Alessandro Valiani & Valerio Caleffi

Transcription

1 An engineering approach to the source terms treatment in shallow water equations Alessandro Valiani & Valerio Caleffi Dipartimento di Ingegneria Università degli Studi di Ferrara - ITALY Roscoff, September 19-23, 2011

2 Overview of recent years research activity in Numerical Hydraulics at ENDIF, University of Ferrara: DFB technique (Divergence Form for Bed Slope Source Term) for 1D and 2D SWE Well-balancing in CWENO schemes for SWE on fixed bed Well-balancing in CWENO schemes for SWE on movable bed Well-balanced bottom discontinuities treatment for high-order SWE in a WENO context A digression: some useful analytical tools for SW Flows Towards compact schemes: a balanced HWENO scheme for SWE Towards compact schemes: Balancing RKDG methods on domains with curved boundaries

3 Divergence Form for Bed Slope Source Term in Shallow Water Equations Valiani & Begnudelli, 2006, ASCE Journal of Hydraulic Engineering, 132 (7), Valiani & Begnudelli, 2008, Closure, ASCE Journal of Hydraulic Engineering, 2008, 134 (5),

4 SWE are extensively used in numerical modeling of rivers, estuaries, lakes and coastal areas, as well as river flooding due to dam break or banks failure. In real life, complex geometry and uneven topography are commonly encountered. The proper numerical integration of bed slope source term is fundamental to obtain correct results. Fundamental literature: Fractional step method (Toro, 1999) Upwind Method(Bermudez & Vázquez, 1994, Hubbard & Garcia- Navarro, 2000) Wave-propagation method(leveque, 1998)

5 In the present method (DFB), the bed slope source term is expressed as the divergence of a proper tensor, just like it happens for the momentum flux tem. 2D SWE: h U= Uh ; Vh U + = tt Bed Slope and friction Source terms: F S F= ( E, G) Uh Vh 2 2 h = U g ; UVh + ; 2 = 2 UVh 2 h Vh+ g 2 E G S= gh( S0x Sfx) gh( S0y S ) zb S0x= x zb S0 y= y 0 fy U U + V S fx= 2 C gh V U + V S fy= 2 C gh

6 SIMPLE IDEAS inspiring the method CONSERVATION is easier to obtain if we express complex quantities as the divergence of some other quantities: Complicated things = div (different things) No matter how complex the bed elevation is: Null velocity water elevation is a constant Hydraulic jump over flat bed must satisfy: Total force conservation AND mass conservation Uniform flow must satisfy: Bed Slope = Friction Slope

7 BED SLOPE SOURCE TERMS: May be rewritten as:

8 BED SLOPE SOURCE TERMS: PHYSICAL MEANING Π Π +Π Π = M M 1x 2x bx fx 1x 2x Π Π 1x 2 x Π bx Π fx M M 1x 1x = static forces over surfaces 1, 2 = x comp. of bottom pressure forces = bottom friction = momentum fluxes over surfaces 1, 2

9 BED SLOPE SOURCE TERMS: PHYSICAL MEANING Π Π +Π Π = M M 1x 2x bx fx 1x 2x lim 1 1 Π = Π = x ρg h1 ; 2x ρg h2 Π = ρg h x sin α; Π = ρg h S x bx * fx * f 2 2 1x= ρ U1 h1 2x= ρ U2h2 M ; M z z z + z gh = lim g η = x x 2 b b b2 b1 * * x 0 x gh2 U2h 2 gh1 U1 h = x 2 2 z b = gh * gh* S f x ( η z ) ( η z ) 2 2 g * b2 * b1 1 2 lim gh x = = x x η η = *

10 BED SLOPE SOURCE TERMS: TWO-DIMENSIONAL APPROACH z b 1 2 gh = gh x x 2 η= cost z b 1 2 gh gh = y y 2 η= cost H matrix defined as: 1 2 gh zb= g h 2 η= cost= η H= Eb Gb = gh gh 2 S0 * 0 z b = gh = H x η= η * zb gh y 2D SWE: U + F = H + S t η= η * f

11 FINITE VOLUME DISCRETISATION t U= dv+ dv f dv V V V η= η + F H S * V V H η= η dv= dv g hr nxrds H n r * V η= η = * r 1 2 = η= η * g hr nyrdsr r= 1 2 η= η * two estimations required: η = zg+ h * h = η z r * r

12 NUMERICAL SIMULATIONS Still water in irregular channel Error on water elevation (IC = SW) without DFB -3 err O(10 ) with DFB -16 err O(10 )

13 NUMERICAL SIMULATIONS Still water in irregular channel Error on water elev. (IC = perturb.) SGM (Surface Gradient Method), Zhou et al. (2001), is applied.

