Partitioned Methods for Multifield Problems

Transcription

1 Partitioned Methods for Multifield Problems Joachim Rang, Joachim Rang Partitioned Methods for Multifield Problems Seite 1

2 Contents Blockform of linear iteration schemes Examples Joachim Rang Partitioned Methods for Multifield Problems Seite 2

3 Contents Blockform of linear iteration schemes Examples Joachim Rang Partitioned Methods for Multifield Problems Seite 3

4 Motivating example: Heat equation Consider the one-dimensional heat equation with Dirichlet boundary conditions u u = exp( t)(x 2 + 2) Exact solution: u(t) = 1 + exp( t)x 2 space discretisation: central differences time discretisation: fully implicit Runge Kutta method with s internal stages Aim: Compute one timestep Joachim Rang Partitioned Methods for Multifield Problems Seite 4

5 Introduction Coupled linear system: A A 1n.. A n1... A nn x 1. x n = b 1. b n, where A ij R n i,n j and b i, x i R n i. Let x = (x 1,..., x n ) and b = (b 1,..., b n ). Moreover A is the matrix containing of the blocks A ij. Let the matrix A be regular Joachim Rang Partitioned Methods for Multifield Problems Seite 5

6 Introduction monolythical approach: Solve Ax = b by a direct or an iterative method. Partitioned method: Solve A ii x i = b n A ij x j j=0 j i is solved w.r.t. x i. Since the other components of x are not known in general, an iterative loop is needed to get an accurate numerical solution Joachim Rang Partitioned Methods for Multifield Problems Seite 6

7 Example 2 2-block system: A 11 x 1 + A 12 x 2 = b 1 (1) A 21 x 1 + A 22 x 2 = b 2. In the first step we solve (1) w.r.t. x 1, i. e. x 1 = A 1 11 (b 1 A 12 x 2 ). Next we communicate the solution x 1 from software 1 to software 2. Then we can solve equation (1) w.r.t. x 2 and get x 2 = (A 22 A 21 A 1 11 A 12) 1 (b 2 A 21 A 1 11 b 1). (2) Joachim Rang Partitioned Methods for Multifield Problems Seite 7

8 Block Jacobi method 1. Set ν := 0 and choose an initial guess x (0) = (x (0) 2. Solve (1) w.r.t. x 1, i. e. 3. Solve (2) w.r.t. x 2, i. e. 4. Set ν := ν Compute the residuum A 11 x (ν+1) 1 = b 1 A 12 x (ν) 2. A 22 x (ν+1) 2 = b 2 A 21 x (ν) 1. r (ν) := Ax (v) b. 1, x(0) 2 ). 6. If r (ν) is sufficiently small, then stop. Otherwise compute x (v+1). This method is called Block Jacobi method or simultaneous iteration method Joachim Rang Partitioned Methods for Multifield Problems Seite 8

9 Block-Gauß Seidel method 1. Set ν := 0 and choose an initial guess x (0) = (x (0) 2. Solve (1) w.r.t. x 1, i. e. 3. Solve (2) w.r.t. x 2, i. e. 4. Set ν := ν Compute the residuum A 11 x (ν+1) 1 = b 1 A 12 x (ν) 2. A 22 x (ν+1) 2 = b 2 A 21 x (ν+1) 1. r (ν) := Ax (v) b. 1, x(0) 2 ). 6. If r (ν) is sufficiently small, then stop. Otherwise compute x (v+1). This method is called Block Gauß Seidel method or sucessive iteration method Joachim Rang Partitioned Methods for Multifield Problems Seite 9

10 Example Let ( 2 1 A 11 = 1 2 ( 1 1 A 21 = 1 1 ) ( 10 1, A 12 = ), A 22 = ( ) ) and let b 1 = ( ) ( 13, b 2 = 14 ) Joachim Rang Partitioned Methods for Multifield Problems Seite 10

