Mathematical and computational modeling for the generation and propagation of waves in marine and coastal environments

Transcription

1 Mathematical and computational modeling for the generation and propagation of waves in marine and coastal environments Maria Kazolea School of Environmental Engineering Technical University of Crete September 27, 213 MINISTRY OF EDUCATION & RELIGIOUS AFFAIRS, CULTURE & SPORTS

2 Motivation Discretize depth-integrated equations that model free surface flows, using mass and momentum conservation, by (unstructured) FV schemes. Most popular (applied): Nonlinear Shallow Water Equations (SWE)

3 Motivation Discretize depth-integrated equations that model free surface flows, using mass and momentum conservation, by (unstructured) FV schemes. Most popular (applied): Nonlinear Shallow Water Equations (SWE) Limitation: Not applicable for wave propagation in intermediate / deeper waters (dispersion has an effect on free surface flow) Popular Boussinesq-type (BT) models for intermediate water depths: Peregrine s standard equations (1967) but for h L 1 5

4 Motivation Discretize depth-integrated equations that model free surface flows, using mass and momentum conservation, by (unstructured) FV schemes. Most popular (applied): Nonlinear Shallow Water Equations (SWE) Limitation: Not applicable for wave propagation in intermediate / deeper waters (dispersion has an effect on free surface flow) Popular Boussinesq-type (BT) models for intermediate water depths: Peregrine s standard equations (1967) but for h 1 L 5 Madsen and Sörensen s (MS) equations (1992) Nowgu s equations (1993) for Beji and Nadaoka (BN) equations (1996) h L 1 2

5 Motivation Discretize depth-integrated equations that model free surface flows, using mass and momentum conservation, by (unstructured) FV schemes. Most popular (applied): Nonlinear Shallow Water Equations (SWE) Limitation: Not applicable for wave propagation in intermediate / deeper waters (dispersion has an effect on free surface flow) Popular Boussinesq-type (BT) models for intermediate water depths: Peregrine s standard equations (1967) but for h 1 L 5 Madsen and Sörensen s (MS) equations (1992) h Nowgu s equations (1993) for 1 L 2 Beji and Nadaoka (BN) equations (1996) Gobbi, Kirby and Wei BT model (2) Variety of BT models that include higher-order nonlinear and dispersive terms: P.A. Madsen et al. (22-29), Lynett et al. (24-21).

6 (Some) Numerical Works Until recently, Finite-Differences (FD) was the predominant method based on the work of Wei & Kirby (1995), for 1D & 2D computations

7 (Some) Numerical Works Until recently, Finite-Differences (FD) was the predominant method based on the work of Wei & Kirby (1995), for 1D & 2D computations More recently hybrid FV/FD schemes in 1D & 2D: Nwogu s, MS, BN, Serre Green-Green Naghdi S-GN equations using: Riemann solvers (Roe s and HLL-type), MUSCL-type reconstructions. Erduran et al. (25 & 27), Cienfuegos et al (26 & 27), Tonelli & Petti (29), Shiach & Mingham (29), Roeber et al. (21), Bonneton et al. (21), Kazolea & Delis (211), Dutykh et al. (211).

8 (Some) Numerical Works Until recently, Finite-Differences (FD) was the predominant method based on the work of Wei & Kirby (1995), for 1D & 2D computations More recently hybrid FV/FD schemes in 1D & 2D: Nwogu s, MS, BN, Serre Green-Green Naghdi S-GN equations using: Riemann solvers (Roe s and HLL-type), MUSCL-type reconstructions. Erduran et al. (25 & 27), Cienfuegos et al (26 & 27), Tonelli & Petti (29), Shiach & Mingham (29), Roeber et al. (21), Bonneton et al. (21), Kazolea & Delis (211), Dutykh et al. (211). MS equations in 2D Tonelli & Petti (29 & 21), two-layer BT equations Lynnet et al., (26-21), TVD Boussinesq solver Shi, Kirby et. al. (212).

9 (Some) Numerical Works Until recently, Finite-Differences (FD) was the predominant method based on the work of Wei & Kirby (1995), for 1D & 2D computations More recently hybrid FV/FD schemes in 1D & 2D: Nwogu s, MS, BN, Serre Green-Green Naghdi S-GN equations using: Riemann solvers (Roe s and HLL-type), MUSCL-type reconstructions. Erduran et al. (25 & 27), Cienfuegos et al (26 & 27), Tonelli & Petti (29), Shiach & Mingham (29), Roeber et al. (21), Bonneton et al. (21), Kazolea & Delis (211), Dutykh et al. (211). MS equations in 2D Tonelli & Petti (29 & 21), two-layer BT equations Lynnet et al., (26-21), TVD Boussinesq solver Shi, Kirby et. al. (212). 2D Finite Element (FE) on unstructured meshes: Walkey & Berzins (22), Sorensen et al. (24), Escilsson & Sherwin (26) and Engsig-Karup et al. (28).

10 (Some) Numerical Works Until recently, Finite-Differences (FD) was the predominant method based on the work of Wei & Kirby (1995), for 1D & 2D computations More recently hybrid FV/FD schemes in 1D & 2D: Nwogu s, MS, BN, Serre Green-Green Naghdi S-GN equations using: Riemann solvers (Roe s and HLL-type), MUSCL-type reconstructions. Erduran et al. (25 & 27), Cienfuegos et al (26 & 27), Tonelli & Petti (29), Shiach & Mingham (29), Roeber et al. (21), Bonneton et al. (21), Kazolea & Delis (211), Dutykh et al. (211). MS equations in 2D Tonelli & Petti (29 & 21), two-layer BT equations Lynnet et al., (26-21), TVD Boussinesq solver Shi, Kirby et. al. (212). 2D Finite Element (FE) on unstructured meshes: Walkey & Berzins (22), Sorensen et al. (24), Escilsson & Sherwin (26) and Engsig-Karup et al. (28). FV for unstructured meshes: Only one work by Asmar and Nwogu (26) using a low-order staggered scheme

11 Outline Physical problem set up

12 Outline Physical problem set up Discretization of Nwogu s and Madsen and Sørensen s extened BT models in 1D Numerical results in 1D

13 Outline Physical problem set up Discretization of Nwogu s and Madsen and Sørensen s extened BT models in 1D Numerical results in 1D Comparison of two 2D FV schemes on triangles