14 NUMERICAL SIMULATIONS Still water over Gaussian bump (Leveque, 1998) Comparison: DFB vs. Leveque (1998)

15 NUMERICAL SIMULATIONS: steady motion parabolic bump Subcritical flow Transcritical flow

16 NUMERICAL SIMULATIONS: steady motion parabolic bump Transcritical flow with shock

17 NUMERICAL SIMULATIONS Run-up of a solitary wave over a conic island (Briggs et al. 1995) Rectangular basin 26 x 27.6m Smooth bottom Island: base radius 3.6m, top radius 1.1m, height m. Solitary wave elevation: 2 Cs( t T) η() t = η0+ H wsech l s being: cs= gh H w /( 2h0) ( ) l = h 4 c h / 3H gh H = m s 0 s 0 w 0 w

18 NUMERICAL SIMULATIONS Run-up of a solitary wave over a conic island (Briggs et al. 1995) (A) Fig. A) Velocity field using DFB. Spurious results are highly reduced both downstream and at the side of the island. (B) Fig. B)Velocity field without DFB. Source term computed using centered finite differences. Spurious results are clearly observable near the island. Momentum flux and source term due to bed slope are not completely balanced.

19 NUMERICAL SIMULATIONS Run-up of a solitary wave over a conic island (Briggs et al. 1995) Water elevation was measured by 3 probed: A) Upstream the island B) At a side of the island Comparison with: Briggs et al. (1995), experimental, circles Liu et a. (1995), numerical, continuous Bradford and Sanders (2002), numerical, dashed Proposed approach (DFB), squared

20 Vantaggi del metodo proposto: CONCLUSIONS on DFB 3 Reference cases are exactly satisfied: a) Total force conservation and mass conservationin hydraulic jump over flat bottom b) C -Property (constant water surface elevation ) is satisfied for any uneven topography c) Bottom Slope = Friction Slope for uniform flow Extreme simplicity and low computational cost Superconvergence Very good results in practical test cases Applicability to different numerical techniques

21 Fourth-order balanced source term treatment in central WENO schemes for shallow water equations Caleffi V., Valiani A., Bernini A., 2006, Journal of Computational Physics, 218(1),

22 Numerical modeling of Shallow Water Equation with Source Terms. Focusing on two particular aspects: Increasing computational efficiency of numerical methods, applying to high order schemes, used on quite coarse grids. Reducing numerical errors, due to the treatment of source term due to bed slope, using well-balanced approaches: i.e. source term treatment must satisfy the so-called C-property, that is still water and null velocity must be preserved exactly.

23 Main features of the numerical model Mathematical model consists of SWE system with BED SLOPE SOURCE TERM; Space and time accuracy is of 4 order: Accuracy in space is obtained by WENO reconstruction of variables; Original treatment of source term (high order and well-balanced). A STAGGERED GRID is used, to obtain a simple computation of fluxes; Accuracy in time is obtained using a scheme Runge-Kutta schemed, coupled with the NCE (Natural Continuous Extension) technique.

24 Mathematical Model of the Shallow Water Equations u f ( u) + = s ( u, x ), u ( x,0) = u0 ( x ); t x Uh 0 h u x t f u s u x z Uh U h + g gh 2 x (, ) = ; ( ) = 2 ; (, ) = ; 2 h

25 Numerical model Integration procedure: At the current time, t n, cell-averaged values of solution are known; A half-cell staggered grid is introduced; Integrating between x j, x j + 1 and The balance law becomes: t, t + n n+1 1 where: n j+ 1 u j 1/ 2 u x t x n dx j 1 tn ( ( )) ( ) x tn tn ( ) ( ) u n+ 1 u n + 1 f u x 1 * j 1/ 2 j 1/ 2 j 1, t f u x j, t d t + S t j 1/ 2 d t + = t + n 1 x x * j+ 1 + = (, ) ; S j + 1/ 2 ( t) = s ( x, t ) dx x x j At the following time step t n+1 the solution is known in terms of cell-averages on the staggered grid.

26 Numerical model Integration in time: Simpson rule (4th order) 3 n+ 1 n t n n n+ β l u j+ 1/ 2 = u j+ 1/ 2 N l f ( uˆ ( x j+ 1, t + βl t) ) f ( uˆ ( x j, t + βl t) ) + xs j+ 1/ 2 x l= 1 n+ β 1 x j+ 1 T T l n s j+ 1/ 2 = s ( x, t + βl t) d x; N = [ 1/ 6,2 / 3,1/ 6 ] ; β = [ 0,1/ 2,1] x x j t n+1 Runge-Kutta(4th order) 4 i n+ 1 n ( i) ( i) n ( j) ˆ ˆ i ˆ ˆ ij i= 1 j= [ ] u = u + t b k with u = u + t a k b = 1/ 6,1/ 3,1/ 3,1/ 6 ; a = 0 except for a = a = 1/ 2 and a = 1 mn t n+1/2 NCE (4th order) n+ 1/ 2 ˆ = 4 n ( i) ˆ + i = [ 5/ 24,1/ 6,1/ 6, 1/ 24] T i= 1 u u t B k B Kuttafluxes: k = ( f + s) j x j T

27 Numerical model The key problem is: How can we obtain balancing between the gradient of momentum flux and the bed slope source term? 1. Using SGM techniques (Surface Gradient methods- Zhou et al. 2001) 2. Balanced reconstruction of Kutta flux 3. Balanced integration of Bed Slope Source Term

