Hydroinformatik II: Partielle Differentialgleichungen (PDEs)

Transcription

1 Hydroinformatik II: Partielle Differentialgleichungen (PDEs) 1 Helmholtz Centre for Environmental Research UFZ, Leipzig 2 Technische Universität Dresden TUD, Dresden Dresden, 15. Mai /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 Inhalte Link zur letzten Vorlesung: Dimensionsanalyse Konzept Partielle Differentialgleichungen Klassifikation Beispiele Anfangs- und Randbedingungen TEXfür den Beleg 4/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

5 Dimensionsanalyse Navier-Stokes-Gleichung 5/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

6 Mathematical Classification (1.5) A common formulation of a PDE in R 3 is L(ψ) = F (t, x i, ψ, ψ,..., n ψ ) = 0, i = 3 (1) x i x n i where L is s differential operator. Second-order PDE with two independent variables are given by A 2 ψ x 2 + B 2 ψ x y + C 2 ψ y 2 + D ψ x + E ψ y + F ψ + G = 0 (2) Second-order PDEs with more independent variables can be classified by examination of the eigenvalues of the matrix a ij. ψ 2 a ij + G = 0 (3) x i x j i j 6/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

7 Mathematical Classification (1.5) PDE type Discriminant Eigenvalues Canonical form Example Elliptic B 2 4AC < 0 λ > 0 2 ψ ξ ψ η 2 = 0 Laplace complex equal signs equation characteristics Parabolic B 2 4AC = 0 λ = 0 2 ψ η 2 = G Diffusion, Burgers equations Hyperbolic B 2 4AC > 0 λ < 0 2 ψ ξ 2 2 ψ η 2 = 0 Wave real different signs equation characteristics 7/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

8 General Balance Equation (1.1.7) Integral form dψ Ω dt dω = ψ Ω t dω + (vψ)dω (D ψ ψ)dω = Ω Ω Q ψ dω Ω (4) Differential form dψ dt = ψ t + (vψ) (Dψ ψ) = Q ψ (5) 8/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

9 PDE: Definition dψ dt = ψ t + (vψ) (Dψ ψ) = Q ψ (6) A common formulation of a PDE in R 3 is L(ψ) = F (t, x i, ψ, ψ,..., n ψ ) = 0, i = 3 (7) x i x n i where L is s differential operator. 9/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

10 PDE: Klassifikation dψ dt = ψ t + (vψ) (Dψ ψ) = Q ψ (8) Physical problem Math. Examples problem Equilibrium problems Elliptic Irrotational incompressible flow equations Inviscid incompressible flow Steady state heat conduction Propagation problems Parabolic Unsteady viscous flow (infinite propagation speed) equations Transient heat transfer Propagation problems Hyperbolic Wave propagation (vibration) (finite propagation speed) equations Inviscid supersonic flow Parabolisch: Diffusion, Gerinne (nichtlinear) Elliptisch: Grundwasser (stationär) 10/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

11 PDE: Elliptic Equation 1-D d 2 ψ dx 2 = 0 (9) ψ = ax + b (10) 11/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

12 PDE: Elliptic Equation 2-D The prototype of an elliptic equation is the Laplace equation. 2 ψ x ψ y 2 = 0 (11) By substitution it can be easily verified that the exact solution of the Laplace equation is ψ = sin(πx)exp( πy) (12) 12/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

13 PDE: Elliptic Equation 2-D 13/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

14 PDE: Parabolic Equation 1-D ψ t = ψ α 2 x 2 (13) Multiple solutions: ψ(t, x) = sin( παx)exp( πt) (14) ψ(t, x) = sin(πx)exp( π 2 t) (15) 14/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

15 PDE: Parabolic Equation 1-D Abbildung: Solution of a parabolic equation 15/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

16 PDE: Hyperbolic Equation 1-D 2 ψ 2 t c2 2 ψ 2 x = 0 (16) ψ(t, x) = a cos( πct L ) sin(πx L ) (17) 16/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

17 17/24 Abbildung: Prof. Dr.-Ing. habil. Characteristics Olaf Kolditz of Hydroinformatik a hyperbolic II - SoSe equation 2015 PDE: Hyperbolic Equation 1-D - Characteristics A ψ t B ψ x = 0 (18)

18 18/24 Prof. Abbildung: Dr.-Ing. habil. Domains Olaf Kolditz of ahydroinformatik hyperboliciiequation - SoSe 2015 PDE: Hyperbolic Equation 1-D - Characteristics A ψ t B ψ x = 0 (19)

19 PDE: Equation Types The following table gives typical examples of balance equations for the denoted quantities and their PDE types. Physical meaning Equation structure Examples Continuity Mass/energy Momentum 2 ψ x ψ y 2 = 0 ψ t + u ψ x ψ α 2 x 2 = 0 ψ t + ψ ψ x x [α(ψ) ψ x ] = 0 Laplace equation Fokker-Planck equation Navier-Stokes equation 19/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

20 Boundary Conditions I The following table gives an overview on common boundary condition types and its mathematical representation. Tabelle: Boundary conditions types Type of BC Mathematical Meaning Physical Meaning Dirichlet type ψ prescribed value potential surface Neumann type ψ prescribed flux stream surface Cauchy type ψ + A ψ resistance between potential and stream surface 20/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

21 Boundary Conditions II To describe conditions at boundaries we can use flux expressions of conservation quantities. Tabelle: Fluxes through surface boundaries Quantity Mass Momentum Energy Flux term ρv ρvv σ ρev λ T 21/24 Prof. Dr.-Ing. habil. Olaf Kolditz Hydroinformatik II - SoSe 2015

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