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Unformatted text preview: MAE 290B - Homework # 5 Numerical Methods in Science and Engineering Prof. Alison Marsden Due date: Tues March 1, 2011 Problem 1 - Eigenvalues of a Tridiagonal Matrix. Let T be an ( N- 1) × ( N- 1) tridiagonal matrix, B [ a,b,c ]. Let D ( N- 1) be the determinant of T . (a) Show that D ( N- 1) = bD ( N- 2)- acD ( N- 3) . (b) Show that D ( N- 1) = r ( N- 1) sin θ sin( Nθ ), where r = √ ac and 2 r cos θ = b . Hint: use induction. (c) Show that the eigenvalues of T are given by λ j = b + 2 √ ac cos α j where α j = jπ N j = 1 , 2 ,...,N- 1 Problem 2 - Modified Wavenumber. Use the modified wavenumber analysis to show that application of the second order upwind spatial differencing scheme ∂ 2 φ ∂x 2 | j =- φ j +3 + 4 φ j +2- 5 φ j +1 + 2 φ j Δ x 2 would lead to numerical instability. Problem 3 - Convection - Diffusion. Consider the convection diffusion equation ∂T ∂t + u ∂T ∂x = α ∂ 2 T ∂x 2 ≤ x ≤ 1 with the boundary conditions T (0 ,t ) = 0 T (1 ,t ) = 0...
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This note was uploaded on 03/13/2011 for the course MAE 290B taught by Professor Marsden during the Winter '11 term at UCSD.
- Winter '11