28 Numerical Model SGM (Surface Gradient method) Technique: The reconstructed vector variable is µ : η h + z µ b = = vh vh Surface water elevation Specific discharge Without SGM With SGM n reconstruction n u j u j + 1/2 u µ µ u n η = h + zb n reconstruction n h = η zb n j j j+ 1/2 j+ 1/2

29 Numerical Model Balanced reconstruction of the flux derivative, coupled with the bed source term K j ( x, u ( x, t) ) ( vh) ( vh) j 2 2 ( + 1/ 2 ( η ) ) + 1/ 2 ( η ) 0 = ( ) ( ) 1/ 2g ( j z) ( j z j ) 2 2 v h g z v h g z η η j It satisfies two analytical conditions: K j 1. = k j = ( fx + s) j 2. x x= x j j ( ) = 0 when η = const. and = 0 K x x hv When a reconstruction is defined, such that, starting from the point values K j, it gives an approximation of its derivatives on x j (see, e.g., the standard reconstruction of flux derivatives), then: eq. 1. ensures that such derivative is an approximation of Kutta fluxes. eq.2. ensures the balancing between flux derivative and bed source term, in case of quiescent fluid.

30 Numerical model Balanced reconstruction of the integral in space of bed source term [2] j 1/ 2 x dz j+ 1 b s gh d x + = x j dx Integration by parts + h = η z 1 x j+ 1 dη η η d [2] 2 2 s j+ 1/ 2 = g z j+ 1 z j g j+ 1z j+ 1 j z j + gz x 2 x j dx 1 2 [2] 2 2 η η ξ x x ( x ) ( ) j+ 1 s j+ 1/ 2 = g z j+ 1 z j g j+ 1z j+ 1 j z j + x dx x j x ξ = gz dη dxx The two forms are equivalent from the analytical point of view, but they are not from the numerical point of view. The latter form is convenient because in practical problems water elevation is a much more regular surface than bottom elevation. Two WENO reconstructions are needed: ˆ η j reconstruction ˆ η ' j ˆ ˆ ' reconstruction ξ j = gz jη j ξ j + 1/2

31 Accuracy test case Unsteady flow over sinusoidal bump Bottom: I. C: 2 ( ) = sin ( π x) z x cos( 2π x) (,0) = 0 + ; ( ) = ( π x) h x h e ( ) vh x,0 sin cos 2. Periodic Boundary Conditions Results a t= 0.1 s.

32 Accuracy test case Unsteady flow over sinusoidal bump Bottom: 2 ( ) = sin ( π x) z x Initial conditions: cos( 2π x) (,0) = 0 + ; ( ) = ( π x ) h x h e Periodic Boundary Conditions Results a t= 0.1 s. ( ) vh x,0 sin cos 2. Centered WENO N of cells L 1 order L 2 order order E E E E E E E E E E E E

33 Test case to verify balancing (C-property) h vh L 1 L 2 L 1 L E E 15 8.E E E E 13 Still water over smooth bottom bottom elevation: Initial conditions: ( ) ( x ) z x = 5e m ( x ) ( ) η,0 = 10 vh x,0 = 0. m Steady flow over parabolic bump Parabolic Bump H max = 0.2 m Axis x= 10 m Boundary conditions: upstream q = 0.18 m 2 /s downstream h = 0.33 m Physical domain dimensions: L = 25 m B = 1 m 250 cells

34 Test case on unsteady flow: rectangular pulse over bump Initial condition: Bump 0.25 cos ( 10π ( x 1 2) ) 1 m if x m z ( x) + = 0 elsewhere Computational domain: L = 1 m B = 1 m η ( x) 1+ ε m se x m vh = 0 = 1 elsewhere 3 ε = 1.0 m

35 Test case on unsteady flow: rectangular pulse over bump

36 Conclusions on high order CWENO for SWE with Source Terms Central WENO schemes are versatile, efficient and robust tools to numerically integrate hyperbolic system of conservation laws: as a consequence, they are particularly attractive for hydraulic engineering application of SWE with bed source terms. Concerning our original contribution, we propose a new approach for the treatment of bed slope source term, which obtains the C-property preservation in the case of quiescent fluid over uneven bottom. Such an approach preserves the 4 order accuracy in time and space, and can be theoretically applied to any WENO scheme(i.e. upwind ). Different test cases are analyzed, to verify the previously described properties: accuracy, C-property preservation, good discontinuities resolution. The natural following step consists of considering the mobile bed case.

37 High-order balanced CWENO scheme for movable bed shallow water equations Caleffi V., Valiani A., Bernini A., 2007, Advances in Water Resources, 30(4),

38 Aim of the research: Extend the CWENO scheme for the application to movable bed problems Advantage of CWENO schemes: CWENO schemes do not need either Riemann problem resolution at the interface of neighbouringcells or the knowledge of the eigenstructure of the adopted system of conservation laws. Simple structure of the code. Strong reduction of computational time. Simplification in the management of applicative cases (interaction between solid and liquid phase, secondary currents in curvilinear channels, etc.) in which the eigenstructure is strongly different from the standards available in literature.

