Numerical Solution of Initial Boundary Value Problems. Jan Nordström Division of Computational Mathematics Department of Mathematics

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1 Numerical Solution of Initial Boundary Value Problems Jan Nordström Division of Computational Mathematics Department of Mathematics

2 Overview Material: Notes + GUS + GKO + HANDOUTS Schedule: 5 lectures + 3 exercises + 2 seminars Mail: jan. nordstrom@liu.se Examination: Mandatory HW, 3 Bologna points

3 Schedule Times Monday Tuesday Wednesday 09:00-10:00 Seminar 1 Seminar 2 10:00-10:30 Introduction coffee/tea coffee/tea 10:30-12:00 Lecture 1 Excersise 1 Excersise 3 12:00-13:00 Lunch Lunch Lunch 13:00-14:30 Lecture 2 Lecture 4 Lecture 5 14:30-15:00 coffee/tea coffee/tea coffee/tea 15:00-16:30 Lecture 3 Exercise 2 Seminar 3 HW of PhD students preparation during seminars. Credit 3 points, European/Bologna system.

4 General structure and principles The Big Picture Relate the PDE to the numerical approximation. Semi-discrete approximation in space, time is left continuous. All approximations of the form: U t + AU = F. What can we say about A based on knowledge of A + A T? High order finite differences, the SBP-SAT technique. Extension to finite volume + discontinuous Galerkin techniques.

5 Course material Slides + possible additional handouts+references to relevant articles. GUS: High Order Difference Methods for Time Dependent PDE, Bertil Gustafsson, Springer-Verlag GKO: Time Dependent Problems and Difference Methods, Bertil Gustafsson, Heinz-Otto Kreiss, Joseph Oliger, John Wiley & Sons, 1995.

6 Lecture 1

7 Well posed problems U t + PU = F, x Ω, t 0 BU = g x Ω, t 0 u = f x Ω, t = 0 (1a) (1b) (1c) data U = dependent variable D = differential operator in space B = boundary operator F = forcing function g = boundary data f = initial data

8 Equation (1) is Well-Posed if U exists and satisfies U 2 I K ( f 2 II + F 2 III + IV) g 2. (2) K independent of data F, f, g. A small K is good! Why is (2) important? Consider perturbed problem V t = PV + F + δf, x Ω, t 0 BV = g + δg x Ω, t 0 V = f + δf x Ω, t = 0. (3a) (3b) (3c)

9 (3)-(1) W = V U, P = linear operator. W t + PW = δf, x Ω, t 0 BW = δg x Ω, t 0 W = δf x Ω, t = 0 (4a) (4b) (4c) Apply (2) to (4) W 2 I K ( δf 2 II + δf 2 III + δg 2 IV). (5) W = V U small if K, δf, δf, δg small! Uniqueness follows from (5).

10 Figure: A good numerical approximation possible. Choice of numerical method next step. Figure: A good numerical approximation NOT possible. Change problem, in practice boundary conditions

11 Existence u x = 0 u = constant u(0) = a u(1) = b a b too many b.c. s! Uniqueness u xx = 0 u = c 1 + c 2 x u(0) = a u = a + c 2 x too few b.c. s! Boundedness u = a + c 2 x no bound too few b.c. s!

12 Example u t = u x, x 0, t 0 Bu = g, x = 0, t 0 u(x, 0) = 0, x 0, t = 0 P = x, B = 1 + β x Laplace sû + û x = 0 û = c 1 e sx i) β = 0, c 1 = ˆq û = ˆqe sx, Well posed ii) β 0 c 1 (1 βs) = ˆq û = ˆq 1 βs e sx, Ill posed

13 Nonlinear problems (see Kreiss and Lorenz 1989) Linearization principle: A non-linear problem is well-posed at u if the linear problem obtained by linearizing all the functions near u are well-posed. Localization principle: If all frozen coefficient problems are well-posed, then the linear problem is also well-posed. U t + UU x = 0, Nonlinear U t + Ū(x, t)u x = 0, Linear U t + ŪU x = 0, Frozen coefficients

14 Summary of well-posedness A problem is well-posed if A solution exists (correct number of b.c.) The solution is bounded by the data (correct form of b.c.). The solution is unique. A nonlinear problem is related to well-posedness through the Linearization and Localization principles. If a problem is not well-posed, do NOT discretize. Modify first to get well-posedness. In practice, change b.c.!

15 Initial value problems for periodic solutions using Fourier transforms Consider the Cauchy problem on x U t = P( / x)u + F U(x, 0) = f (x). (6a) (6a) Definition: The problem (10) is well-posed if there is a unique solution satisfying where K, α are constants. t ) u 2 K 2 e ( f 2αt 2 + F 2 dt, (7) 0

16 As an example, consider U t + AU x = BU xx where A, B = constant and symmetric. Fourier transform ˆP(iω) = (iωa + ω 2 B), Û t = ˆP(iω)Û, Û(0) = ˆf, Û = e ˆP(iω)tˆf Theorem (6) is well-posed if and only if there are k, α such that Proof: e ˆP(iω)t Ke αt. (8) U 2 = 1 Û 2 = e ˆP(iω)t 2 max 2π 1 ˆf 2 K 2 e 2αt f 2. 2π

17 Definition: The Petrovski condition is satisfied if the eigenvalues λ(ω) of ˆP(iω) satisfy Re(λ(ω)) α. (9) α = constant, independent of ω, α = 0 if no zero order terms. Theorem: The Petrovski condition is necessary for well-posedness. It is sufficient if there is a constant K and matrix T such that T 1 ˆPT = diagonal and T 1 T K for all ω. Same proof as above.

18 Periodic difference approximations d dt U j = QU j + F j U j (0) = f j (10a) (10b) where U j, F j, f j are vectors and Q is a matrix. Def: The Petrovski condition is satisfied if the eigenvalues of the symbol ˆQ(ξ) satisfy where ξ = ωh < π. Re(λ(ξ, h)) α, (11)

19 Theorem: The problem (10) is stable in the semi-discrete sense if (11) is valid and ˆQ can be diagonalized using a similarity transform with a bounded condition number. Note similarity with PDE. As an example, consider the heat equation. Expand in Fourier-series d dt U j = QU j = U j+1 2U j + U j 1 h 2 U j = f j. U j = 1 Û ω e iωx j. 2π

20 The problem separates into d dtûω = ˆQÛ ω Û w = ˆf ω Û ω = e ˆQtˆfω. Exactly as in continuous case. Let ωh = ξ. We have Q = D + D Qe iωx j = eiωx j+1 2e iωx j + e iωx j 1 h 2 = e iωx (eiξ 2 + e iξ ) j h 2 = e iωx (eiξ/2 e iξ/2 ) j h ) 2 = e iωx sin (ξ/2)2 j ( 4 h 2 = e iωx j ˆQ.

21 The Von Neumann condition on the time-step comes from the specific time-advancement scheme. The Petrovski, eigenvalue condition is more general, and fundamental. Example of Von Neumann condition using Euler forward: Û n+1 = (1 + t ˆQ)Û n = ˆQÛ n ˆQ 1, condition on time-step.

22 Summary of theory for initial value problems The continuous/semi-discrete problem is well-posed/stable if The Petrovski (Von Neumann) condition is satisfied. ˆQ = TΛT 1 can be diagonalized and T T 1 K. Stability in semi-discrete form well-posedness for PDE

23 Exercises/Seminars Discuss the difference between the Petrovski and Von Neumann condition. Discuss the use of energy-methods for periodic problems. Show that if A + A T 0, then no positive real parts in eigenvalues of A.