Parabolic Problem2011 - 10/13/2011 Math Modeling II (Fall...

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Unformatted text preview: 10/13/2011 Math Modeling II (Fall 2009, Xin Li) 1 15.1 Parabolic Problem Finite Difference Method Classification of PDEs General form of linear second-order PDEs with two independent variables Linear PDEs: a, b, c,.,g = f(x,y) only We will treat the model equations (Heat equations, ) g fu eu du cu bu au y x yy xy xx roots) complex (2 Elliptic , ac 4 b root) double (1 Parabolic , ac 4 b roots) real (2 Hyperbolic , ac 4 b 2 2 2 10/13/2011 Math Modeling II (Fall 2009, Xin Li) 2 Heat Equation: Parabolic PDE Heat transfer in a one-dimensional rod T t , (t) g u(a,t) (t) g ,t) u( a x f(x), ) u(x, T t a, x t u x u c ; 2 1 2 2 x = 0 x = a g 1 (t) g 2 (t) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Discretize the solution domain in space and time with h = x and k = t Time ( j index ) space ( i index ) x t 10/13/2011 Math Modeling II (Fall 2009, Xin Li) 3 Initial and Boundary Conditions 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Initial conditions : u(x,0) = f(x) u(0, t) = g 1 (t) u(a, t) = g 2 (t) Explicit Euler method Heat Equation Finite-difference (i,j) (i+1,j) (i-1,j) (i,j+1) u(x,t) x x t t x i x i+1 x i-1 t j t j+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 10/13/2011 Math Modeling II (Fall 2009, Xin Li) 4 Explicit Method Explicit Euler method for heat equation Rearrange jk m T t k ih n a x h j i t , / x , / ) 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 j i j i j i u u u u u u h ck u u , 1 , , 1 , 1 , , 1 2 , 1 , ) 2 1 ( ) 2 ( 2 2 x t c h ck 5 . Stability: Explicit Euler Method Stable Unstable (negative coefficients) j i j i j i j i u u u u , 1 , , 1 1 , ) 2 1 ( 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 . 5 . 5 . 4 . 2 . 4 . 4 . 1 . 8 . 1 . 1 . 01 . 98 . 01 . 01 . 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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This note was uploaded on 11/27/2011 for the course MAP 4371 taught by Professor Xli during the Fall '11 term at University of Central Florida.

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Parabolic Problem2011 - 10/13/2011 Math Modeling II (Fall...

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