39 Mathematical Model of the Shallow Water Equations 1D shallow water equations for a movable bed: u t f + = s x with: u conservative variables vector; f flux vector; s source terms vector. vh 1 ( 1 λ ) q 0 η + s z u = vh f = v h + gh s = gh x z 1 ( 1 λ ) q s ; 2 ; ; h water depth q s bed load sediment transport v vertically averaged velocity λ bed porosity z bottom elevation η water level

40 Mathematical Model of the sediment transport The sediment transport evaluation is case dependent. Two models are used in this study: qs = Av 3 This simple power law is selected in order to allow the comparison with numerical results proposed by Črnjarić-Žicet al., 2004 in previous numerical works carried out on the same topic. q = β v v s ( ) r cr To allow the comparison with analytical solutions by Lyn & Altinakar 2002.

41 Numerical Model The numerical model is very similar to the previously discussed one, only the function used to achieve a balanced reconstruction of the flux derivative (coupled with the bed source term) is quite different: K j ( x, u ( x, t) ) ( vh) ( vh) j 2 2 ( + 1/ 2 ( η ) ) + 1/ 2 ( η ) 0 = ( ) ( ) 1/ 2g ( j z) ( j z j ) 2 2 v h g z v h g z η η j Becomes ( + ( λ ) s ) + ( λ ) ( s ) ( ) ( ) j ( 1 ( 1 λ ) qs ) ( 1 ( 1 λ ) qs ) vh 1 1 q vh 1 1 q 0 j K (, (, )) / 2 (( ) 2 ( ) 2 j x u x t = v h + gh v h + gh + g η j z η j z j ) 0 j

42 Test Cases Observations: Simulations are carried out following 2 steps: 1. A preliminary run is made considering fixed bed until the hydrodynamic stationary conditions are achieved; 2. A second run with movable bed follows the first. The previously obtained hydrodynamic stationary conditions are adopted as initial conditions for the second run. Boundary conditions: Hydrodynamicvariables in subcritical flows are imposed at boundary cells by decoupling water flow from sediment flow and following the procedure suggested by Sanders (2002), in order to avoid reflection of non physical undulations. Note that the decoupling hypothesis is necessary only at the boundaries and not elsewhere. A discharge is imposed at the inlet and a depth is imposed at the outlet.

43 Accuracy test case (Xing & Shu 2005, Unsteady flow over sinusoidal bump modified) Bottom: Initial conditions: Solid discharge: 2 ( ) = sin ( π x) z x cos( 2π x) (,0) = 0 + ; ( ) = ( π x) h x h e ( ) vh x,0 sin cos q = Av with: A = 0.2 [ s / m] and: λ = 0.2; s Water Level Periodic Boundary Conditions Results a t= 0.1 s. N of cells L 1 order L 2 order order e e e e e e e e e e e e Bottom N of cells L 1 order L 2 order order e e e e e e e e e e e e

44 Subcritical flow over movable bed test case (Lyn & Altinakar, 2002) ( ) 2 1/2 An analytical solution is available, in nearly critical flows, i.e. 1 Fr = O ψu, starting from: vu hu 0 h uɶ uɶ 1 q + JU = 0 with: JU = g vu g ; uɶ = v and ψu = t x ( 1 λ) hu v 0 huψ U 0 z using a perturbation analysis in the small parameter ψ U, the following eigensystemis achieved: ψ U l1 = v ; 2 U l 2, = v ; 2 ± U 2 2FrU 4 FrU 4 FrU Fr U ( ) Li = 1, li vu / g,1 vu / g i = 1,...,3 then, the system can be analytically solved using the classical characteristics method. b U

45 Subcritical flow over movable bed test case ( ) x 2 z x = zmaxe with 15m < x < 15m r 4 ( ) β q = β v v with = ; r = 2.65; s cr 3 h = 1 m; λ = 0.4; ψ = ; U U FrU = 0.96;

46 Subcritical flow over movable bed test case ( ) x 2 z x = zmaxe with 15m < x < 15m r 4 ( ) β q = β v v with = ; r = 2.65; s cr 3 h = 1 m; λ = 0.4; ψ = ; U U FrU = 0.96;

47 Supercritical flow over movable bed test case ( ) x 2 z x = zmaxe with 15m < x < 15m r 4 ( ) β q = β v v with = ; r = 2.65; s cr 3 h = 1 m; λ = 0.4; ψ = ; U U FrU = 1.04;

48 Supercritical flow over movable bed test case ( ) x 2 z x = zmaxe with 15m < x < 15m r 4 ( ) β q = β v v with = ; r = 2.65; s cr 3 h = 1 m; λ = 0.4; ψ = ; U U FrU = 1.04;

49 z b ( x) Numerical Approximations of Hyperbolic Systems Convection-diffusion of a bump on movable bed with low/high solid discharge Initial conditions: q= 10 m 2 /s; h= 10 m; L = 1000 m; B = 1 m; 250 cells; ( ) 2 sin π x if 300m x 500m = 0 otherwise Intense sediment transport A = 1 m -1 s 2 Moderate sediment transport A = m -1 s 2 qs = Av 3

50 Convection-diffusion of a step on movable bed with a low discharge Initial conditions: q= 10 m 2 /s; h= 10 m; A = 1 m -1 s 2 (intense sed. tr.) Computational domain geometry: same configuration of the previous cases, with bottom ( π ) ( ) ( ) zb x = 1 exp x x0 5 +, x0 = 400 m 1

51 Conclusion 1. In this work the applications of a CWENO scheme to shallow water equations over a moving bed is presented. 2. The method is competitive with respect to other high resolution schemes for solving complex system of conservation laws, mainly for its computational efficiency and for its generality. 3. The code is validated using test cases on movable bed, obtaining very good results in terms of: accuracy; well-balancing property; suitability to correctly reproduce the amplitude and the celerity of water and sand waves.