11 Result Joachim Rang Partitioned Methods for Multifield Problems Seite 11

12 Blockform of linear iteration methods Notation: Let A, B R n,n be given matrices. A B, if a ij b ij holds for all i, j {1,..., n}. A B, if a ij b ij holds for all i, j {1,..., n}. A is called nonnegative, if A 0, A is called positive, if A > 0, A is called monotone, if Ax 0 implies x 0. Theorem: A matrix A is monotone, if and only if A is regular with A 1 0. Proof: see Axelsson (1996) Joachim Rang Partitioned Methods for Multifield Problems Seite 12

13 Definition A matrix A with A = (a ij ) is called a (nonsingular) M-Matrix, if a ij 0 for i j and if it is monotone, i. e. A 1 0. By M(A) = (m ij (A)) we denote the comparison matrix given by { aij, if i = j m ij (A) := a ij, if i j. If M(A) is a M-Matrix, then A is called H-matrix Joachim Rang Partitioned Methods for Multifield Problems Seite 13

14 Example Let Then A = M(A) = is a M-Matrix and it follows that A is an H-matrix Joachim Rang Partitioned Methods for Multifield Problems Seite 14

15 Splittings types of splittings: Let C, R R n,n be matrices. Then the matrix A = C R is called: a regular splitting, if C is monotone and R 0 (see Varga (1963)) a weak regular splitting, if C is monotone and C 1 R 0 (see Ortega and Rheinboldt (1970)) a nonnegative splitting, if C is regular and C 1 R 0 (see Beauwens (1979) and Song (1991)) a convergent splitting, if C is regular and ρ(c 1 R) < Joachim Rang Partitioned Methods for Multifield Problems Seite 15

16 Block-iterative methods Let A be a M-matrix with a ij 0 for i j. Moreover let A be of the form D 1 A A 1n A 21 D 2... A 2n A =....., A 1n A n2... D n where D i R n i,n i. It can be shown that D i is regular and D 1 i 0. Let D A := diag(d 1,..., D n ). Then we have D 1 A Joachim Rang Partitioned Methods for Multifield Problems Seite 16

17 Theorem Let A = C R be a weak regular splitting. Then the splitting is convergent if and only if A is monotone. Proof: Axelsson (1996) Joachim Rang Partitioned Methods for Multifield Problems Seite 17

18 Block Jacobi method Splitting: R = DA A, i. e. C := D A. Then it follows C 1 = D 1 A 0 and the matrix C is monotone. Moreover we have R 0. All these conclusions show us that C R is a regular splitting. (Block) Jacobi method or simultaneous iteration method: D A x (k+1) = (D A A)x (k) + b Joachim Rang Partitioned Methods for Multifield Problems Seite 18

19 Block Gauß Seidel method Splitting: A := D L U with l ij = a ij for i < j and u ij = a ij for i > j. Let C := D L. Then we have C 1 = (D L) 1 = [ D ( I D 1 L )] 1 = ( I D 1 L ) 1 D 1 = ( D 1 L ) i D 1 0. Moreover we have U 0 and finally we have proven that A = C R with R = U is a regular splitting. This approach is called (block) Gauß Seidel method or sucessive iteration method and is given by i=0 (D L)x (k+1) = Ux (k) + b Joachim Rang Partitioned Methods for Multifield Problems Seite 19

20 Contents Blockform of linear iteration schemes Examples Joachim Rang Partitioned Methods for Multifield Problems Seite 20

21 Heat equation space discretisation: n = 20 time stepsize: τ = Joachim Rang Partitioned Methods for Multifield Problems Seite 21

22 Heat equation space discretisation: n = 50 time stepsize: τ = Joachim Rang Partitioned Methods for Multifield Problems Seite 22

23 Heat equation: spectral radius space discretisation: n = Joachim Rang Partitioned Methods for Multifield Problems Seite 23

24 Example Consider the problem { u = f, x (0, 1) u = 0, x {0, 1} Exact solution: Then: u(x) = x(1 x) (x 1/4) + 1/10 f(x) = x x x (80x 2 40x + 13) 2 Space discretisation: central differences Joachim Rang Partitioned Methods for Multifield Problems Seite 24

25 Monolythical approach discrete system: A h u h = f h n u h u CPU E E E Joachim Rang Partitioned Methods for Multifield Problems Seite 25