14 Outline Physical problem set up Discretization of Nwogu s and Madsen and Sørensen s extened BT models in 1D Numerical results in 1D Comparison of two 2D FV schemes on triangles A new unstructured FV scheme for BT equations Numerical results in 2D

15 Outline Physical problem set up Discretization of Nwogu s and Madsen and Sørensen s extened BT models in 1D Numerical results in 1D Comparison of two 2D FV schemes on triangles A new unstructured FV scheme for BT equations Numerical results in 2D Conclusions

16 Physical problem setup η: free surface elevation; h: steel water level; H = η + h: total water depth; b: bottom topography; L: wave length; A: wave amplitude Intermediate water: Deep water: h L > < h L 1 2 Shallow water: h L 1 2

17 Physical problem setup η: free surface elevation; h: steel water level; H = η + h: total water depth; b: bottom topography; L: wave length; A: wave amplitude Intermediate water: Deep water: h L > < h L 1 2 Shallow water: h L 1 2

18 Mathematical models: Nwogu s equations Vector conservative form (for both models): U t + F(U ) x = S(U), [ ] [ H Hu U = P, F(U) = Hu gh2 ] [, U = H Hu ]. (1) Using z a =.53753h as optimal reference depth (Roeber et al., 21).

19 Mathematical models: Nwogu s equations Vector conservative form (for both models): U t + F(U ) x = S(U), [ ] [ H Hu U = P, F(U) = Hu gh2 ] [, U = H Hu ]. (1) Using z a =.53753h as optimal reference depth (Roeber et al., 21). P = Hu + Hz a ( za 2 u xx + (hu) xx ) [ Velocity function]

20 Mathematical models: Nwogu s equations Vector conservative form (for both models): U t + F(U ) x = S(U), [ ] [ H Hu U = P, F(U) = Hu gh2 ] [, U = H Hu ]. (1) Using z a =.53753h as optimal reference depth (Roeber et al., 21). P = Hu + Hz a ( za 2 u xx + (hu) xx ) [ Velocity function] S(U) = S b + S f + S d [Source terms]

21 Mathematical models: Nwogu s equations Vector conservative form (for both models): U t + F(U ) x = S(U), [ ] [ H Hu U = P, F(U) = Hu gh2 ] [, U = H Hu ]. (1) Using z a =.53753h as optimal reference depth (Roeber et al., 21). P = Hu + Hz a ( za 2 u xx + (hu) xx ) [ Velocity function] S(U) = S b + S f + S d [Source terms] S b = [ ghb x ] T, S f = [ ghs f ], S d = [ ψ C uψ C + ψ M R b ] u u S f = nm 2 Friction force, n H 4/3 m = Manning coeff.

22 Mathematical models: Nwogu s equations Vector conservative form (for both models): U t + F(U ) x = S(U), [ ] [ H Hu U = P, F(U) = Hu gh2 ] [, U = H Hu ]. (1) Using z a =.53753h as optimal reference depth (Roeber et al., 21). P = Hu + Hz a ( za 2 u xx + (hu) xx ) [ Velocity function] S(U) = S b + S f + S d [Source terms] S b = [ ghb x ] T, S f = [ ghs f ], S d = [ ψ C uψ C + ψ M R b ] u u S f = nm 2 Friction force, n H 4/3 m = Manning coeff. ( ψ M = H t z za ) ( ) ( ) a 2 u z xx + (hu) xx, a ψc = 2 h2 hu 2 6 xx + z a + h h(hu) 2 xx x R b parametrization of wave breaking characteristics

23 Mathematical models: Madsen & Sørensen s equations P = Hu (B )h2 (Hu) xx 1 3 h x(hu) x [ Velocity function]

24 Mathematical models: Madsen & Sørensen s equations P = Hu (B )h2 (Hu) xx 1 3 h x(hu) x [ Velocity function] S(U) = S b + S f + S d [Source term] where now S d = [ ψ R b ] and

25 Mathematical models: Madsen & Sørensen s equations P = Hu (B )h2 (Hu) xx 1 3 h x(hu) x [ Velocity function] S(U) = S b + S f + S d [Source term] where now S d = [ ψ R b ] and ψ = Bgh 3 η xxx 2h 2 h x Bgη xx B = 1 determines the dispersion properties of the system. 15

26 Numerical Model Advective part and topography source: Well-balanced FV formulation Dispersive terms: Finite differences.

27 Numerical Model Advective part and topography source: Well-balanced FV formulation Dispersive terms: Finite differences. Roe s Riemann solver is used (Roe, 1981).

28 Numerical Model Advective part and topography source: Well-balanced FV formulation Dispersive terms: Finite differences. Roe s Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994, Delis et al., 28).

29 Numerical Model Advective part and topography source: Well-balanced FV formulation Dispersive terms: Finite differences. Roe s Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994, Delis et al., 28). High-order spatial accuracy: fourth order MUSCL-type scheme (Yamamoto et al., 1998).

30 Numerical Model Advective part and topography source: Well-balanced FV formulation Dispersive terms: Finite differences. Roe s Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994, Delis et al., 28). High-order spatial accuracy: fourth order MUSCL-type scheme (Yamamoto et al., 1998). For dispersive terms: fourth order FD of first-order spatial derivatives, second and third-order FD for second and third-order derivatives. Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29)

31 Numerical Model (cont.) Special treatment wet/dry fronts: Identify dry cells: through a tolerance parameter Consistent depth reconstruction: satisfy h = b to high-order on x x wet/dry fronts Satisfy an extended C-property: Redefinition of the bed slope, numerical fluxes are computed assuming temporarily zero velocity at wet/dry faces (Brufau et al., 24) Time Integration (should at least match the order of truncation errors from dispersion terms): Third order Adams-Basforth predictor and fourth-order Adams-Moulton corrector stage.

