Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen

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1 Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen Introduction to PDEs and Numerical Methods Lecture 6: Numerical solution of the heat equation with FD method: method of lines, Euler forward, Euler backward, the Theta method, and their stability Dr. Noemi Friedman,

2 Overview of the course Introduction (definition of PDEs, classification, basic math, introductory examples of PDEs) Analytical solution of elementary PDEs (Fourier series/transform, separation of variables, Green s function) Numerical solutions of PDEs: Finite difference method Finite element method PDE lecture Seite 2

3 Overview of this lecture Finite difference operators The heat equation Analytical solution Semidiscretization- Method of Lines (spatial discretization) Time discretization Euler foward Euler backward The theta-method Consisteny, stability, convergence PDE lecture Seite 3

4 We only analyse now the hom. equation - why can we do that? d u = Au(t) + f dt Instead of analyzing stability of the inhomogenous case, we discretize the homogenous one. We can do that, because of the following. Let s take a stationary function u 0 for which the equation: d dt u 0 = Au 0 + f = 0 holds. And let s suppose, that the solution can be written in the form u t = u 0 + v(t) r.h.s: Au t + f = Au 0 + f + Av t = Av t l.h.s: = 0 d dt u t = d dt u 0 + v t = d v t dt d v t = Av t dt Dr. Noemi Friedman PDE lecture Seite 4

5 Heat equation Comparism of the solutions: analytical and method of lines Homogeneous equation (no internal source term): d u = Au(t) dt Analytical solution with hom. BC: u x, t = j d j e ω j 2 βt sin ω j x ω j = jπ l j = 0,,2.. as t u 0 With method of lines: u t = i c i e λ i(t t 0 ) v i λ i = 2β2 h 2 cos iπ N > 0 < as t u Dr. Noemi Friedman PDE lecture Seite 5

6 2.) Euler forward - explicit Euler forward: approximate time derivate with the forward difference u j (t) t = β2 h 2 u j 2u j + u j+ + O(h 2 ) u j,n = u j,n+ u j,n + O() u n = u n+ u n = Au n u j,n+ u j,n = β2 h 2 u j,n 2u j,n + u j+,n + O h 2 + O() Stable? (does it give decaying solution?) B = β2 h Dr. Noemi Friedman PDE lecture Seite 6

7 2.) Euler forward d-tour on eigenvalues and eigenvectors Stable? (does it give decaying solution?) u n+ = Bu n u n+2 = B 2 u n u n+3 = B 3 u n u n+k = B k u n Some discussion about egienvalues and eigenvectors: Av i = λ i v i AV = DV V AV = D V AV is a diagonal matrix, with the eigenvalues in the diagonal. D = V = λ λ 2 λ n λ n v v 2 v n v n Dr. Noemi Friedman PDE lecture Seite 7

8 Eigenvalues, eigenvectors change of basis (coordinate system) How to write an arbitrary vector, u, in a new basis v (), v (2), v (3) What is V AV? u = u e + u 2 e 2 + u 3 e 3 u = u v () + u 2 v (2) + u 3 v (3) v () () () () = v e + v 2 e2 + v 3 e3 v (2) (2) (2) (2) = v e + v 2 e2 + v 3 e3 v (3) (3) (3) (3) = v e + v 2 e2 + v 3 e3 u = u () () () v e + v 2 e2 + v 3 e3 + + u 2 (2) (2) (2) v e + v 2 e2 + v 3 e3 + + u 3 (3) (3) (3) v e + v 2 e2 + v 3 e3 + v (3) v (2) e 2 u e e 3 u v () u = e u v () + u2 v (2) + u3 v (3) u 2 e 2 () (2) (3) u v 2 + u2 v 2 + u3 v 2 e 3 () (2) (3) u v 3 + u2 v 3 + u3 v Dr. Noemi Friedman PDE lecture Seite 8 u 3

9 Eigenvalues, eigenvectors change of basis (coordinate system) Some discussion about egienvalues and eigenvectors: v () v (2) v (3) u u u 2 u 3 = v () v 2 () v 3 () v (2) v 2 (2) v 3 (2) v (3) v 2 (3) v 3 (2) u u 2 u 3 u = e u v () + u2 v (2) + u3 v (3) u 2 e 2 () (2) (3) u v 2 + u2 v 2 + u3 v 2 e 3 () (2) (3) u v 3 + u2 v 3 + u3 v u = V u u = V u u 3 Let s consider now the respresentation of an operator in the new basis: Au = b AV u = V b V AV is the representationa of the A operator in the new basis defined by v (), v (2), v (3) V AV u = V V b V AV u = b Dr. Noemi Friedman PDE lecture Seite 9

10 2.) Euler forward d-tour on eigenvalues and eigenvectors V AV is the representationa of the A operator in the new basis defined by v (), v (2), v (3) If v (), v (2), v (3) are the eigenvectors of A, this representation is a diagonal matrix. Let s go back to the problem of the Euler forward method used for solving the heat equation Stable? (does it give decaying solution?) u n+ = Bu n u n+2 = B 2 u n u n+k = B k u n Let s check instead in the basis defined by the eigenvectors: u i = V u i u n+ = V BV u n u n+2 = V B 2 V u n u n+k = V B k V u n V B k V = V BB BBV = V BVV BVV VV BVV BV = D k = λ k λ 2 k D D λ n k Dr. Noemi Friedman PDE lecture Seite 0

11 2.) Euler forward stability analysis Stable? (does it give decaying solution?) B = β2 h When for all the eigenvalues λ i < then response is decaying with time. β2 a = 2 h 2 λ i = a + 2b cos iπ N b = β2 h 2 = 2 β2 h 2 cos iπ N 2 β2 h 2 cos iπ N < i =.. N Dr. Noemi Friedman PDE lecture Seite

12 2.) Euler forward stability analysis Stable? (does it give decaying solution?) When for all the eigenvalues λ i 2 β2 h 2 cos iπ N < i =.. N < then response is decaying with time. 2 β2 h 2 cos iπ N < 2 β2 h 2 cos iπ N < allways satisfied < 2 β2 h 2 cos iπ N 0< 2 Max value: 2 2 β2 h 2 cos iπ N > 2 β2 h 2 cos iπ N < 2 cos iπ N < h2 β 2 2 < h2 β 2 > Dr. Noemi Friedman PDE lecture Seite 2 < h2 2β 2 Method is not unconditionally stable, only stable if criteria is satisfied!

