Coupled Surface and Saturated/Unsaturated Ground Water Flow in - PowerPoint PPT Presentation

Coupled Surface and Saturated/Unsaturated Ground Water Flow in Heterogeneous Media Heiko Berninger ∗ , Ralf Kornhuber, and Oliver Sander eve ∗ , Freie Universit¨ Universit´ e de Gen` at Berlin and Matheon Multiscale Simulation & Analysis in Energy and the Environment , Radon Special Semester 2011

Outline Richards equation with homogeneous equations of state: Berninger, Kh. & Sander 10 • solver friendly finite element discretization: linear efficiency and robustness Heterogeneous equations of state: Thesis of Berninger 07, Berninger, Kh. & Sander 07,09,... • nonlinear domain decomposition: linear efficiency and robustness Coupled Richards and shallow water equations: Ern et al. 06, Sochala et al. 09, Dawson 08, Berninger et al. 11 • continuous of mass flow and discontinuous pressure (clogging) • mass conserving discretization (discontinuous Galerkin, ...) Dedner et al. 09, ... • Steklov–Poincar´ e formulation and substructuring • numerical experiments All computations made with Dune Bastian, Gr¨ aser, Sander, ...

Outline Richards equation with homogeneous equations of state: Berninger, Kh. & Sander 10 • solver friendly finite element discretization: linear efficiency and robustness Heterogeneous equations of state: Thesis of Berninger 07, Berninger, Kh. & Sander 07,09,... • nonlinear domain decomposition: linear efficiency and robustness Coupled Richards and shallow water equations: Ern et al. 06, Sochala et al. 09, Dawson 08, Berninger et al. 11 • continuous of mass flow and discontinuous pressure (clogging) • mass conserving discretization (discontinuous Galerkin, ...) Dedner et al. 09 • Steklov–Poincar´ e formulation and substructuring • numerical experiments All computations made with Dune Bastian, Gr¨ aser, Sander, ...

Runoff Generation for Lowland Areas γE γE vadose γSP γSP saturated h γD mathematical challenges: • saturated/unsaturated ground water flow: non-smooth degenerate pdes l Signorini-type bc (seepage face) • coupling subsurface and surface water: heterogeneous domain decomposition • uncertain parameters (permeability, ...): stochastic pdes Forster & Kh. 10, Forster 11

Saturated/Unsaturated Groundwater Flow: Richards Equation ∂ ∂t θ ( p ) + div v ( x, p ) = 0 , v ( x, p ) = − K ( x ) kr( θ ( p )) ∇ ( p − ̺gz ) equations of state: (Brooks & Corey, Burdine) � θ − θ m � − ε � 3+ 2 � � p ε θ m + ( θ M − θ m ) ( p ≤ p b ) θ ( p ) = kr( θ ) = p b θ M − θ m θ M ( p ≥ p b ) 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 saturation vs. pressure: p �→ θ ( p ) relative permeability vs. saturation: θ �→ kr( θ )

Saturated/Unsaturated Groundwater Flow: Richards Equation ∂ ∂t θ ( p ) + div v ( x, p ) = 0 , v ( x, p ) = − K ( x ) kr( θ ( p )) ∇ ( p − ̺gz ) equations of state: (Brooks & Corey, Burdine) � θ − θ m � − ε � 3+ 2 � � p ε θ m + ( θ M − θ m ) ( p ≤ p b ) θ ( p ) = kr( θ ) = p b θ M − θ m θ M ( p ≥ p b ) 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 p b , ε → 0 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 saturation vs. pressure: p �→ θ ( p ) relative permeability vs. saturation: θ �→ kr( θ )

Homogeneous Equations of State Alt & Luckhaus 83, Otto 97 � p Kirchhoff Transformation: κ ( p ) := kr( θ ( q )) dq = ⇒ ∇ κ ( p ) = kr( θ ( p )) ∇ p 0 3 1 0.95 0 0.21 −1 −4/3 0 −3 −1 0 3 −2 −4/3 −1 0 2 M ( u ) := θ ( κ − 1 ( u )) generalized pressure: u := κ ( p ) separation of ill–conditioning and numerical solution: semilinear variational equation � � u ( t ) ∈ H 1 ∀ v ∈ H 1 � � 0 (Ω) : M ( u ) t v dx + K ∇ u − kr( M ( u )) ̺ge z ∇ v dx = 0 0 (Ω) Ω Ω

Solver-Friendly Discretization lumped implicit/explicit-upwind discretization in time, finite elements S j ⊂ H 1 0 (Ω) : � � u n +1 I S j ( M ( u n +1 τK ∇ u n +1 ∈ S j : ) v ) dx + ∇ v dx = ℓ u n j ( v ) ∀ v ∈ S j j j j Ω Ω equivalent convex minimization problem u j ∈ S j : J ( u j ) + φ j ( u j ) ≤ J ( v ) + φ j ( v ) ∀ v ∈ S j J ( v ) = 1 quadratic energy 2 ( τK ∇ v, ∇ v ) − ℓ u n j ( v ) φ j ( v ) = � Φ( v ( p )) h p � convex, l.s.c., proper functional = Ω I S j (Φ( v )) dx nonlinear convex function Φ : R → R ∪ { + ∞} with ∂ Φ = M

