del2 - Discrete Laplacian

Syntax

Definitions

If the matrix U is regarded as a function u(x,y) evaluated at the point on a square grid, then 4*del2(U) is a finite difference approximation of Laplace's differential operator applied to u, that is:

where:

in the interior. On the edges, the same formula is applied to a cubic extrapolation.

For functions of more variables u(x,y,z,...), del2(U) is an approximation,

where N is the number of variables in u.

Description

L = del2(U) where U is a rectangular array is a discrete approximation of

The matrix L is the same size as U with each element equal to the difference between an element of U and the average of its four neighbors.

L = del2(U) when U is an multidimensional array, returns an approximation of

where N is ndims(u).

L = del2(U,h) where H is a scalar uses H as the spacing between points in each direction (h=1 by default).

L = del2(U,hx,hy) when U is a rectangular array, uses the spacing specified by hx and hy. If hx is a scalar, it gives the spacing between points in the x-direction. If hx is a vector, it must be of length size(u,2) and specifies the x-coordinates of the points. Similarly, if hy is a scalar, it gives the spacing between points in the y-direction. If hy is a vector, it must be of length size(u,1) and specifies the y-coordinates of the points.

L = del2(U,hx,hy,hz,...) where U is multidimensional uses the spacing given by hx, hy, hz, ...

Tips

MATLAB software computes the boundaries of the grid by extrapolating the second differences from the interior. The algorithm used for this computation can be seen in the del2 program file code. To view this code, type:

type del2

Examples

The function:

has:

For this function, 4*del2(U) is also 4.

[x,y] = meshgrid(-4:4,-3:3);
U = x.*x+y.*y  
U =
    25     18     13     10      9     10     13     18     25
    20     13      8      5      4      5      8     13     20
    17     10      5      2      1      2      5     10     17
    16      9      4      1      0      1      4      9     16
    17     10      5      2      1      2      5     10     17
    20     13      8      5      4      5      8     13     20
    25     18     13     10      9     10     13     18     25

V = 4*del2(U)
V =
     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      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      4      4      4      4
     4      4      4      4      4      4      4      4      4

See Also

diff | gradient

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