Hyperbolic Problem2010

# Hyperbolic Problem2010 - 11/1/2010 Hyperbolic Problem...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 11/1/2010 Hyperbolic Problem Finite Difference Methods Advection Equation • Follow this link: http://www.cse.illinois.edu/iem/pde/discadvc/ • Here u=u(x,t) and c=c(x,t) • Using forward in time and central in space: Math Modeling II (Fall 2009, Xin Li) 1 11/1/2010 FD for Advection Equation • Rewrite: • where • This is unstable for all σ>0 • Lax method: replace u(x,t) by [u(x+h,t)+u(x‐h,t)]/2. (Note the misprint in the textbook!) • Upwind method: use backward/forward if c>0 or c<0, resp. • Lax‐Wendroff method: 2nd order in time/space Wave Equation • Model equation • Replace by finite differences (of order 2) Math Modeling II (Fall 2009, Xin Li) 2 11/1/2010 FD for the Wave Equation • This is an explicit method: where ρ=k2/h2 . The method is stable if ρ<1. • The equation comes with boundary conditions: • We can discretize them as: Incorporating the Boundary Conditions • Note that the above implies u(x,k)=u(x,0)=f(x). This use of O(k) approximation leads to low accuracy. • An alternative is to use [u(x,k)‐u(x,‐k)]/k to replace ut: [u(x,k)‐u(x,‐k)]/k=0. So, u(x,‐k)=u(x,k). • Use this in (when t=0): Math Modeling II (Fall 2009, Xin Li) 3 11/1/2010 Second order to first order • We can transform the wave equation to a first order system of equations (of hyperbolic type) • Let • Then • In matrix notation Homework 1. P. 604: 1 You are supposed to use the finite difference scheme, not any analytic solution. 2. P.604: 2 This should be quick and uses the chain rule. 3. P.604: 4 More differentiation. Math Modeling II (Fall 2009, Xin Li) 4 ...
View Full Document

## This note was uploaded on 11/27/2011 for the course MATH 3484 taught by Professor Staff during the Fall '10 term at University of Central Florida.

Ask a homework question - tutors are online