Algebraic Solution: Monotone Multigrid Kh. 99, 02, Gr¨ aser & Kh. 07 • given iterate u ν j • fine grid smoothing: − successive 1D minimization of J + φ j in direction of nodal basis functions of S j : u ν 1 step of nonlinear Gauss–Seidel iteration → smoothed iterate ¯ j • coarse grid correction: 1.8 M ( u ) 1.6 u ν − Newton linearization of M ( u ) at ¯ j 1.4 − constrain corrections to smooth regime of M 1.2 1 1 step of damped MMG → new iterate u ν +1 � � � � � � 0.8 j 0.6 0.4 ⇒ ( J + φ j )( u ν +1 ) ≤ ( J + φ j )( u ν = j ) 0.2 j 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 uν ¯ j ( p ) u Theorem: (for V –cycle) Global convergence and asymptotic multigrid convergence ∀ − p b , ε ≥ 0 (robustness).

Algebraic Solution: Monotone Multigrid Kh. 99, 02, Gr¨ aser & Kh. 07 • given iterate u ν j • fine grid smoothing: − successive 1D minimization of J + φ j in direction of nodal basis functions of S j : u ν 1 step of nonlinear Gauss–Seidel iteration → smoothed iterate ¯ j Signorini condition • coarse grid correction: 1.8 M ( u ) 1.6 u ν − Newton linearization of M ( u ) at ¯ j 1.4 − constrain corrections to smooth regime of M 1.2 1 1 step of damped MMG → new iterate u ν +1 � � � � � � 0.8 j 0.6 0.4 ⇒ ( J + φ j )( u ν +1 ) ≤ ( J + φ j )( u ν = j ) 0.2 j 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 uν ¯ j ( p ) 0 u Theorem: (for V –cycle) Global convergence and asymptotic multigrid convergence ∀ − p b , ε ≥ 0 (robustness).

Solver–Friendly Discretization • Kirchhoff transformation • discretization: discrete minimization problem for u j • algebraic solution by monotone multigrid (descent method!) • inverse discrete Kirchhoff transformation

Solver–Friendly Discretization • Kirchhoff transformation: u = κ ( p ) • discretization: discrete minimization problem for u j • algebraic solution by monotone multigrid (descent method!) • inverse discrete Kirchhoff transformation

Solver–Friendly Discretization • Kirchhoff transformation: u = κ ( p ) • discretization: discrete minimization problem for u j • algebraic solution by monotone multigrid (descent method!) • inverse discrete Kirchhoff transformation

Solver–Friendly Discretization • Kirchhoff transformation: u = κ ( p ) • discretization: discrete minimization problem for u j • algebraic solution by monotone multigrid • inverse discrete Kirchhoff transformation

Solver–Friendly Discretization • Kirchhoff transformation: u = κ ( p ) • discretization: discrete minimization problem for u j • algebraic solution by monotone multigrid p j = I j ( κ − 1 ( u j )) • discrete inverse Kirchhoff transformation:

Reinterpretation and Convergence Analysis Berninger, Kh. & Sander 10 reinterpretation in terms of physical variables: inexact finite element discretization with special quadrature points convergence properties: M ( u j ) → M ( u ) in H 1 (Ω) generalized variables: u j → u and in L 2 (Ω) physical variables p j → p and θ j ( p j ) = I j ( M ( u j )) → θ ( p )

Reinterpretation and Convergence Analysis Berninger, Kh. & Sander 10 reinterpretation in terms of physical variables: inexact finite element discretization with special quadrature points convergence properties: M ( u j ) → M ( u ) in H 1 (Ω) generalized variables: u j → u and in L 2 (Ω) physical variables p j → p and θ j ( p j ) = I j ( M ( u j )) → θ ( p )

Experimental Order of L 2 -Convergence model problem: time discretized Richards equation without gravity physical parameters: Ω = (0 , 2) × (0 , 1) , sandy soil → ε , θ m , θ M , p b , n triangulation; uniformly refined triangulation T 11 (8 394 753 nodes) 1 10 1 0.1 0.1 0.01 0.01 0.001 L2-error L2-error 0.001 0.0001 0.0001 1e-05 1e-05 1e-06 1e-06 1e-07 1e-07 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 mesh size h mesh size h generalized pressure u physical pressure p

Experimental Order of H 1 -Convergence model problem: time discretized Richards equation without gravity physical parameters: Ω = (0 , 2) × (0 , 1) , sandy soil → ε , θ m , θ M , p b , n triangulation; uniformly refined triangulation T 11 (8 394 753 nodes) 10 10 1 1 H1-error H1-error 0.1 0.1 0.01 0.01 0.001 0.001 0.0001 0.001 0.01 0.1 1 0.0001 0.001 0.01 0.1 1 mesh size h mesh size h generalized pressure u physical pressure p

Evolution of a Wetting Front in a Porous Dam physical parameters: Ω = (0 , 2) × (0 , 1) , sand → ε , θ m , θ M , p b , n triangulation; uniformly refined triangulation T 4 (216 849 nodes) initial wetting front wetting front for t = 100 s pressure p j