32 Numerical Model (cont.) Special treatment wet/dry fronts: Identify dry cells: through a tolerance parameter Consistent depth reconstruction: satisfy h = b to high-order on x x wet/dry fronts Satisfy an extended C-property: Redefinition of the bed slope, numerical fluxes are computed assuming temporarily zero velocity at wet/dry faces (Brufau et al., 24) Time Integration (should at least match the order of truncation errors from dispersion terms): Third order Adams-Basforth predictor and fourth-order Adams-Moulton corrector stage. Extract depth averaged velocities, u, from the velocities functions P by solving a tridiagonal system,

33 Numerical Model (cont.) Special treatment wet/dry fronts: Identify dry cells: through a tolerance parameter Consistent depth reconstruction: satisfy h = b to high-order on x x wet/dry fronts Satisfy an extended C-property: Redefinition of the bed slope, numerical fluxes are computed assuming temporarily zero velocity at wet/dry faces (Brufau et al., 24) Time Integration (should at least match the order of truncation errors from dispersion terms): Third order Adams-Basforth predictor and fourth-order Adams-Moulton corrector stage. Extract depth averaged velocities, u, from the velocities functions P by solving a tridiagonal system, Thomas algorithm.

34 Numerical Test & Results in 1D: (Solitary wave propagation) Two cases: A/ =.1 Dx =.5 C r =.4 (MS left, Nwogu s right) η/h η/h x(m) Analytical t=5s t=15s x(m) Analytical t=5s t=15s η/h x Ct η/d x Ct η/a Nwogu MS

35 Solitary wave propagation Two cases: A/h =.2 Dx =.5 C r =.4 (MS left, Nwogu s right) η/h η/h x(m) x(m).2 Analytical t=5s t=15s.2 Analytical t=5s t=15s η/h.1 η/h x Ct x Ct η/a Nwogu MS

36 Head on collision of two solitary waves A/h =.3 Dx =.1m C r =.4, x [, 3].3 η/h.2.1 η/h Analytical.6 Nwogu MS.4.2 η/h Analytical Nwogu MS x/d

37 Regular wave propagation over a submerged bar T = 2.2s, H =.2, h/l =.11, Dx =.4m.1 Depth(m) Bar 5 1 x(m)

38 Regular wave propagation over a submerged bar T = 2.2s, H =.2, h/l =.11, Dx =.4m η(m).4.2 WG 4 MS Nwogu Experimental η(m).4.2 WG t(s) t(s)

39 Regular wave propagation over a submerged bar T = 2.2s, H =.2, h/l =.11, Dx =.4m η(m).4.2 WG 4 MS Nwogu Experimental η(m).4.2 WG t(s) t(s).4.4 WG 6 WG 7 η(m).2 η(m) t(s) t(s)

40 Regular wave propagation over a submerged bar T = 2.2s, H =.2, h/l =.11, Dx =.4m η(m).4.2 WG 4 MS Nwogu Experimental η(m).4.2 WG t(s) t(s).4.4 WG 6 WG 7 η(m).2 η(m) t(s) t(s).4.4 WG 8 WG 1 η(m).2 η(m) t(s) t(s)

41 Mathematical Model: The NSWE U t U = + H( U) = L( U) on Ω [, t] R 2 R +, H Hu Hv, H(U) = [F, G] = Hu Hu gh2 Huv Hv Huv Hv gh2, R 1 = [ L(U) = [R 1 + R 2 + S b ] gh b(x,y) x ] T and R 2 = [ gh b(x,y) y ] T. S = [ ghs f x ghs f y ] T with S f x = n2 m u u H 4 3 and Sy f = n2 m v u, H 4 3

42 NCFV approach C P U t dω + HdΩ = SdΩ C P C P UdΩ + H ñdl = SdΩ t C P C P C P

43 NCFV approach C P U t dω + HdΩ = SdΩ C P C P Introducing the flux vectors ) Φ PQ = (Fñ x + Gñ y dl and Φ P,Γ = C PQ Hence, FV scheme reads U P = 1 Φ PQ 1 t C P C Q K P Φ P,Γ + 1 C P P UdΩ + H ñdl = SdΩ t C P C P C P C P Γ C P. (Fñ x + Gñ y ) dl (S b + S d + S f )dω

44 NCFV approach C P U t dω + HdΩ = SdΩ C P C P Introducing the flux vectors ) Φ PQ = (Fñ x + Gñ y dl and Φ P,Γ = C PQ Hence, FV scheme reads U P = 1 Φ PQ 1 t C P C Q K P Φ P,Γ + 1 C P P Φ PQ Numerical flux UdΩ + H ñdl = SdΩ t C P C P C P C P Γ C P. (Fñ x + Gñ y ) dl (S b + S d + S f )dω

45 NCFV approach: Gradient & divergence formulas Green -Gauss linear reconstruction on Ω P C P = 1 3 Ω P wda = Ω P ( w) P = wñdl Ω P 1 ) 1 (w P + w Q n PQ C P 2 Q K P

46 NCFV approach: Gradient & divergence formulas Green -Gauss linear reconstruction on Ω P C P = 1 3 Ω P wda = Ω P Ω P udω = ( u) P = ( w) P = 3 u ñdl = Ω P 2 (u P + u Q ) n PQ Q K P 1 (u P + u Q ) n PQ 2 C P Q K P wñdl Ω P 1 ) 1 (w P + w Q n PQ C P 2 Q K P

47 NCFV approach: Gradient & divergence formulas Green -Gauss linear reconstruction on Ω P C P = 1 3 Ω P wda = Ω P ( w) P = wñdl Ω P 1 ) 1 (w P + w Q n PQ C P 2 Q K P

48 NCFV approach: Gradient & divergence formulas Green -Gauss linear reconstruction on Ω P C P = 1 3 Ω P wda = Ω P ( w) P = wñdl Ω P 1 ) 1 (w P + w Q n PQ C P 2 Q K P w L i,pq = w i,p LIM ( ( w i ) upw P r PQ, ( w i ) cent r PQ ) w R i,pq = w i,q 1 2 LIM ( ( w i ) upw Q r PQ, ( w i ) cent r PQ ) ( w i ), cent r PQ = w i,q w i,p, ( w i ) P, upw r PQ = 2( w i ) P ( w i ) cent

49 CCFV approach U p T p = t q K(p) Φ q + SdΩ. T p Φ q Numerical flux. Linear reconstruction Naive calculation (at point D) (w i,p ) L D = w i,p + r pd r pq LIM ( ( w i ) upw p r pq, ( w i ) cent r pq ) ; (w i,q ) R D = w i,q r Dq r pq LIM ( ( w i ) upw q r pq, ( w i ) cent r pq ),

50 CCFV approach U p T p = t q K(p) Φ q + SdΩ. T p Φ q Numerical flux. Linear reconstruction Naive calculation (at point D) (w i,p ) L D = w i,p + r pd r pq LIM ( ( w i ) upw p r pq, ( w i ) cent r pq ) ; (w i,q ) R D = w i,q r Dq r pq LIM ( ( w i ) upw q r pq, ( w i ) cent r pq ), Corrected calculation (at point M) (Delis et al., 211) w L i,p = (w i,p ) L D + r DM ( w i ) p, w R i,q = (w i,q ) R D + r DM ( w i ) q.