52 Well-balanced bottom discontinuities treatment for high-order shallow water equations WENO scheme Caleffi & Valiani, 2009, ASCE Journal of Engineering Mechanics, 135 (7),

53 The context This work is part of the extensive study on high order WENO schemes for SWE with source terms. The final aim is numerical modeling for engineering problems. Available work Reliable WENO schemes are considered as applicable in terms of: very good efficiency, very good resolution of discontinuities, stability; Moreover, well balanced source terms treatment are available (i.e. satisfying C- property). Open problems Strongly uneven geometries are typical of real problems; a lot of engineering problems are characterized by the presence of singularities, such as bores propagating on wet bed, or sheer geometrical singularities, like bottom discontinuities.

54 Inthiswork A 1D FV WENO scheme is proposed, 4th order accurate in time and space, to numerically integrate SWE with source term due to the bottom slope, including bottom discontinuities. The main original contribution consists in the new approach for managing bottom discontinuities. The method is (physically) based on a proper correction of numerical flux, obtained from the energy conservation principle and from a momentum balance whichtaketheforceonthebottomstepsintoaccount. When the bottom is continuous, a high-order extension of the DFB method is used. This is a further original contribution. Some classical test cases are used, to verify the order of accuracy, the C-property preservation, and the good resolution which can be obtained, in both cases of continuous and discontinuous bottom.

55 The numerical model- introduction Classical Shallow Water Equations: 0 h vh ut + fx = s dz, vh = t x gh / 2 + v h gh dx Are: integratedovereachcell I j =[x j-1/2, x j+1/2 ] discretized in space

56 The numerical model the WENO reconstruction of variables The spatial discretization requires the estimate of point values of conservative variables at the cell boundaries, starting form the cell-averaged values of such variables. Similarly, to compute the source term, half-cell averaged values of water elevation are necessary: left and right half cell are considered. To compute such values, WENO reconstructions are used (see Qiu & Shu (2002) and references herein) SGMmethod(Zhouetal.,2001)isapplied,sothatthewatersurfaceelevation (= piezometric head),η=z+h, and the specific discharge, q = vh, are chosen as fundamental quantities to perform variable reconstructions.

57 The numerical model- introduction Formally, SWE in a semi-discretized form can be written as: du j 1 R L j j j dt = x F F + S = L ( u x t),,, 1) 2) S j Integralonacellofthesourceterm ( uˆ, uˆ + 1, zˆ, zˆ + 1 ) ( uˆ 1, uˆ, zˆ 1, zˆ ) R R R L R L Fj = F j j j j j L L R L R L Fj = Fj j j j j Numerical fluxes, to be better specified in the following, relative to the right interface(r) and to the leftinterface(l)ofthej-thcell. It is worth noting that not only conservative variables, but also bed elevation values, relative to the same interface, but to different adjacent cells can be different. THE BED ELEVATION CAN BE DISCONTINUOUS.

58 The numerical model time integration Several methods, available in literature, concerning integration in time of the semidiscretized equation are tested. Finally, the Strong Stability Preserving Runge-Kutta SSPRK(5,4) scheme [five steps, fourth order accurate], by Spiteri& Ruuth(2002) is selected. Coupled with a proper spatial WENO discretization, such a method allows to obtain very efficient non oscillatory schemes.

59 Numerical Model flux evaluation The flux evaluation is standard for a flat continuous bottom F = R L (, u ) F u j+ 1/2 j j+ 1

60 ?

61 Analytical inversion specific energy water depth relationship Valiani& Caleffi(Idra 2008, ADWR 2008). qˆ Γ0 π ± 2α E η2, 3 = 1+ 2cos 2 3 h = h a ( ) 3 α = arctan Γ 1 0 R* L* F = F uˆ, uˆ j+ 1/2 j+ 1/2 hc, Ec Γ0 E c ( ) j+ 1/2 j j+ 1 sub 2 sup = η h 3 c = η h c F 0 * * = ( ˆ, uˆ + ) + φ ( uˆ j ) φ ( uˆ j ) R R L j F u j j 1 R R*

62 The numerical model computation of fluxes For the j-th cell, numerical fluxes have the following expression: 0 R R* L* L R* L* F j = F ( uˆ j, uˆ j+ 1 ) + R R* ; F F φ ( uˆ j ) φ ( uˆ j ) ( uˆ uˆ ) 0 =, +. φ ( uˆ j ) φ ( uˆ j ) j j 1 j L L* Where: 2 2 ( ) 2 φ u = gh + v h TOTAL FORCE (divided by ρ) A similar approach is proposed by Audusse et al. (2004) and Noelle et al. (2006): in these previous works fluxes corrections physically correspond to static forces balances only. On the contrary, we suggest to consider a complete momentum balance, including dynamic forces.

