Improving convergence of QuickFlow

Transcription

1 Improving convergence of QuickFlow A Steady State Solver for the Shallow Water Equations Femke van Wageningen-Kessels January 22,

2 Background January 22,

3 Background: software WAQUA QuickFlow - since 1970 s - - since slow convergence time-dependent solution not well suited for stationary solution fast convergence only stationary solution uses WAQUA output as initial solution January 22,

4 Goal & content How does QuickFlow work? Shallow water equations Spatial discretization Time iteration methods Test problems Improvements to QuickFlow Planning January 22,

5 2D Shallow water equations Navier Stokes equations Reynolds equations 2D Shallow water equations: U + U U t x + V U y V t + U V x + V V y ζ fv + g x τ bottom,x ρ 0 H ζ + fu + g y τ bottom,y ρ 0 H ν H ν H ( 2 U x U y 2 ( 2 V x V y 2 ) ) = 0 = 0 ζ t + HU x + HV y = 0 Shallow water equations on curvilinear grid January 22,

6 Spatial discretization: grid 5 n 4 3 V -velocity depth water level U-velocity one computational cell dex m January 22,

7 Spatial discretization: methods Momentum equations: Third order upwind: U V x Central difference and V U y Continuity equations: Finite volume January 22,

8 Spatial discretization: boundaries replacements discharge or velocity boundary closed free slip water level boundary January 22,

9 Spatial discretization: moving boundary ents dary H δ U = 0 H surface bottom January 22,

10 Time iteration: WAQUA - ADI stage 1: V -momentum explicit du dt = A 1 (u l, u l+1/2 ) stage 2: U-momentum & continuity explicit du dt = A 2 (u l+1/2, u l+1 ) Stationary: du dt = 1 2 [A 1(u, u) + A 2 (u, u)] A(u) January 22,

11 Time iteration: Euler Backward du dt = A(u) un+1 u n t = A(u n+1 ) t M (u n+1 u n ) + A(u n+1 ) b = 0 Advantages: easy implementation unconditionally stable January 22,

12 Time iteration: Newtons method F(x) tm(u n+1 u n ) + A(u n+1 ) b = 0 x k+1 = x k α ( F (x k ) ) 1 F(x k ) Advantages: easy implementation global convergence (line search) January 22,

13 Test problem: Chézy 0.5 Solution by WAQUA 1.2 Rectangular gutter grid cells Y coordinate [km] [m] inflow: velocity boundary outflow: water level boundary X coordinate [km] January 22,

14 Test problems: real Lek Randwijk January 22,

15 Test problem: Lek MAPS.sep at T=Inf, stationary Relatively easy Y coordinate [km] [m] grid cells inflow: Difference WAQUA QuickFlow x X coordinate [km] discharge boundary outflow: water level boundary Y coordinate [km] [m] January 22, X coordinate [km] 7

16 Test problem: Randwijk Difference WAQUA QuickFlow Relatively complex grid cells Y coordinate [km] [m] inflow: discharge boundary outflow: X coordinate [km] water level boundary 0.0 January 22,

17 Improvements: damping BC s hard open boundaries nonphysical reflection of waves long run-up time Remedie: damping boundary conditions January 22,

18 Improvements: semi-explicit time Example: advection equation u t = u u x Euler Backward & upwind u n+1 m u n m t = u n+1 m u n+1 m u n m 1 x Alternative u n+1 m u n m t = u n m u n+1 m u n m 1 x Implementation in Euler Backward or in Newton January 22,

19 Improvements: quasi-newton Approximate Jacobian F (x 0 ) B k Advantages: x k+1 = x k ( B k) 1 F ( x k ) reduce costs for computing F reduce costs for LU-decomposition of F possibly faster convergence (depends on method) January 22,

20 Improvements: adaptive time stepping complicated area small time step or large time step for continuity equation: ζ l+1 ζ l t + x (Hu) = 0 t x (Hu) = 0 January 22,

21 Plan Resolve large errors near open boundaries Inventorise & solve problems Chézy Lek Randwijk January 22,

22 Conclusion Software: WAQUA & QuickFlow Modelling river flow: Shallow water equations Spatial discretization Time iteration: ADI, Euler Backward, Newton Improvements: Damping boundary conditions Semi-explicit time iteration Quasi-Newton Adaptive time stepping January 22,

23 Questions? January 22,