51 CCFV approach:gradient formulas Three element (compact stencil) gradient w i,p = 1 C c p q,r K(p) r q 1 2 (w i,q + w i,r )n q,r Extended element (wide stencil) gradient w i,p = 1 ) 1 (w Cp w i,l + w i,r n lr 2 l,r K (p) r l

52 CCFV approach:gradient formulas Three element (compact stencil) gradient w i,p = 1 C c p q,r K(p) r q 1 2 (w i,q + w i,r )n q,r Extended element (wide stencil) gradient w i,p = 1 ) 1 (w Cp w i,l + w i,r n lr 2 l,r K (p) r l Boundary conditions: use the theory of characteristics for weak formulation for the NCFV and ghost cells for CCFV scheme

53 Topography source discretization Wel-ballanced schemes introduce topography source flux vectors S b : S b (U )dxdy = S bq (CCFV) T p q K(p) S b (U )dxdy = (NCFV) C P Q K P S bpq S b = ( 1 2 P I ) Λ Λ 1 P 1 Sb where (for 1st order scheme): S b q = g HL + H R ( b R b L) n qx 2 S b PQ = g HL + H R ( b R b L) n PQx 2 g HL + H R ( b R b L) n qy g HL + H R ( b R b L) n PQy 2 q 2 PQ

54 Topography source discretization Wel-ballanced schemes introduce topography source flux vectors S b : S b (U )dxdy = S bq (CCFV) T p q K(p) S b (U )dxdy = (NCFV) C P Q K P S bpq S b = ( 1 2 P I ) Λ Λ 1 P 1 Sb + S /2nd order scheme, correction b term S b q = g HL + H p ( ) b L b p nqx 2 S b PQ = g HL + H P ( ) b L b P npqx 2 g HL + H p ( ) b L b p nqy g HL + H P ( ) b L b P npqy 2 2.

55 Topography source discretization (wet/dry) Extended C-property, (Castro et al, 25) In the MUSCL scheme for hydrostatic conditions we must have, at i-cell b L b i = (h L h i ) ( B) i = ( h) i If in the gradient calculation, of a wet cell, a dry node is involved we correct the h L and/or h R by imposing h L = h i (b L b i ) and/or h R = h j (b R b j ) Redefine the bed value in the dry node for emerging bed situations in Sb to maintain hydrostatic conditions, (Brufau et al, 22). For water in motion over emerging slopes: If h L > ɛ wd and h R ɛ wd and h L < (b B b L ), set temporarily for the wet i-cell u L = v L =

56 Numerical Results Grids used: Consistently refined grids, for N =degrees of freedom, characteristic length h N = L x L y /N Equivalent meshes Scheme NCFV CCFVc1 CCFVc2 CCFVw1 CCFVw2 Description Node-Centered FV Scheme Cell-Centered FV compact (naive) reconstruction stencil Cell-Centered FV compact reconstruction stencil (corrected) Cell-Centered FV wide (naive) reconstruction stencil Cell-Centered FV wide reconstruction stencil (corrected)

57 Traveling vortex: Convergence results The traveling vortex solution

58 Comparison 1. Convergence behavior to second order: NCFV is not grid dependent/ CCFV depents on the grid used.

59 Comparison 1. Convergence behavior to second order: NCFV is not grid dependent/ CCFV depents on the grid used. 2. Edge based limiting procedure in the MUSCL reconstruction: inadequate for the CCFV schemes if the proposed correction not used.

60 Comparison 1. Convergence behavior to second order: NCFV is not grid dependent/ CCFV depents on the grid used. 2. Edge based limiting procedure in the MUSCL reconstruction: inadequate for the CCFV schemes if the proposed correction not used. 3. CCFVw2 has an almost identical behavior with the NCFV scheme is achieved, but with the extra computational cost introduced.

61 Comparison 1. Convergence behavior to second order: NCFV is not grid dependent/ CCFV depents on the grid used. 2. Edge based limiting procedure in the MUSCL reconstruction: inadequate for the CCFV schemes if the proposed correction not used. 3. CCFVw2 has an almost identical behavior with the NCFV scheme is achieved, but with the extra computational cost introduced. 4. Ghost cells for the boundary treatment in CCFV formulations can lead to an order reduction.

62 Comparison 1. Convergence behavior to second order: NCFV is not grid dependent/ CCFV depents on the grid used. 2. Edge based limiting procedure in the MUSCL reconstruction: inadequate for the CCFV schemes if the proposed correction not used. 3. CCFVw2 has an almost identical behavior with the NCFV scheme is achieved, but with the extra computational cost introduced. 4. Ghost cells for the boundary treatment in CCFV formulations can lead to an order reduction. 5. Wet/dry treatment: accurate for both FV approaches on all grid types.

63 Comparison 1. Convergence behavior to second order: NCFV is not grid dependent/ CCFV depents on the grid used. 2. Edge based limiting procedure in the MUSCL reconstruction: inadequate for the CCFV schemes if the proposed correction not used. 3. CCFVw2 has an almost identical behavior with the NCFV scheme is achieved, but with the extra computational cost introduced. 4. Ghost cells for the boundary treatment in CCFV formulations can lead to an order reduction. 5. Wet/dry treatment: accurate for both FV approaches on all grid types.