63 The numerical model the balanced discretization of the source term The4 orderextensionof2 orderdfbmethod(valiani&begnudelli,2006)isused. S R L ( η z ) ( η z ) g 2 2 ˆ ˆ = ; 2 x j j j j j The extension to fourth order accuracy is obtained using a proper numerical extrapolation (Noelle et al., 2006). Each cell is considered as a whole, and then it is divided into two half-cells, obtaining the following quadrature expressions: S S S j (2) (1) j j 4S S = with: 3 g R L ( ˆ ) ( ˆ η z η z ) ; 2 x g L C L L g R R R C ( η zˆ ) ( η zˆ ) ( η zˆ ) ( η zˆ ) ; 2 x 2 x 2 2 (1) j = j j j j (2) j = j j j j + j j j j

64 Accuracy test case (Xing & Shu, 2005) Second order standard model: FVM Balanced Godunov type HLL flux MUSCL predictor corrector Accuracy analysis cells L 1 order L 2 order order E E E E E E E E E E E E

65 Test case for accuracy (Xing & Shu, 2005) -Efficiency 4 th order n cells L 1 order L 2 order order E E E E E E E E E sd order n cells L 1 order L 2 order order E E e E E e E E e Fourth order cells -> 7.03 [s] 14% Secondorder-6561 cells -> [s] 100%

66 Test case for C-property (balancing) Still water preservation on continuous and discontinuous bottom ( ) ( x ) z x = 5e m ( ) z x 4 m 4 < x < 8 m = 0 m altrimenti h vh Bottom L 1 L 2 L 1 L 2 Continuos 3.20E E E E E E 15 Discontinuous 1.80E E E E E E 11 L L

67 Test case: shock wave moving towards a step

68 Test case: shock wave moving towards a step

69 Conclusions on bottom discontinuities Concerning SWE with source term due to bed elevation variability, A well balanced FV WENO method is presented. Itis4 orderaccurateintimeandspace. Time integration and conservative variable reconstruction are performed using well stated techniques. Two original ingredients are introduced: 1. A numerical flux correction, based on physical reasoning (momentum balance), to treat bottom discontinuities; 2. Ahighorder(4 )extensionofdfb2 nd ordermethodtotreatsourceterm,incase of continuous bottom elevation; Model validation against significant test cases gives very good responses.

70 SOME USEFUL ANALYTICAL RESULTS on FREE SURFACE FLOWS Depth energy and depth force relationships in open channel flows: Analytical findings Valiani & Caleffi, AWR, Advances in Water Resources 31 (2008)

71 We make reference to the work: Shallow water equations with variable topography in the resonance regime By Philippe G. LeFlochand Mai DucThanh Concerning section 2.2, Equilibrium states, such authors Consider the problem of finding a certain RHS, given a prescribed LHS. 0 0 ( ) [ ] ( ) [ ] State U = h, u, a left hand state State U = h, u, a, a a right hand state We want to find out h and u in terms of U and a, solving: 2 2 u u0 a + h + = a0 + h0 + 2 g 2 g applying also mass conservation as: u h = u h 0 0

72 According classical hydraulic engineering language, we use the terms Y h u h = u h q E = h u as the water depth 2 2 g as the unit width discharge as the specific energy For the U0 state, we call critical depth the 2 2 1/3 u0 h 0 min ( 0 ) = value: Yc h U g and, finally, we make any variable as non dimensional, by dividing for the corresponding critical value: Y η = ; Γ = Y c E E c

73 SOME USEFUL ANALYTICAL RESULTS on FREE SURFACE FLOWS Water Depth (Y), Specific Energy (E), Total Force (F) Q = q b, discharge (prescribed) The subscript c means critical conditions: Minimum E, for the given Q Minimum F, for the given Q depth ND energy force Y E F η =, Γ =, Φ = Y E F c c c 2 2 Γ = + ; Φ = η η 3 3 η 3 3 η

74 Analytical inversion of the Energy-Depth relationship It can be shown that, given a prescribed value of the ndspecific energy Γ0, greater then the critical value (Γ0 1), three values of nddepth exist, satisfying the relation Γ=Γ0. Such three values are the following: Γ 2 Γ 2 π ± α 0 0 η1 = 1 2cos ; 2,3 1 2cos 2 α η = α = Γ a real number. Being: arctan ( 3 ) 0 1 It is simple to show that: The 1 root is negative (to discard) The 2 root is positive and 1 (supercritical flow) The 3 root is positive and 1 (subcritical flow)

75 Analytical inversion of the Total Force-Depth relationship It can be shown that, given a prescribed value of the ndtotal force Φ0, greater then the critical value (Φ0 1), three values of nddepth exist, satisfying the relation Φ =Φ0. Such three values are the following: cos θ η ; η2,3 2 0 cos π ± θ = Φ = Φ 3 3 Being: θ = ( ) 3 Φ 0 arctan 1 a real number. It is simple to show that: The 1 root is negative (to discard) The 2 root is positive and 1 (supercritical flow) The 3 root is positive and 1 (subcritical flow)

76 A classical example: the sluice gate For a given discharge, the downstream flow depth is imposed by gate opening. Computing the corresponding specific energy and using the proper inverse relationship allows to find the upstream depth.

