`del2`

Discrete Laplacian of an array.

Category: Mathematics > Numerical Integration and Differential Equations

Syntax

L = del2(U)

Returns the discrete Laplacian of U with unit spacing, scaled by 1/4 for a matrix (1/2n for n dimensions).

L = del2(U, h)

Uses the spacing h in all directions.

L = del2(U, hx, hy, ...)

Uses a separate spacing for each dimension.

Description

In the interior, each value is the mean of the second differences along every dimension, which approximates one fourth of the Laplacian d2u/dx2 + d2u/dy2 for a matrix. Near the edges, the stencil is extrapolated linearly from the neighbors.

Multiply by 4 (for a matrix) to get the plain Laplacian. For a quadratic function, the discrete Laplacian is exact and constant.

Input arguments

U: Numeric vector, matrix or N-D array of samples on a regular grid.
h, hx, hy: Scalar spacings, or coordinate vectors.

Output arguments

L: Array the same size as U.

Example

Laplacian of x^2 + y^2

[X, Y] = meshgrid(0:4);
U = X.^2 + Y.^2;
4*del2(U)

Output

ans = 5×5

     4     4     4     4     4
     4     4     4     4     4
     4     4     4     4     4
     4     4     4     4     4
     4     4     4     4     4

del2