64 An unstructured FV scheme for BT Equations Vector conservative form for Nwogu s equations: U t + H( U ) = L( U ) on Ω [, t] R 2 R +, U vector of the new variables, U = [H, Hu, Hv] T and H = [F, G] H U = P 1, L(U) = [S b + S d + S f ] P 2 [ ] [ ] P1 z 2 with P = = H a P ( u) + 2 z a ( hu) + u and 2 ψ c S b = ghb x, S d = uψ c + ψ Mx, S f = Sx f + R bx ghb y vψ c + ψ My Sy f + R by

65 Vector conservative form (cont.) [( ) ( z 2 ψ c = a 2 h2 h ( u) + z a + h ) ] h ( hu) 6 2 ψ M = [ ψmx ψ My ] = H t z 2 a 2 ( u) + H tz a ( hu)

66 Vector conservative form (cont.) [( ) ( z 2 ψ c = a 2 h2 h ( u) + z a + h ) ] h ( hu) 6 2 ψ M = [ ψmx ψ My ] = H t z 2 a 2 ( u) + H tz a ( hu) R b = [R bx, R by ] T = parametrization of wave breaking characteristics where: [ ν ] T R bx = R bx, where R bx = ν(hu) x 2 ((Hu) y + (Hv) x ) and [ ν ] T R by = R by, where R by = 2 ((Hu) y + (Hv) x ) ν(hv) y. where ν = Bδ 2 b Hη t is the eddy viscosity coefficient with < B < 1 and δ b is a mixing length coefficient.

67 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

68 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

69 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

70 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

71 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

72 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

73 Numerical Model: Spatial discretization Advective (nonlinear) part and topography source term: Well-balanced FV formulation. Roe s approximate Riemann solver is used (Roe, 1981). Upwinding of the topography source term (Bermudez et al., 1994). High-order spatial accuracy: third-order MUSCL-type scheme (Barth, 1993). Satisfy the C-property (flow at rest) to higher spatial order: Addition of an extra term to bed upwinding (Hubbard and Garcia-Navarro, 2 and Delis and Nikolos, 29). Special treatment of wet/dry fronts. Dispersion terms: consistent FV approximations based on gradient and divergence computations (Kazolea et al., 212).

74 Discretization of the dispersive terms (mass equation) Integral averaging: (ψ c ) P = 1 [( z 2 a C P C P 2 h2 6 = 1 { [( z 2 a C P 2 h2 6 Q K P C PQ ) ( h ( u) + z a + h ) ] h ( hu) dω 2 ) ] [( h ( u) ñdl + z a + h ) ] } h ( hu) ñdl C PQ 2 C PQ C PQ ( ) [( ) ] z 2 a 2 h2 z 2 h ( u) ñdl a 6 2 h2 h [ ( u) n PQ ] 6 M, M ( z a + h 2 ) [( h ( hu) ñdl z a + h ) ] h [ ( hu) n PQ ] 2 M M K PQ := {R N R is a vertex of M PQ and RQ M PQ } ( w) M = 1 M PQ R,Q K PQ R Q 1 2 (w R + w Q ) n RQ

75 Discretization of the dispersive terms (momentum equations) 1 ( uψ c + ψ M ) dω = u P ψ c dω + 1 ψ M dω. C P C P C P C P C P C P The ψ c is discretized as before and the second term takes the discrete form: (ψ M ) P = 1 C P = 1 C P [ za 2 H t 2 ψ M dω = 1 C P C P z H t CP 2 ] P CP H t z 2 a a 1 ( u)dω + 2 C P [ ( u)] P + [H t z a ] P [ ( hu)] P, 2 ( u) + H tz a ( hu)dω H t z a ( hu)dω C P

76 Time integration Consider the semi-discrete scheme: U P = L (U) t Time Integration (match the order of truncation errors from dispersion terms): Use 3rd order explicit Strong Stability-Preserving Runge-Kutta (SSP-RK): U (1) P = U (n) P + t n L ( U (n)) ; U (2) = 3 P 4 U(n) + 1 P 4 U(1) + P t n 1 4 L ( U (1)) ; U (n+1) = 1 P 3 U(n) + 2 P 3 U(2) + P t n 2 3 L ( U (2)) Time step t n estimated by a CFL stability condition as t n R P = CFL min P ( u2 + v 2 + ) n c P

77 Velocity field recovery From new solution variables P = [P 1, P 2 ] T At each step in the RK scheme a linear system MV = C with M R 2N 2N and C = [P 1 P 2 P N ] T, has to be solved to obtain the velocities V = [u 1 u 2 u N ] T. Each two rows of the system read as [ ] H (i) z 2 (i) a P 2 ( u) + z a ( hu) + u P = P (i), i = 1, 2, n + 1. P

78 Velocity field recovery From new solution variables P = [P 1, P 2 ] T At each step in the RK scheme a linear system MV = C with M R 2N 2N and C = [P 1 P 2 P N ] T, has to be solved to obtain the velocities V = [u 1 u 2 u N ] T. Each two rows of the system read as [ ] H (i) z 2 (i) a P 2 ( u) + z a ( hu) + u P = P (i), i = 1, 2, n + 1. P Important to (a) keep the unknown information needed at the minimum possible level (i.e neighboring nodes) and (b) exploit already computed geometrical information. (z H a) P 2 P 2 1 C P ( u) M n PQ + (z a) P C Q K P P ( hu) M n PQ + u P = P P Q K P

79 Velocity field recovery From new solution variables P = [P 1, P 2 ] T At each step in the RK scheme a linear system MV = C with M R 2N 2N and C = [P 1 P 2 P N ] T, has to be solved to obtain the velocities V = [u 1 u 2 u N ] T. Each two rows of the system read as [ ] H (i) z 2 (i) a P 2 ( u) + z a ( hu) + u P = P (i), i = 1, 2, n + 1. P Important to (a) keep the unknown information needed at the minimum possible level (i.e neighboring nodes) and (b) exploit already computed geometrical information. (z 2 a) P 2 C P (z H a) P 2 P 2 1 C P A Q u Q + A P u P + (z a) P C Q K P P ( u) M n PQ + (z a) P C Q K P P ( hu) M n PQ + u P = P P Q K P B Q u Q + B P u P + Iu P = 1 P P, P = 1... N H Q K P P

80 Solution of the linear system The matrix M is sparse and structurally symmetric but is also mesh dependent.

81 Solution of the linear system The matrix M is sparse and structurally symmetric but is also mesh dependent. Matrix M is stored in the compressed sparse row (CSR) format.

82 Solution of the linear system The matrix M is sparse and structurally symmetric but is also mesh dependent. Matrix M is stored in the compressed sparse row (CSR) format. The ILUT preconditioner from SPARSKIT package is used.