77 A classical example: the width reduction due to bridge piers For a given discharge, and given width ratio ( strong contraction), a certain specific energy is necessary to cross the piers. Upstream and downstream flow depth, corresponding to such energy, can be computed by the inverse relationship. Total force inversion gives immediately the depth upstream the jump.

78 HWENO A new well-balanced Hermite weighted essentially nonoscillatory scheme for shallow water equations Caleffi, 2010, International Journal For Numerical Methods In Fluids, DOI: /fld.2410

79 The context This work is developed in the context of the high-order accuracy schemes for the shallow water equationswith bottom slope source term. In particular, the class of well-balanced and compact methods is take into account. The state of the art The state of the art is represented by the Runge-Kutta Discontinuous Galerkin schemes(e.g. Cockburn & Shu, 2001) and the PNPM methods(dumbseret al., 2008), while the Hermitian WENO schemes are recently proposed as an interesting alternative (e.g. Qiu& Shu, 2004). In this work The attention is focused on the well balancing of a fourth-order accurate HWENO schemes and a satisfying source term treatment is achieved. Cockburn, B. & Shu, C.W., 2001, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, Journal of Scientific Computing, 16(3), pp Qiu, J.X. & Shu, C.W., 2004, Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case. J. Comput. Phys.,193, DumbserM, BalsaraD, Toro EF, MunzCD. A unified framework for the construction of one-step finite-volume and discontinuous Galerkin schemes. J Comput Phys 2008;227:

80 Mathematical Model for the HWENO Scheme To obtain the compactness of the HWENOschemes both the conservative variables and their derivatives are evolved in time, while in the original WENOschemes only the conservative variables are evolved. Classical shallow water equations u + f ( u) = s( u, x), u( x,0) = u ( x); t x 0 u 0 u( x, t) = ; f ( u) = ; s( u, x) = ; 2 u1 2 2 u2 u1 u 2 g gu1 z + x u1 2 with: u = h; u = Uh; 1 2

81 Mathematical Model for the HWENO Scheme Derivative of the shallow water equations (with respect to x) v + g( u, v) = r( u, v, x), v( x,0) = v ( x); t x 0 v depth derivative bottom derivative 2 v1 2 0 (, ) = ; (, ) = u ; (,, ) = ; 2 u2 v 2 gu1 v1 2 v 2 gv1 zx gu1ζ + x u1 u 1 v x t g u v r u v x depth bottom slope derivative ( ) ( ) ( ) with: v = u x; v = u x; ζ = z x; g u, v = f ' u u = f ' u v x

82 Numerical Model Working in the classical framework of the Godunov type - Finite Volume methods (on uniform grid), the semi-discretized form of the mathematical model becomes: Cell-averaged variables and derivatives Cell-averaged source terms du j 1 ( ˆ ˆ dv ) j 1 = f ˆ ( ˆ ˆ ) ˆ j+ 1/2 f j 1/2 + s j; = g j+ 1/2 g j 1/2 + rj; dt x dt x Numerical Fluxes

83 Numerical Model The HLL Riemann solveris used to evaluate the numerical fluxes, together with the evaluation of the wave celerity based on the two rarefaction approximation ( ) ˆ G ( ) fˆ = F u, u ; g = u, u, v, v ; j + 1/ 2 j + 1/ 2 j + 1/ 2 j + 1/ 2 j + 1/ 2 j + 1/ 2 j + 1/ 2 j + 1/ 2 HWENO Reconstruction of the Conservative Variable at the cell-interface HWENO Reconstruction of the Conservative Variable Derivative at the cell-interface The selected time integration scheme, specific for conservation laws, is the five step, fourth-order accurate, Strong Stability Preserving Runge-Kutta, SSPRK(5,4), by Spiteri & Ruuth(2002). The Surface Gradient Method (Zhou et al.,2001) is applied to simplify the wellbalancing achievement, therefore the reconstructed quantities are: η = h + z water level ξ = h x + z x water level derivative

84 HWENO Reconstructions u, u, u, v, v, v u, u, v, v + + Starting from: j 1 j j + 1 j 1 j j + 1 j + 1/ 2 j + 1/ 2 j + 1/ 2 j + 1/ 2 Point values Starting from: η, η, η, ξ, ξ, ξ j 1 j j + 1 j 1 j j + 1 η, η, ξ, ξ R L R L j j j j Half-cell averages For example, we consider the reconstruction for the half-cell average of water level. ξ η ξ j 1 η j 1 ξ j ξ j + 1 R η j η j η j + 1 j 1 j j + 1

