • dimensions - (double) Number of grid dimensions (1, 2, or 3).

Methods