Hydroinformatik II: Finite Differenzen Methode

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1 Hydroinformatik II: Finite Differenzen Methode 1 Helmholtz Centre for Environmental Research UFZ, Leipzig 2 Technische Universität Dresden TUD, Dresden Dresden, 05. Juni /24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

2 Vorlesungsplan Hydroinformatik II SoSe 2015 # Datum Thema Einführung, Grundlagen: Kontinuumsmechanik Grundlagen: Kontinuumsmechanik/Hydromechanik Maifeiertag HW: Einführung in Qt (Installation) Grundlagen: Partielle Differentialgleichungen / TEX Grundlagen: Numerische Methoden Pfingsten Numerik: Finite Differenzen Methode Grundlagen: Diffusionsprozesse Numerik: Übung explizite FDM Numerik: Implizite FDM Gerinnehydraulik: Theorie - Grundlagen Gerinnehydraulik: Saint-Venant Gleichung (= HSA) Gerinnehydraulik: Programmierung, Übung Kurs-Zusammenfassung und Abschluss 2/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

3 Konzept 3/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

4 Fahrplan Vorlesung Grundlagen der Finite Differenzen Methode Approximation methods Finite difference method FDM (Ch. 3) Taylor series expansion Derivatives Diffusion equation (Finite element method FEM Hydrosystemanalyse) 4/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

5 Näherungsverfahren 5/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

6 FDM Anwendungen - MODFLOW 6/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

7 Ableitungen 7/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

8 Taylor-Reihe in time u n+1 = u n +1 = m=0 in space m=0 t m m! x m m! [ m ] u n t m [ m ] u n x m (1) (2) 8/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

9 Trunkation u n+1 = u n + t [ ] u n + t2 t 2 [ 2 ] n u t 2 + 0( t 3 ) (3) u n +1 = u n + x [ ] u n + x 2 x 2 [ 2 ] n u x 2 + 0( x 3 ) (4) 9/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

10 1. Ableitung [ ] u n t = un+1 u n t t 2 [ 2 ] n u t 2 + 0( t 2 ) (5) [ ] u n = un +1 un x x x 2 [ 2 ] n u x 2 + 0( x 2 ) (6) 10/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

11 Differenzen-Schemata Forward difference approximation [ ] u n = un +1 un + 0( x) (7) x x Backward difference approximation [ ] u n = un u 1 n + 0( x) (8) x x Central difference approximation [ ] u n = un +1 un 1 + 0( x 2 ) (9) x 2 x 11/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

12 Zentrale Differenzen [ u u+1 n = u n + x x [ u u 1 n = u n x x ] n ] n + x x 2 2 [ 2 u x 2 [ 2 u x 2 ] n ] n + 0( x 3 ) 0( x 3 ) (10) Central difference approximation [ ] u n = un +1 un 1 + 0( x 2 ) (11) x 2 x 12/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

13 Ableitungen 13/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

14 2. Ableitung [ 2 ] n u x 2 1 x ( [ u x ] n +1 [ u x ] n ) un +1 2un + u n 1 x 2 (12) [ 2 ] n u x 2 = un +1 2un + u 1 n x 2 + x 2 12 [ 4 ] n u x (13) 14/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

15 Übersicht Differenzenverfahren 15/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

16 Diffusionsgleichung u t α 2 u x 2 = 0 (14) 16/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

17 Analytical solution for diffusion equation (Skript 5.2.2) Diffusion equation u t α 2 u x 2 = 0 (15) Analytical solution u = sin(πx)e αt2 (16) K: validity Übung 17/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

18 Übersicht Differenzenverfahren 18/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

19 Explizite FDM - FTCS Verfahren (Skript 3.2.2/4.1) PDE for diffusion processes u t u α 2 x 2 = 0 (17) forward time / centered space [ ] u n un+1 u n [ 2 ] n u t t x 2 un 1 2un + u+1 n x 2 (18) substitute u n+1 u n α un 1 2un + u+1 n t x 2 = 0 (19) FTCS scheme for diffusion equations u n+1 = u n + α t x 2 (un 1 2u n + u n +1) Ne = α t x 2 (20) 19/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

20 Eigenschaften numerischer Verfahren Analysis of approximation schemes consists of three steps: Develop the algebraic scheme, Check consistency of the algebraic approximate equation, Investigate stability behavior of the scheme. 20/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

21 Eigenschaften numerischer Verfahren Analysis of approximation schemes consists of three steps: Develop the algebraic scheme, u n+1 = u n + α t x 2 (un 1 2u n + u+1) n (21) Check consistency of the algebraic approximate equation, lim ˆL(u n ) L(u[t n, x ]) = 0 (22) t, x 0 Investigate stability behavior of the scheme. Ne = α t 1/2 (23) x 2 21/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

22 Lösung des FTCS Schemas Algebraische Schema u n+1 = u n + α t x 2 (un 1 2u n + u n +1) (24) Resultierendes Gleichungssystem u n+1 = Au n, n = 0, 1, 2,... (25) A = 1 2 α t x 2 α t x 2 α t x α t α t x 2 x α t x 2, u n = u n 2 u n 3... u n np 2 u n np 1 22/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

23 Explizite und implizite Differenzenverfahren 23/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

24 Implizites Differenzenverfahren: Next Lecture Algebraische Schema: [ 2 ] n+1 u x 2 un+1 1 2un+1 + u n+1 +1 x 2 (26) u n+1 u n t α un+1 1 2un+1 + u n+1 +1 x 2 = 0 (27) α t x 2 ( un un+1 u n+1 +1 ) + un+1 = u n (28) 24/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

25 Implizites Differenzenverfahren: Next Lecture Algebraische Schema: [ 2 ] n+1 u x 2 un+1 1 2un+1 + u n+1 +1 x 2 (26) u n+1 u n t α un+1 1 2un+1 + u n+1 +1 x 2 = 0 (27) α t x 2 ( un un+1 u n+1 +1 ) + un+1 = u n (28) 24/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

26 Implizites Differenzenverfahren: Next Lecture Algebraische Schema: [ 2 ] n+1 u x 2 un+1 1 2un+1 + u n+1 +1 x 2 (26) u n+1 u n t α un+1 1 2un+1 + u n+1 +1 x 2 = 0 (27) α t x 2 ( un un+1 u n+1 +1 ) + un+1 = u n (28) 24/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015