Hyperbolic Problem2011 - ρ<1 • The equation comes...

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10/24/2011 Math Modeling II (Fall 2009, Xin Li) 1 Hyperbolic Problem Finite Difference Methods Advection Equation Follow this link: http://www.cse.illinois.edu/iem/pde/discadvc/ 𝜕? 𝜕? = −𝑐 𝜕? 𝜕𝑥 Here ? = ?(𝑥, ?) and 𝑐 = 𝑐(𝑥, ?) Using forward in time and central in space: 1 𝑘 ? 𝑥, ? + 𝑘 − ? 𝑥, ? = −𝑐 1 2ℎ ? 𝑥 + ℎ, ? − ?(𝑥 − ℎ, ?)]
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10/24/2011 Math Modeling II (Fall 2009, Xin Li) 2 FD for Advection Equation Rewrite: where This is unstable for all σ >0 Lax method: replace ?(𝑥, ?) by spatial average ? 𝑥 + ℎ, ? + ? 𝑥 − ℎ, ? /2 . (Note the many misprints on page 602 in the textbook!) Upwind method: use backward /forward FD in space if c>0 or c<0, resp. Lax-Wendroff method: 2 nd order in time/space Wave Equation Model equation Replace by finite differences (of order 2)
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10/24/2011 Math Modeling II (Fall 2009, Xin Li) 3 FD for the Wave Equation This is an explicit method: where ρ =k 2 /h 2 . The method is stable if
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Unformatted text preview: ρ <1 . • The equation comes with boundary conditions: • We can discretize them as: Incorporating the Boundary Conditions • Note that the above implies ? °, ± = ? °, 0 = ²(°) . This use of ³(±) approximation leads to low accuracy. • An alternative is to use ´ µ,¶ −´ µ,−¶ 2¶ to replace ? ? : ´ µ,¶ −´ µ,−¶ 2¶ = 0 . So, ? °, ± = ? °, ·± . • Use this in (when ¸ = 0 ): 10/24/2011 Math Modeling II (Fall 2009, Xin Li) 4 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 • Now, ideas used for advection equation can be used. 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....
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  • Fall '11
  • Xli
  • Partial differential equation, finite difference, Finite difference method, Finite differences, Numerical differential equations, advection equation

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