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Finite Difference Method

Knowing the value of a two-dimensional function  at each node in the mesh, your objective is to calculate the partial derivatives  and at node

[Note that, in this example, the mesh sizes in x and y are identical (h); strictly speaking, this need not be true.  In some applications, we may need more resolution along the x- or y-axis; we could then use separate mesh sizes hx and hy.]

By definition, the partial derivative of a function  with respect to x and the partial derivative with respect to y is

If we applied these formula to our grid values, we would get the finite difference expressions

Note that these are approximations to the values of the partial derivatives since we’re not taking the limit as h goes to zero; but as h becomes smaller, the approximations should improve.

With this background, here’s your assignment:

• Assume the function f is defined as f(x, y) = 3 tan x cos y
• Use differentiation rules to find the exact partial derivatives  and ,  and evaluate those exact partial derivatives at (1.56, -2.1).
• Use the finite difference formulas to estimate  and ,   at (1.56, -2.1) for three different values of the mesh size

• • h = 0.01
  • h = 0.001
  • h = 0.0001

• Use your calculated values to fill in this table:

• Answer the following questions:

For which partial derivative (  or  is the finite difference approximation consistently more accurate? Why do you think the finite difference approximation for the other partial derivative is consistently less accurate?  (Hint: What happens to the tan x function near x = 1.56?). Under what real-world conditions might we see such poor approximations?