Parabolic Problem2010 - 10/29/2010 Parabolic Problem...

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10/29/2010 Math Modeling II (Fall 2009, Xin Li) 1 Parabolic Problem Chapter 15, Section 1 Classification of PDEs y General form of linear second-order PDEs with two independent variables 0 g fu eu du cu bu au = + + + + + + y Linear PDEs: a, b, c,….,g = f(x,y) only y x yy xy xx < = > roots) complex (2 Elliptic , 0 ac 4 b root) double (1 Parabolic , 0 ac 4 b roots) real (2 Hyperbolic , 0 ac 4 b 2 2 2 y We will treat the model equations (Heat equations, …)
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10/29/2010 Math Modeling II (Fall 2009, Xin Li) 2 Heat Equation: Parabolic PDE y Heat transfer in a one-dimensional rod x = 0 x = a a x f(x), ) u(x, T t a, x t u x u c < < = < < = 0 0 0 0 ; 2 2 g 1 (t) g 2 (t) T t , (t) g u(a,t) (t) g ,t) u( = = 0 0 2 1 9 10 Discretize the solution domain in space and time with h = Δ x and k = Δ t t 2 3 4 5 6 7 8 0 1 01234567891 0 Time ( jindex ) space ( i index ) x
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10/29/2010 Math Modeling II (Fall 2009, Xin Li) 3 Initial and Boundary Conditions 9 10 Explicit Euler method 3 4 5 6 7 8 u(0, t) = g 1 (t) u(a, t) = g 2 (t) 0 1 2 01234567891 0 Initial conditions : u(x,0) = f(x) Heat Equation y Finite-difference (i,j+1) u(x,t) t t j+1 (i,j) (i+1,j) (i-1,j) x x t x x i x t j i i+1 i-1 ) 2 ( ) ( 1 , 1 , , 1 2 , 1 , j i j i j i xx j i j i t u u u h c cu u u k u + + + = = Forward-difference Central-difference at time level j
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10/29/2010 Math Modeling II (Fall 2009, Xin Li) 4 Explicit Method y Explicit Euler method for heat equation = = Δ = ih n a x h i x , / y Rearrange = = Δ = jk m T t k j t , / ) u u 2 u ( h c ) u u ( k 1 cu u j , 1 i j , i j , 1 i 2 j , i 1 j , i xx t + + + = = j i j i j i j i j i u u u h ck u u , 1 , , 1 2 , 1 , ) 2 ( + + + + = ck Δ j i j i j i u u u , 1 , , 1 ) 2 1 ( + + + = σ 2 2 x t c h Δ = = 5 . 0 0 < Stability: Explicit Euler Method y Stable j i j i j i j i u u u u , 1 , , 1 1 , ) 2 1 ( + + + + = y Unstable (negative coefficients) j i j i j i j i j i j i j i j i j i j i j i j i j i j i j i u u u u u u u u u u u u u u u , 1 , 1 1 , , 1 , , 1 1 , , 1 , , 1 1 , , 1 , , 1 1 , 5 . 0 5 . 0 5 . 0 4 . 0 2 . 0 4 . 0 4 . 0 1 . 0 8 . 0 1 . 0 1 . 0 01 . 0 98 . 0 01 . 0 01 . 0 + + + + + + + + + = = + + = = + + = = + + = = j i j i j i j i j i j i j i j i j i j i j i j i u u u u u u u u u u u u , 1 , , 1 1 , , 1 , , 1 1 , , 1 , , 1 1 , 100 199 100 100 10 19 10 10 1 + + + + + + + = = + = = + = =
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10/29/2010 Math Modeling II (Fall 2009, Xin Li) 5 Hands Hands-on Example on Example: : Explicit Explicit Method Method y Heat Equation (Parabolic PDE) 40 0 2 u(x,0) 1 x 0 ; + = = xx t x cu u y Assume c = 0.5, h = 0.25, k = 0.05 60 ) , 1 ( , 20 t) u(0, 2 = = t t e t u e 2 12 34 0 20 + 40 x 60e -2t 20e -t 0 1 Example (continued) y Explicit Euler method h ck 2 2 2 1 4 . 0 ) 25 . 0 ( ) 05 . 0 )( 5 . 0 ( + + = = = σ y First step: t = 0.05 j i j i j i j i j i j i j i u u u u u u u , 1 , , 1 , 1 , , 1 1 , 4 . 0 2 . 0 4 . 0
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Parabolic Problem2010 - 10/29/2010 Parabolic Problem...

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