83 Solution of the linear system The matrix M is sparse and structurally symmetric but is also mesh dependent. Matrix M is stored in the compressed sparse row (CSR) format. The ILUT preconditioner from SPARSKIT package is used. The reverse Cuthill McKee (RCM) algorithm is also employed to reorder the matrix elements as to minimize the matrix bandwidth.

84 Solution of the linear system The matrix M is sparse and structurally symmetric but is also mesh dependent. Matrix M is stored in the compressed sparse row (CSR) format. The ILUT preconditioner from SPARSKIT package is used. The reverse Cuthill McKee (RCM) algorithm is also employed to reorder the matrix elements as to minimize the matrix bandwidth. System is solved using Bi-Conjugate Gradient Stabilized method (BiCGStab), with tolerance 5 1 6

85 Solution of the linear system The matrix M is sparse and structurally symmetric but is also mesh dependent. Matrix M is stored in the compressed sparse row (CSR) format. The ILUT preconditioner from SPARSKIT package is used. The reverse Cuthill McKee (RCM) algorithm is also employed to reorder the matrix elements as to minimize the matrix bandwidth. System is solved using Bi-Conjugate Gradient Stabilized method (BiCGStab), with tolerance Convergence to the solution was obtained in one or two steps with the numerical solution for the velocities at the previous time step given as initial guess.

86 Boundary conditions and the internal source function Wall (reflective) boundary condition: u ñ = for x Ω By conservation of mass (no loss or gain through the wall) [ ( ) ( z 2 HdΩ + Hu + a t Ω Ω 2 h2 h ( u) + z a + h ) ] h ( hu) ñdl = 6 2 Define the normal boundary advective flux in weak form, 1 Φ P,Γ = 2 g(h ) 2 n P,1x by the method of characteristics 1 2 g(h ) 2 n P,1y

87 Boundary conditions and the internal source function Wall (reflective) boundary condition: u ñ = for x Ω By conservation of mass (no loss or gain through the wall) [ ( ) ( z 2 HdΩ + Hu + a t Ω Ω 2 h2 h ( u) + z a + h ) ] h ( hu) ñdl = 6 2 Define the normal boundary advective flux in weak form, 1 Φ P,Γ = 2 g(h ) 2 n P,1x by the method of characteristics 1 2 g(h ) 2 n P,1y Absorbing boundaries: should dissipate the energy of incoming waves ( ) x d(x) 2 Sponge layer is defined: m(x) = 1, L L s 1.5L, L s

88 Boundary conditions and the internal source function Wall (reflective) boundary condition: u ñ = for x Ω By conservation of mass (no loss or gain through the wall) [ ( ) ( z 2 HdΩ + Hu + a t Ω Ω 2 h2 h ( u) + z a + h ) ] h ( hu) ñdl = 6 2 Define the normal boundary advective flux in weak form, 1 Φ P,Γ = 2 g(h ) 2 n P,1x by the method of characteristics 1 2 g(h ) 2 n P,1y Absorbing boundaries: should dissipate the energy of incoming waves ( ) x d(x) 2 Sponge layer is defined: m(x) = 1, L L s 1.5L, Internal source function for regular waves (Wei et al., 1993) added to the mass equation S(x, t) = D exp ( γ(x x s ) 2) sin(λy ωt) L s

89 Wave breaking models Eddy viscosity wave breaking model (R b ) P = = 1 C P 1 C P R b dω = 1 C P C P C P [ ] Rbx ñ dl 1 R by ñ C P C PQ [ R by ] dω R bx [ ] Rbx n PQ R by n PQ M

90 Wave breaking models Eddy viscosity wave breaking model (R b ) P = Hybrid models = 1 C P 1 C P R b dω = 1 C P C P C P [ ] Rbx ñ dl 1 R by ñ C P C PQ [ R by ] dω R bx [ ] Rbx n PQ R by n PQ Hybrid(ɛ) BT degenerate into NSWE as dispersive terms become negligible. Criterion: ɛ = A h.8 In post breaking region ɛ <.4 in order to switch NSWE/BT. M

91 Wave breaking models Eddy viscosity wave breaking model (R b ) P = Hybrid models = 1 C P 1 C P R b dω = 1 C P C P C P [ ] Rbx ñ dl 1 R by ñ C P C PQ [ R by ] dω R bx [ ] Rbx n PQ R by n PQ Hybrid(ɛ) BT degenerate into NSWE as dispersive terms become negligible. Criterion: ɛ = A h.8 In post breaking region ɛ <.4 in order to switch NSWE/BT. New Hybrid criteria: η t γ gh, γ [.35,.65], η x tan(φ c ). Distinguish the different breaking waves. Find non-breaking undular bores checking the Froude number. Extend the computational region of the NSWE. M

92 Suppression of the dispersive terms Not clear how the switching between the two models is implemented

93 Suppression of the dispersive terms Not clear how the switching between the two models is implemented Discontinuity at the switching point BT/NSWE, causing spurious oscillations.

94 Suppression of the dispersive terms Not clear how the switching between the two models is implemented Discontinuity at the switching point BT/NSWE, causing spurious oscillations. Matrix M can not be change.

95 Suppression of the dispersive terms Not clear how the switching between the two models is implemented Discontinuity at the switching point BT/NSWE, causing spurious oscillations. Matrix M can not be change. Need of a clever implementation.

96 Suppression of the dispersive terms Not clear how the switching between the two models is implemented Discontinuity at the switching point BT/NSWE, causing spurious oscillations. Matrix M can not be change. Need of a clever implementation. Methodology. Starting with the solution vector U n P, P = 1,..., N, at time t n, 1. H is computed C P using the BT model (named from now on H n+1 ). BT 1.1 If breaking is on for N br < N cells additional solution vector H n+1 ψ BT c named H n+1. BT/SW 2. P n+1 BT is computed C P / t H n+1 Hn+1 BT Hn for the ψ t n+1 M. 2.1 N br cells P n+1 BT/SW Pn+1 BT 3. MV = C,with C = [P n+1 1, P n+1 2,, P n+1 ψ c ψ M [(Hu), (Hv)] n+1,t N ]T BT un+1 BT 4. Final solution: H n+1 BT/SW, Pn+1 BT/SW, un+1 BT/SW un+1 BT vector with its values at the breaking nodes replaced by those of u n+1 SW.