13 2.) Euler forward summary Euler forward: approximate time derivate with the forward difference u j (t) t = β2 h 2 u j 2u j + u j+ + O(h 2 ) u j,n = u j,n+ u j,n + O() u n = u n+ u n = Au n u j,n+ u j,n = β2 h 2 u j,n 2u j,n + u j+,n + O h 2 + O() Stable? (does it give decaying solution?) the absolut values of the eigenvalues of matrix B can not be greater than one. λ j = 2 β2 h 2 cos N π N λ j < h2 2β Dr. Noemi Friedman PDE lecture Seite 3

14 3.) Euler backward (implicit) derivation Euler backward: approximate time derivate with the backward difference u j (t) t = β2 h 2 u j 2u j + u j+ + O(h 2 ) u j,n+ = u j,n+ u j,n + O() u n+ = u n+ u n = Au n+ u j,n+ u j,n = β2 h 2 u j,n+ 2u j,n+ + u j+,n+ + O h 2 + O() u n = u n+ Au n+ = (I A)u n+ (Stability criteria: unconditionaly stable, to be shown later) Dr. Noemi Friedman PDE lecture Seite 4

15 4.) Theta method (implicit) derivation Theta method (Crank-Nicolson) u j (t) t = β2 h 2 u j 2u j + u j+ + O(h 2 ) u j,n+θ = u j,n+ u j,n + O( p ) u j,n = u j,n+ u j,n u n = u n+ u n + O() = Au n Euler f. u j,n+ u j,n = θ β2 h 2 u j,n+ 2u j,n+ + u j+,n+ + + θ β2 h 2 u j,n 2u j,n + u j+,n + O h 2 + O( p ) u j,n+ = u j,n+ u j,n u n+ = u n+ u n + O() = Au n+ Euler b. u n+θ = u n+ u n = θau n+ + θ Au n (I θa)u n+ =(I + θ A)u n B B 2 θ = 0 Euler forward θ = Euler backward Dr. Noemi Friedman PDE lecture Seite 5

16 4.) Theta method (implicit) stability analysis Stability criteria: B = (I θa) B u n+ = B 2 u n B 2 = (I + θ A) ) B and B 2 are both tridiagonal sym. matrices B and B 2 have the same eigenvectors v i = v i v i N v i k = sin ikπ N 2) The representation in the basis defined by the eigenvectors V B V u n+ = V B 2 V u n D λ (B ) λ 2 (B ) D 2 λ (B 2 ) λ 2 (B 2 ) u n+ = u n λ n (B ) λ n (B 2 ) Dr. Noemi Friedman PDE lecture Seite 6

17 4.) Theta method (implicit) stability analysis Stability criteria: B u n+ = B 2 u n B = (I θa) a b u n+ = λ (B 2 ) λ (B ) λ 2 (B 2 ) λ 2 (B ) u n B = b a b b a λ n (B 2 ) λ n (B ) B 2 = (I + θ A) a 2 b 2 λ i B = a + 2b cos iπ N = + 2β2 h 2 θ cos iπ N λ i B 2 = a 2 + 2b 2 cos iπ N = 2β2 h 2 θ cos iπ N B 2 = b 2 a 2 b 2 b 2 a Dr. Noemi Friedman PDE lecture Seite 7

18 u n+ = Numerical solution of the heat equation 4.) Theta method (implicit) stability analysis λ (B 2 ) λ (B ) λ 2 (B 2 ) λ 2 (B ) λ n (B 2 ) λ n (B ) u n 2r 0< C i 2 2β λ i (B 2 ) 2 λ i (B ) = h 2 θ cos iπ N + 2β2 h 2 θ cos iπ N r β2 h 2 2r 0< C i 2 < < 2r θ C i + 2rθC i < C i cos iπ N Dr. Noemi Friedman PDE lecture Seite 8

19 4.) Theta method (implicit) stability analysis ) Right hand side of the equation: < 2r θ C i + 2rθC i > 0 2rθC i < 2r θ C i / 2 2rθC i < 2rC i + 2rθC i /+2rθC i 2 < 2rC i + 4rθC i > r 2θ C i /: ( 2) r β2 h 2 2) Left hand side of the equation: 2r θ C i + 2rθC i < > 0 2r θ C i < + 2rθC i 2rC i > 0 unconditionally satisfied 2θ 0 (θ 0.5) 2θ > 0 (θ < 0.5) > r 2θ C i < 0 unconditionally satisfied C i 2θ > r h 2 β 2 C i 2θ > max 2 max Dr. Noemi Friedman PDE lecture Seite 9 h 2 2β 2 2θ >

20 Summary: Instationary heat equation what to solve? Stability checking from eigenvalue analysis: Method of lines Euler forward method find the eigenvalues (λ j ) and eigenvectors (v j ) of matrix A u n+ = I + A B u n u n+ = Bu n Euler backward method u n = I A B u n+ Theta method B u n+ = u n Solve system of equations for u n+ I θa u n+ = I + θ A u n B θ u n+ = B 2θ u n B θ B 2θ Dr. Noemi Friedman PDE lecture Seite 20