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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
• 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 ...
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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.
- Fall '10