HW5 - MAE 290B - Homework # 5 Numerical Methods in Science...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This note was uploaded on 03/13/2011 for the course MAE 290B taught by Professor Marsden during the Winter '11 term at UCSD.

Page1 / 2

HW5 - MAE 290B - Homework # 5 Numerical Methods in Science...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online