MATLAB Function Reference Volume 1: A - E - Bad Request

2del2
Purpose Discrete Laplacian
Syntax L = del2(U)
L = del2(U,h)
L = del2(U,hx,hy)
L = del2(U,hx,hy,hz,...)
Definition If the matrix U is regarded as a function uxy ( , ) 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:
l
where:
l ij
∇ 2 u 1
---------- --
4 4
d2u dx 2
--------- d2u dy 2
⎛ ⎞
= = ⎜ + --------- ⎟
⎝ ⎠
1
= -- ( u
4 i + 1, j + ui – 1, j + ui, j + 1 + ui, j – 1)
– ui, in the interior. On the edges, the same formula is applied to a cubic
extrapolation.
For functions of more variables uxyz… ( , , , ) , del2(U) is an approximation,
l
∇ 2 u 1
---------- --------
2N 2N
d2u dx 2
--------- d2u dy 2
--------- d2u dz 2
⎛ ⎞
= = ⎜ + + --------- + …⎟
⎝ ⎠
where N is the number of variables in u.
Description L = del2(U) where U is a rectangular array is a discrete approximation of
∇
l
2 u 1
---------- --
4 4
d2u dx 2
--------- d2u dy 2
⎛ ⎞
=
= ⎜ + --------- ⎟
⎝ ⎠
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.
del2
2-413