d n U i dx n dx n δ n U i

Transcription

1 Last time Taylor s series on equally spaced nodes Forward difference d n U i d n n U i h n + 0 h Backward difference d n U i d n n U i h n + 0 h Centered difference d n U i d n δ n U i or 2 h n + 0 h2 for odd n Requires n + points and in general n U i = n- U i ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3

2 For greater accuracy one must add points Uncentered: n + m points for dn U i d n i + 0 h m Centered: n+(m-) points for dn U i d n i + 0 h m m must be even For unequal spacing the centered approimation looses its meaning and accuracy is reduced to uncentered approimations Alternative to Taylor Series : Polynomial Fit h αh i - i i + Approimate U via U = a 2 + b + c 3 unknowns (a, b, c) require 3 equations ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 2

3 U i- = ah 2 +b-h +c U i = c U i+ =a(αh) 2 +b(αh) + c h 2 -h 0 0 (αh) 2 αh a b c = U i- U i U i+ Solve for a, b, c, a b c = α U i- -U i +U i+ -U i / α 2 +α h 2 α 2 U i- -U i +U i+ -U i / α 2 +α h U i du d = 2a + b d 2 U d 2 = 2a d 2 U d 2 = 2U i+ - + α U i + αu i- αα + h 2 i.e. The same as truncated Taylor Series ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 3

4 Loss of the leading error term is significant. By retaining the leading error in a Taylor Series one is continually reminded of the accuracy of the approimations and the origin of the formulation, (i.e. PDE) is more easily retained. Alternate use of Error Term d U i ~ U i - d h h d 2 U i + 0 h 2 2 d 2 2 U i h + 0 h 2 du i d U i h - h 2 2 U i h h + 0 h2 U i h - 2 U i 2h +0h2 [ U i+ -U i - 2 U i+2-2u i+ +U i ] h + 0h ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 4

5 du i d -U i+2 +4U i+ 3U i 2h +0h 2 i.e. The same as if derived directly from Taylor Series Difference Formulas for Cross-Derivatives a.) 2-D Taylor Series : U( +, y+ y) = U + + y U + + y U y 2! y, y, y, y + + y Uy, + + hot... 3! y 3 2 Where y + y + 2 y 2 2 y y y etc. ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 5

6 Procedure as in -D case: write Taylor series for all points in terms of U,, y, at point where 2 U y is wanted; mi together to get desired accuracy. b.) Easier: Operate on -D formulas. U U U U = y y 2 y 2 j+ j j+ j 2 U U U j+ U j 2 y 2 y i+ i U U y j i Intuitive ; What is leading error? ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 6

7 From -D: 2 3 U U j+ U j ( y) U + 3 y 2 y 6 y U U j+ U j ( y) U U j+ U j ( y) U 3 3 y y 6 y 2 y 6 y U j+ U j U j+ U j 3 U j+ U j 2 y 2 y U i+ i ( ) 2 y ( y) U 3 3 y y U j+ U j U j+ U j 2 y 2 y U i+ i ( ) U ( y) U 3 3 y y 6 y So the leading error terms are symmetrical. ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 7

8 Consider an Elliptic PDE (Poisson s Eqn.) U = 0 U=f 2 U = g y U = 0 y U = a "Shadow Nodes" A B C PDE : U U U 2U + U U 2U + U g y h k 2 2 i+, j i, j i, j i, j+ i, j i, j + = ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 8

9 Define = h2 k h 2 h 2 h 2 k 2-2 k 2 k 2 = g Computational Molecule gh 2 = -2 + Valid at all interior nodes ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 9

10 U Boundary A : = 0 U - U h = = gh 2 i.e. U = U Boundary B : = gh 2 ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 0

11 Boundary C : U -U h = a U = U + ah = - ah + gh 2 Top Row (nodes 9-2) -2 + = f + gh 2 ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3

12 Corners : e. node = - ah + gh 2 Basic Rule : Type I Boundary : Do not use the PDE Type II, III : Use PDE plus BC, together The node spacing adjacent to the Type II boundaries are spaced at a /2 format. This is NOT required. It IS accurate, however, the unknown values (u) on the boundary are not calculated. This may be a disadvantage. Similarly, if the geometry does not lend itself to this spacing conveniently - do not use it. The alternate strategy is to place nodes directly on the boundary. If nodes are on the boundary then the shadow node contributions do not usually move to the diagonal. ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 2

13 U 2 4 U 2 U 3 6 = U 2 Forcing + BC's ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 3

14 For = : (h = k) -4 = G Solution Strategies gh 2 Direct: - Eact algebraic solution in finite # of steps - Non-repetitive - Complicated coding - Eploit sparse / banded structure - Node numbering : dictates the banded structure - LU Decomposition popular - Preserves bandwidth - Back Substitution step easy relative to decomposition numerous solutions to the same matri can be gotten cheaply. ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 4

15 Iterative: - Eact algebraic solution only after # of steps - Monotonously repetitive - Coding simple; proceed directly from molecule - Eploit sparseness; banded structure irrelevant - Retain double subscripts U ij - Node numbering important: - Dictates order in which iterations proceed - Can determine convergence properties - Point versus "Block" or "Line" methods: Matri Inversion (usually 3-Diag) ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 5

16 y, j -4 = gh 2 = G, i U i+, j + U i-, j + U i, j+ + U i, j- - 4U i, j = G Jacobi : Solve for U i, j U n+ i,j = 4 U i+, j + U i-, j + U i, j+ + U i, j- - G n - Easy - Need 2 arrays U n,u n+ - Iteration independent of node # s / order of calculation Gauss - Seidel : Use latest info within Jacobi U n+ i,j = 4 U n i+, j + U n+ n i-, j + U i, j+ + U n+ i, j- - G ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 6

17 - Easy - Only one array - Ordering makes difference! S.O.R. : Accelerate / Dampen the Gauss - Seidel : U n+ i,j Gauss - Seidel estimate for U n+ i,j n+ U i,j = ωu n+ n i,j + - ω U i,j = n Ui,j + ω U n+ n i,j - U i,j U = ω [ U + U + U + U G ] + ( ω ) U 4 n+ n n+ n n+ n i, j i+, j i, j i, j+ i, j i, j where 0 < ω < 2 ME 525 (Sullivan) - Finite Difference Calculus Cont. - Lecture 3 7