apma4301_IterativeMG08

apma4301_IterativeMG08 - Lecture #9 (Part B) 29 October...

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1of97 Applied Mathematics 4301: Numerical Methods for PDEs Introduction to Iterative Methods of Linear Algebra and Multigrid Wed aft. 4:10-6:40 S. W. Mudd Bldg. 1024 Prof. David Keyes, instructor S. W. Mudd Bldg. 215 [email protected] Yan Yan, teaching assistant [email protected] Lecture #9 (Part B) 29 October 2008
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2of97 Mostly adapted from “A Multigrid Tutorial” • Bill Briggs –(fo rme r SIAM VP for Education) •V a n H e n s o n r Depu ty Director of the ISCR, LLNL) • Steve McCormick –(co -PI in TOPS )
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3of97 1. Model problems • 1-D boundary value problem: •G r i d : • Let & for . σ , < < ) ( = ) ( σ + ) ( x x f x u x u 0 1 0 u u = ) ( = ) ( 0 1 0 x u v i i ) ( x f f i i ) ( x 0 x 1 x 2 x i x = 0 x = 1 This discretizes the variables, but what about the equations? h = 1 N + 1 , x i = ih , i = 0,1, K , N + 1 x N + 1 i = K , N + 1
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4of97 Approximate u ’’(x) via Taylor series • Approximate 2 nd derivative using Taylor series: h O x u h x u h x u h x u x u ) ( + ) ( + ) ( + ) ( + ) ( = ) ( i i i i i + 4 3 2 1 ! 3 ! 2 h O x u h x u h x u h x u x u ) ( + ) ( ) ( + ) ( ) ( = ) ( i i i i i 4 3 2 1 ! 3 ! 2 0 0 h O h x u x u x u x u ) ( + ) ( + ) ( ) ( = ) ( i i i i + 2 2 1 1 2 • Summing & solving: +
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5of97 Approximate equation via finite differences • Approximate the BVP by a finite difference scheme: = σ + + f v h v v v i i i i i + 2 1 1 2 v v 0 = = N+1 0 i = 1,2,…,N σ , < < ) ( = ) ( σ + ) ( x x f x u x u 0 1 0 u u = ) ( = ) ( 0 1 0
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6of97 Discrete model problem Letting and we obtain the matrix equation Av =f ,where A is Nx N , symmetric, positive definite, & v ) , . . . , , ( = v v v 2 1 T N f , . . . , , ( = f f f 2 1 N ) T A = 1 h 2 2 + σ h 2 1 12 + h 2 1 + h 2 O OO 1 + h 2 1 + h 2 v = v 1 v 2 v 3 M v N 1 v N f = f 1 f 2 f 3 M f N 1 f N
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7of97 Stencil notation A= [-1 2 -1] dropping h -2 & σ for convenience QQ Q 2 -1 -1
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8of97 2-D model problem •C o n s i d e r t h e p r o b l e m o n s i d e r t h e g r i d ) , ( = ) , ( h j h i y x y x j i 0 L+1 i 0 M+1 j x y z < < , < < , ) , ( = σ + y x y x f u u u y y x x 1 0 1 0 h x = 1 L + 1 , h y = 1 M + 1 , u=0, when x=0, x=1, y=0, or y=1 σ≥0 u
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9of97 Discretizing the 2-D problem • Let & . Again, using 2 nd - order finite differences to approximate & we arrive at the approximate equation for the unknown , for i =1,2, …L & j =1,2, …, M : • Ordering the unknowns (& also the vector f ) lexicographically by y-lines: y x u v j i j i ) , ( y x f f j i j i ) , ( u x x u y y y x u ) , ( j i = σ + + + + f v h v v v h v v v j i j i y j i j i j i x j i j i j i + , , , + , 2 1 1 2 1 1 2 2 v i , j = 0: i = 0, i = L + 1, j = j = M + 1 v = ( v 1,1 , v 1,2 , K , v M , v 2,1 , v 2,2 , K , v 2, M , K , v L ,1 , v L ,2 , K , v L , M ) T
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