Tut Sheet 06 PDF

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Tutorial Sheet 6...

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Numerical Methods 4H Problem sheet 6

Tutorial questions Question 1. Use the Taylor series expansions of u(x − 2h), u(x − h), u(x + h) and u(x + 2h) to derive the finite difference expression u(x + 2h) − 2u(x + h) + 2u(x − h) − u(x − 2h) d3 u = + O(h2 ). 3 dx 2h3 Question 2. Given the values u(x − h), u(x) and u(x + θh) where 0 < θ ≤ 1, find a second order accurate finite difference formula for u′ (x). Question 3. Two less common molecules for the computation of uxx + uyy are

(a)

1 0 1 0 −4 2h2 1 0

1 0 1

(b)

1 6h2

1 4 4 −20 1 4

1 4 . 1

Determine the order of the error term associated with each of these molecules. Determine also the local truncation error when using the molecules to solve Laplace’s equation.

Further questions Question 4. Given the values u−2 = u(x − 2h), u−1 = u(x − h), u0 = u(x) and uθ = u(x + θh) derive the finite difference formula u′′(x) =

6uθ + (θ − 3)(θ + 1)(θ + 2)u0 − 2θ(θ 2 − 2)u−1 + θ (θ 2 − 1)u−2 + O(h2 ) 2 θ(θ + 1)(θ + 2)h

Question 5. Use the values u(x), u(x ± h) and u(x ± 2h) to calculate a second order accurate finite difference approximation to u(4)(x). Question 6. Construct a finite difference approximation to   du d a(x) ψ(x) = dx dx that is second order accurate in terms of u(x), u(x ± h) and a(x) and a(x ± h)....