EngineeringAppliedExerciseFiniteDifferenceMethod.docx
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
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and the partial derivative with respect to y is
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If we applied these formula to our grid values, we would get the finite difference expressions
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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:
*You must show sufficient detail to support your work to earn credit for your calculations. This can be hand-written work, typed calculations, or excel formulas. To avoid round-off error, retain at least six decimal places in all of your calculations. Complete all trigonometric calculations in radians.
· 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:
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Estimated partial derivatives using finite difference formulas:
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h
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finite difference approx. to
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Exact
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finite difference approx. to
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Exact
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0.01
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0.001
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0.0001
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· 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?
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