Computing Divergence of a vector field numerically

I am trying to compute the divergence at individual points of a vector field numerically. I have had issues with Matlab returning nonzero arrays when it should return all zero's. For example:

[x y] = meshgrid(0:pi/2:2*pi,0:pi/2:2*pi); u = sin(x); v = -y*cos(x); divergence(x,y,u,v)

The divergence of this field evaluates symbolically to 0, so why am I not getting this result numerically?

Thank you

Answer 1

[John D'Errico]

First, divergence computes a NUMERICAL approximation to the gradients necessary. It has no clue as to the real derivatives, or the real functions that created these variables. With such a coarse grid, you would never expect a true zero result, even if 0 is the correct result in symbolic terms.

Having said that, you need to understand the * and .* operators in MATLAB.

• is used for MATRIX multiplication.

.* is used for element-wise multiplication.

There IS a difference.

So, if we use a rather finer grid, and use the proper operator where necessary, you might see something more reasonable.

[x y] = meshgrid(linspace(0,2*pi,100)); u = sin(x); v = -y.*cos(x); divergence(x,y,u,v)

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