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

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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 • 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 ...
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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.

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