85 HWENO Reconstructions Three 2 nd order polynomials, p and a 4 th 0( x), p1( x), p2( x) order polynomial q( x) are defined such that ξ η Numerical Approximations of Hyperbolic Systems j 1 j j p0( x)d x = η ; = 1,0; 0( )d = 1; I j + l l p x x ξ I j x j + l x j p1 ( x)d x = η j l; l = 0,1; p 1( )d = j 1; I + x x ξ j l I + x + x j+ 1 1 p2( x)d x = η j+ l; l = 1,0,1; x I j + l 1 1 q( x)d x = η j l; l = 1,0,1; q ( x)d x = ξ j l; l = 1,1; x I + j l x I + + j+ l

86 HWENO Reconstructions where the solution is regular: 2 x 1/ 2 2 ( R ) = j + R I j pl ( x)d x = η j + ( x ) l = 0,1,2; l x O x j 2 x 1/2 R 4 ( R ) = j + I j q( x)d x = η j + ( x ); q x O x x j where the solution is smooth: 2 l =0 R R ( ) ( ) R η j = γ l I j = I j ; l q where γ, γ, γ. are linear weights where the solution is discontinuous: R we want evaluate η j using the information coming from the more regular polynomials.

87 HWENO Reconstructions This is obtained writing: η R ( ) 2 R j ω l I j l =0 =. l with, the index of smoothness is defined as usual by: 2 i 2i 1 pl ( x) αl γ k l = x d x; l = ; k = ; I i 2 i=1 j x α k ( ε + βk ) k β ω α

88 Source term treatment Numerical Approximations of Hyperbolic Systems Several well-balanced treatments of the classical SWE source term are available in literature. Here the approach presented in (Caleffi & Valiani, 2009) is used. More interesting is the original well-balanced treatment of the source term related to the Shallow Water Equations Derivative: R 1 1 ( ζ ) ( ζ ) r (2) j ˆ = = gh z gh d x = gv z gu d x j, I x x x I 1 x 1 x j j x x x The key idea is the introduction of a function of x: and the following approximation: φ η ξ η ξ ζ * * * * ( x ;, ) = g( z)( ), * * * * ( ζ ) φ η ξ φ η ξ gv1 zx gu1 x d x ( x j 1/2;, ) ( i 1/ 2;, ) I + x j We have analytically proved that this approximation is second-order accurate, consistent and well-balanced.

89 Source term treatment Assuming: Numerical Approximations of Hyperbolic Systems η = η and ξ = ξ * * j The second-order accuratesource term treatment becomes: R j = φ( x ; η, ξ ) φ( x ; η, ξ ) HDFB2 j j+ 1/2 j j j 1/ 2 j j The fourth-order accurate formulation is obtained using a numerical extrapolation (Caleffi& Valiani, 2009): R R with: ( b) ( a) = φ( x + ; η, ξ ) φ( x ; η, ξ ) 4Rj Rj =. = + 3 ( a) j j 1/ 2 j j j 1/ 2 j j HDFB4 ( b) L R Rj j φ φ φ = φ( x ; η, ξ ) φ( x ; η, ξ ) L L L L L j j j j 1/ 2 j j φ = φ( x ; η, ξ ) φ( x ; η, ξ ) R R R R R j+ 1 j j j j j

90 Applications Accuracy analysis - Unsteady flow over a sinusoidal bump (Xing& Shu, 2005) vh( x,0) = sin(cos(2 π x)) h( x,0) = h + e 0 ( π x) cos 2 2 z( x) = sin ( π x) With periodic boundary conditions.

91 Applications Accuracy analysis - Unsteady flow over a sinusoidal bump (Xing & Shu, 2005) HWENO Scheme free surface level. 1 2 Cells time [s] L order L order L order E E E E E E E E E E E E DGRK Scheme (Xing & Shu, 2006) free surface level. 1 2 Cells time [s] L order L order L order E E E E E E E E E E E E Xing,Y. & Shu, C.W., 2006, A new approach of high order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, Communications in Computational Physics, pp

92 Applications C-property Quiescent flow over a non-flat bottom (Xing & Shu, 2005) 2 ( x 5) 2 5 z( x) = 5e quantity η U h 1 2 L L L 4.69E E E E E E-13

93 Applications Pulse over a bump(leveque, 1998) ( π x ) 0.25 cos 10 ( 1 / 2) 1 m if x 1/ 2 < 0.1 m; z( x) = + 0 otherwise.

94 Applications Pulse over a bump small amplitude (10 3 )

95 Applications Pulse over a bump large amplitude (10 1 )

96 Conclusions on HWENO scheme This work regards the development of a well-balanced, 4 th order accurate, compact HWENO schemefor the integration of the SWE. In particular, several aspects of the scheme, originally proposed in the context of the gas dynamics, are modified and extended to allow the application to SWE. We have obtained: a very good compactness of the scheme; the C-property fulfillment; An high-order accuracy; a good computational efficiency; a good stability. Conversely, further improvements are necessary to obtain good, problem independent, non-oscillatory properties of the scheme. A new index of smoothness may be introduced?

97 RKDG Balancing RKDG methods on domains with curved boundaries Work in progress

98 General conclusion Any engineering approach cannot do without mathematics and physics, so suggestions are welcome

99 Acknowledgments Many Thanks to the Organizing Committee for this invitation Many Thanks to our colleagues Anna Bernini and Lorenzo Begnudelli, who are co-authors in some works presented here. Thank you for your attention!

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