97 Numerical Results:Spatial accuracy Solitary wave: A/h =.1, (x, y) [, 3] [, 5m] Reference solution of N = 232, 849 nodes

98 Solitary wave interaction with a vertical circular cylinder Area: (x, y) = [ 4, 1m] [,.55m], A/h =.25, N = 1, 69

99 Solitary wave interaction with a vertical circular cylinder Wave Gauges:

100 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

101 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

102 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

103 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

104 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

105 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

106 Run-up of a solitary wave on a conical island (Briggs et al. 1995) Area: (x, y) = [ 5, 28m] [, 3m], A/h =.18, N = 52, 191, CFL =.8 Using mesh h enrichment

107 Run-up of a solitary wave on a conical island (cont) Time series of surface elevation at wave gauges around the island:

108 Run-up of a solitary wave on a conical island (cont) Time series of surface elevation at wave gauges around the island: Experimental measurements and numerical runup around the island: Simulation 28min on a single 2.4GHz Intel Core 2 Quad Q66 processor

109 Wave propagation over a semicircular shoal (Whalin 1971) T = 2.s, h/l =.117, A/h =.165, kh =.735 and S = 1.198

110 Wave propagation over a semicircular shoal (Whalin 1971) T = 2.s, h/l =.117, A/h =.165, kh =.735 and S = Free surface and spatial evolution of the 1st, 2nd and 3rd harmonic

111 Solitary wave run-up on a plane beach (Synolakis, 1987) Area: (x, y) = [ 2, 6m] [, 1m], A/h =.28, N = 8, 816, CFL =.4, n m =.1 H(m) Bed New Hybrid Eddy Visc. Experimental t g/h =

112 Solitary wave run-up on a plane beach (Synolakis, 1987) Area: (x, y) = [ 2, 6m] [, 1m], A/h =.28, N = 8, 816, CFL =.4, n m =.1 H(m) Bed New Hybrid Eddy Visc. Experimental H(m) t g/h = t g/h =

113 Solitary wave run-up on a plane beach (Synolakis, 1987) Area: (x, y) = [ 2, 6m] [, 1m], A/h =.28, N = 8, 816, CFL =.4, n m =.1 H(m) Bed New Hybrid Eddy Visc. Experimental H(m) H(m) t g/h = t g/h = t g/h =

114 Solitary wave run-up on a plane beach (Synolakis, 1987) Area: (x, y) = [ 2, 6m] [, 1m], A/h =.28, N = 8, 816, CFL =.4, n m =.1 H(m) Bed New Hybrid Eddy Visc. Experimental H(m) H(m) H(m) t g/h = t g/h = t g/h = t g/h =

115 Solitary wave run-up on a plane beach (Synolakis, 1987) Area: (x, y) = [ 2, 6m] [, 1m], A/h =.28, N = 8, 816, CFL =.4, n m =.1 H(m) Bed New Hybrid Eddy Visc. Experimental H(m) H(m) H(m) t g/h = t g/h = H(m) t g/h = t g/h = t g/h =

116 Solitary wave run-up on a plane beach (Synolakis, 1987) Area: (x, y) = [ 2, 6m] [, 1m], A/h =.28, N = 8, 816, CFL =.4, n m =.1 H(m) Bed New Hybrid Eddy Visc. Experimental H(m) H(m) H(m) t g/h = t g/h = H(m) t g/h = t g/h = H(m) t g/h = t g/h =7 1 5 x(m)

117 Wave over a bar (Beji and Battjes, 1993) (x, y) = [ 1, 3m] [,.8m], H =.2m, T = 2.2s, N = 4, Depth(m) Bar 5 1 x(m)

118 Wave over a bar (Beji and Battjes, 1993) (x, y) = [ 1, 3m] [,.8m], H =.2m, T = 2.2s, N = 4, 364 Depth(m) Bar 5 1 x(m) H(m).5.4 Free surface.3 l NSW Bar

119 Wave over a bar (Beji and Battjes, 1993) (x, y) = [ 1, 3m] [,.8m], H =.2m, T = 2.2s, N = 4, 364 Depth(m) H(m) Bar 5 1 x(m) H(m) H(m).5.4 Free surface.3 l NSW Bar

120 Wave over a bar (Beji and Battjes, 1993) (x, y) = [ 1, 3m] [,.8m], H =.2m, T = 2.2s, N = 4, 364 Depth(m) H(m) Bar 5 1 x(m) H(m) H(m).5.4 Free surface.3 l NSW Bar H(m).4.3 H(m) x(m) 15 2

121 Wave over a bar (Beji and Battjes, 1993) Wave gauges: η(m) η(m) η(m) η(m) Hybrid (ε) New Hybrid Experimental data t(s)

122 A two-dimensional reef(roeber, 29) η/h Area: (x, y) = [, 83m] [, 1m], A/h =.3, N = 1, 9, n m = t =31.5s Bed NSW BT Experimental Shoaling

123 A two-dimensional reef(roeber, 29) η/h Area: (x, y) = [, 83m] [, 1m], A/h =.3, N = 1, 9, n m = t =31.5s Bed NSW BT Experimental η/h t =34.5s Shoaling During breaking

124 A two-dimensional reef(roeber, 29) η/h Area: (x, y) = [, 83m] [, 1m], A/h =.3, N = 1, 9, n m = t =31.5s Bed NSW BT Experimental η/h η/h t =34.5s t =35.5s Shoaling During breaking Wave jet hits still water

125 .4.2 A two-dimensional reef(roeber, 29) η/h η/h η/h Area: (x, y) = [, 83m] [, 1m], A/h =.3, N = 1, 9, n m = t =31.5s Bed NSW BT Experimental t =34.5s t =35.5s t =38.5s Shoaling During breaking Wave jet hits still water Bore propagation

126 A two-dimensional reef(roeber, 29) η/h.4.2 t =59.5s Hydraulic jump

127 A two-dimensional reef(roeber, 29) η/h.4.2 t =59.5s Hydraulic jump Time series of surface elevation at wave gauges: η/h η/h η/h Gauge at (x =44.25m, y =.5m) Gauge at (x =5.37m, y =.5m) η/h Gauge at (x =48.233m, y =.5m) Gauge at (x =8.m, y =.5m) t(s) 6 8 1

128 A two-dimensional reef(roeber, 29) Spatial snapshots along the centerline: η/h.4 t =34.5s.2 Bed l NSW Free surface η/h.4 t =36.5s.2 η/h η/h t =38.5s t =49.5s.2 η/h η/h t =4.5s t =56.5s