26 Partititioned approach (Block Gauß Seidel method) discrete system: ( A11 A 12 A 21 A 22 ) ( ) ( ) u (1) f (1) = u (2) Blockform of linear iteration schemes Examples f (2) with A 11 = A 22 = h { (A 12 ) ij = (A 21 ) 1, i = n, j = 1 ij = 0, else Joachim Rang Partitioned Methods for Multifield Problems Seite 26

27 Partititioned approach Block Gauß Seidel method: n u h u Iterations CPU ρ E E E Joachim Rang Partitioned Methods for Multifield Problems Seite 27

28 A parallel ansatz Idea: Split the domain into (0, 1/2) and (1/2, 1) At x = 1/2 we require u left (1/2) = u right (1/2), u left(1/2) = u right(1/2), Joachim Rang Partitioned Methods for Multifield Problems Seite 28

29 Time-dependent problem Consider the problem u u = f, (t, x) (0, 1) 2 u = 0, t (0, 1), x {0, 1} u = u 0 (x), x (0, 1) Space discretisation: central differences Time discretisation: backward Euler method discrete problem: (I τa)u h m+1 = u h m + τf(t m+1 ) Here: spectral radius ρ depends on τ Joachim Rang Partitioned Methods for Multifield Problems Seite 29

30 Literature Owe Axelsson. Iterative solution methods. Cambridge Univ. Press., Cambridge, Robert Beauwens. Factorization iterative methods, M-operators and H-operators. Numer. Math., 31: , Andras Meister: Numerik linearer Gleichungssysteme. Eine Einführung in moderne Verfahren. Mit MATLAB-Implementierungen von C. Vömel, Vieweg+Teubner, Wiesbaden, J.M. Ortega and W.C. Rheinboldt. Iterative solution of nonlinear equations in several variables. Academic Press, New York-London, Yongzhong Song. Comparisons of nonnegative splittings of matrices. Linear Algebra Appl., : , R.S. Varga. Matrix iterative analysis. Series in Automatic Computation. Prentice-Hall, Joachim Rang Partitioned Methods for Multifield Problems Seite 30

31 BiCGStab method Iterative methods which convergece faster than the Jacobi, the Gauss Seidel and SOR-method are cg-schemes (conjugient gradient schemes). These methods are based on optimization ideas, i.e. in each iteration step one or more search vectors are computed which minimises the residuum. In this lecture we only introduce the BiCGStab method Joachim Rang Partitioned Methods for Multifield Problems Seite 31

32 BiCGStab method (I) Let x (0) be an initial guess for the solution of the linear system Ax = b and let r (0) = Ax (0) b be the initial residuum. Set r (0) := r (0) and p (0) := r (0). Calculate ρ (0) := (r (0), r (0) ). Let ɛ be a given tolerance. If r (0) ɛ then stop Joachim Rang Partitioned Methods for Multifield Problems Seite 32

33 BiCGStab method (II) Otherwise we iterate from k = 0 to n 2 and compute s (k) = Ap (k), σ (k) = (s (k), r (0) ), stop, if σ (k) = 0. α (k) = ρ(k) σ (k), w(k) = r (k) α (k) s (k), v (k) = Aw (k), ω (k) = (v(k), w k ) (v (k), v (k) ) x (k+1) = x (k) α (k) p (k) ω (k) w (k) r (k+1) = r (k) α (k) s (k) ω (k) v (k) ρ (k+1) = (r (k+1), r (0) ) If r (k+1) ɛr (0) then stop. Otherwise compute the new search vectors β (k) = ρ(k+1) ρ (k) α (k) ω (k), p(k+1) = r (k+1) + β (k) (p (k) ω (k) s (k) ) Joachim Rang Partitioned Methods for Multifield Problems Seite 33

34 Example Consider the problem { u = f, x (0, 1) u = 0, x {0, 1} Exact solution: Then: u(x) = x(1 x) (x 1/4) + 1/10 f(x) = x x x (80x 2 40x + 13) 2 Space discretisation: central differences Joachim Rang Partitioned Methods for Multifield Problems Seite 34

35 Example Dimension: Joachim Rang Partitioned Methods for Multifield Problems Seite 35