129 A three-dimensional reef(lynett et al., 29) Area: (x, y) = [, 45m] [ 13m, 13m], A/h =.5, N = 87, 961

130 A three-dimensional reef(lynett et al., 29) Area: (x, y) = [, 45m] [ 13m, 13m], A/h =.5, N = 87, 961

131 A three-dimensional reef(lynett et al., 29) Time series of surface elevation at wave gauges: η WG 2

132 A three-dimensional reef(lynett et al., 29) Time series of surface elevation at wave gauges: η η WG 2 WG 3

133 A three-dimensional reef(lynett et al., 29) Time series of surface elevation at wave gauges: η η η WG 2 WG 3 WG 6

134 A three-dimensional reef(lynett et al., 29) Time series of surface elevation at wave gauges: η η η η.2 WG 2 WG 3 WG 6 WG t(s)

135 A three-dimensional reef(lynett et al., 29) Time series of velocities at wave gauges: u(m/s) 2 ADV 1 Experimental New Hybrid

136 A three-dimensional reef(lynett et al., 29) Time series of velocities at wave gauges: u(m/s) 2 ADV 1 Experimental New Hybrid ADV 2 2 u(m/s)

137 A three-dimensional reef(lynett et al., 29) Time series of velocities at wave gauges: u(m/s) 2 ADV 1 Experimental New Hybrid ADV 2 2 u(m/s) u(m/s) ADV

138 A three-dimensional reef(lynett et al., 29) Time series of velocities at wave gauges: u(m/s) 2 ADV 1 Experimental New Hybrid ADV 2 2 u(m/s) u(m/s) v(m/s) ADV ADV t(s)

139 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE.

140 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE. Conservative formulation and higher-order FV scheme enhance the applicability of the model without altering its dispersion characteristics.

141 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE. Conservative formulation and higher-order FV scheme enhance the applicability of the model without altering its dispersion characteristics. The well-balanced topography and wet/dry front discretizations provided accurate conservative and stable wave propagation shoaling and run-up.

142 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE. Conservative formulation and higher-order FV scheme enhance the applicability of the model without altering its dispersion characteristics. The well-balanced topography and wet/dry front discretizations provided accurate conservative and stable wave propagation shoaling and run-up. The edge-based structure adopted can provide computational efficiency, since most of the geometric quantities needed can be calculated in a pre-processing stage.

143 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE. Conservative formulation and higher-order FV scheme enhance the applicability of the model without altering its dispersion characteristics. The well-balanced topography and wet/dry front discretizations provided accurate conservative and stable wave propagation shoaling and run-up. The edge-based structure adopted can provide computational efficiency, since most of the geometric quantities needed can be calculated in a pre-processing stage. Different type of wave breaking can be implemented. The New Hybrid model proved more stable and accurate than the others applied in this work.

144 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE. Conservative formulation and higher-order FV scheme enhance the applicability of the model without altering its dispersion characteristics. The well-balanced topography and wet/dry front discretizations provided accurate conservative and stable wave propagation shoaling and run-up. The edge-based structure adopted can provide computational efficiency, since most of the geometric quantities needed can be calculated in a pre-processing stage. Different type of wave breaking can be implemented. The New Hybrid model proved more stable and accurate than the others applied in this work. Relatively straight forward to extend existing NSWE codes that use (unstructured) FV schemes as to include dispersion characteristics for deeper water simulations.

145 Conclusions A 2D unstructured FV scheme numerical model has been developed for solving Nwogu s extended BT equations formulated as to have identical flux terms as to the NSWE. Conservative formulation and higher-order FV scheme enhance the applicability of the model without altering its dispersion characteristics. The well-balanced topography and wet/dry front discretizations provided accurate conservative and stable wave propagation shoaling and run-up. The edge-based structure adopted can provide computational efficiency, since most of the geometric quantities needed can be calculated in a pre-processing stage. Different type of wave breaking can be implemented. The New Hybrid model proved more stable and accurate than the others applied in this work. Relatively straight forward to extend existing NSWE codes that use (unstructured) FV schemes as to include dispersion characteristics for deeper water simulations.

146 References 1. A. I. Delis, M. Kazolea, and N. A Kampanis. A robust high resolution finite volume scheme for the simulation of long waves over complex domain. Int. J. Num. Meth. Fluids, 56:419, A. I. Delis, I. A Nikolos, and M. Kazolea. Performance and comparison of cellcentered and node-centered unstructured finite volume discretizations for shallow water free surface flows. Arch. Comput. Methods Eng., 18:57, M. Kazolea and A. I. Delis. A well-balanced shock-capturing hybrid finite volume-finite difference numerical scheme for extended 1D Boussinesq models. Applied Numerical Mathematics, 67:167186, M. Kazolea, A. I. Delis, I. A Nikolos, and C. E. Synolakis. An unstructured finite volume numerical scheme for extended 2D Boussinesq-type equations.coast. Eng., 69:4266, 212.

147 Thank you for your attention!!

148 Solitary wave run-up on a plane beach Area: x [ 2, 6m] A/h =.4, Dx =.5, CFL =.4, n m =.1.1 H(m).5 t g/h= x(m)

149 Solitary wave run-up on a plane beach Area: x [ 2, 6m] A/h =.4, Dx =.5, CFL =.4, n m = H(m).5 H(m).5 t g/h= x(m) t g/h= x(m)

150 Solitary wave run-up on a plane beach Area: x [ 2, 6m] A/h =.4, Dx =.5, CFL =.4, n m = H(m).5 H(m).5 H(m).5 t g/h= x(m) t g/h= x(m) t g/h= x(m)

151 Solitary wave run-up on a plane beach Area: x [ 2, 6m] A/h =.4, Dx =.5, CFL =.4, n m = H(m).5 H(m).5 H(m).5 t g/h= x(m) t g/h= x(m) t g/h= x(m) H(m).5 t g/h= x(m)

152 Solitary wave run-up on a plane beach Area: x [ 2, 6m] A/h =.4, Dx =.5, CFL =.4, n m = H(m).5 H(m).5 H(m).5 t g/h= x(m) t g/h= x(m) t g/h= x(m) H(m).5 H(m).5 t g/h= x(m) t g/h= x(m)