Comparison of cell-centered and node-centered formulations of a high-resolution well-balanced finite volume scheme: application to shallow water flows

Transcription

1 Comparison of cell-centered and node-centered formulations of a high-resolution well-balanced finite volume scheme: application to shallow water flows Dr Argiris I. Delis Dr Ioannis K. Nikolos (TUC) Maria Kazolea (TUC) Department of Sciences-Division of Mathematics Technical University of Crete (TUC), Chania, Greece CMWR 2010, Barcelona

2 By the words of Morton and Sonar (Acta Numerica, 2007): "The jury was out there for a long time in the judgment between cell-center and vertex-center methods" CMWR 2010, Barcelona 1

3 Overview The 2D nonlinear shallow water model CMWR 2010, Barcelona 2

4 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work CMWR 2010, Barcelona 2

5 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work Finite volumes on triangles: The node-center (NCFV) and cell-center (CCFV) approach, in a unified framework CMWR 2010, Barcelona 2

6 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work Finite volumes on triangles: The node-center (NCFV) and cell-center (CCFV) approach, in a unified framework Considerations and improved linear reconstruction for the CCFV approach CMWR 2010, Barcelona 2

7 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work Finite volumes on triangles: The node-center (NCFV) and cell-center (CCFV) approach, in a unified framework Considerations and improved linear reconstruction for the CCFV approach Topography discretisation and wet/dry front treatment. CMWR 2010, Barcelona 2

8 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work Finite volumes on triangles: The node-center (NCFV) and cell-center (CCFV) approach, in a unified framework Considerations and improved linear reconstruction for the CCFV approach Topography discretisation and wet/dry front treatment. Convergence tests, on different grids, for problems with known analytical solutions CMWR 2010, Barcelona 2

9 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work Finite volumes on triangles: The node-center (NCFV) and cell-center (CCFV) approach, in a unified framework Considerations and improved linear reconstruction for the CCFV approach Topography discretisation and wet/dry front treatment. Convergence tests, on different grids, for problems with known analytical solutions The Malpasset dam case CMWR 2010, Barcelona 2

10 Overview The 2D nonlinear shallow water model Different grids and grid terminology used in this work Finite volumes on triangles: The node-center (NCFV) and cell-center (CCFV) approach, in a unified framework Considerations and improved linear reconstruction for the CCFV approach Topography discretisation and wet/dry front treatment. Convergence tests, on different grids, for problems with known analytical solutions The Malpasset dam case Conclusions CMWR 2010, Barcelona 2

11 MATHEMATICAL MODEL: 2D Non-linear Shallow Water Equations U = U t + F(U) = L(U, x,y) on Ω [0, t] R2 R +, h hu hv, F(U) = [F G] = hu hu gh2 huv hv huv hv gh2, R 1 = [ 0 gh B(x,y) x 0 L(U) = [R 1 + R 2 + S] ] T [ and R 2 = 0 0 gh B(x,y) y ] T and S = [ 0 ghs x f ghs y f ] T, Sf x = n 2 u u 2 + v 2 m, S y h 4/3 f = v u 2 + v 2 n2 m h 4/3 CMWR 2010, Barcelona 3

12 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). CMWR 2010, Barcelona 4

13 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. CMWR 2010, Barcelona 4

14 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. 3. Converge to the correct physical solutions (entropy solutions). CMWR 2010, Barcelona 4

15 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. 3. Converge to the correct physical solutions (entropy solutions). 4. At least second-order accurate on smooth regions of the flow. CMWR 2010, Barcelona 4

16 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. 3. Converge to the correct physical solutions (entropy solutions). 4. At least second-order accurate on smooth regions of the flow. 5. Can handle complex topography and accurate predict wet/dry fronts. CMWR 2010, Barcelona 4

17 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. 3. Converge to the correct physical solutions (entropy solutions). 4. At least second-order accurate on smooth regions of the flow. 5. Can handle complex topography and accurate predict wet/dry fronts. 6. Well-balanced between (numerical) fluxes and sources. CMWR 2010, Barcelona 4

18 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. 3. Converge to the correct physical solutions (entropy solutions). 4. At least second-order accurate on smooth regions of the flow. 5. Can handle complex topography and accurate predict wet/dry fronts. 6. Well-balanced between (numerical) fluxes and sources. 7. Enable various practical inflow/outflow boundary conditions CMWR 2010, Barcelona 4

19 Constructing Numerical Methods: Objectives We need numerical schemes that are: 1. Conservative (conservation of physical quantities). 2. Shock-Capturing with sharp non-oscillatory high-resolution of discontinuities. 3. Converge to the correct physical solutions (entropy solutions). 4. At least second-order accurate on smooth regions of the flow. 5. Can handle complex topography and accurate predict wet/dry fronts. 6. Well-balanced between (numerical) fluxes and sources. 7. Enable various practical inflow/outflow boundary conditions 8. Preserve the positivity of the water depth. CMWR 2010, Barcelona 4

20 Grid Terminilogy (a) Equilateral (Type-I) (b) Orthogonal (Type-II) (c) Orthogonal (Type-III) (d) Distorted (Type-IV) CMWR 2010, Barcelona 5

21 Grid Terminilogy (a) Equilateral (Type-I) (b) Orthogonal (Type-II) (c) Orthogonal (Type-III) (d) Distorted (Type-IV) Major requirement: to enable meaningful asymptotic order of convergence use consistently refined grids, i.e. for N = degrees of freedom, the characteristic length h N = (L x L y )/N, CMWR 2010, Barcelona 5

22 Grid Terminilogy (a) Equilateral (Type-I) (b) Orthogonal (Type-II) (c) Orthogonal (Type-III) (d) Distorted (Type-IV) Major requirement: to enable meaningful asymptotic order of convergence use consistently refined grids, i.e. for N = degrees of freedom, the characteristic length h N = (L x L y )/N, For fair comparison between NCFV and CCFV need to derive equivalent meshes, based on the degrees of freedom N, CMWR 2010, Barcelona 5

23 Grid Terminilogy (a) Equilateral (Type-I) (b) Orthogonal (Type-II) (c) Orthogonal (Type-III) (d) Distorted (Type-IV) Major requirement: to enable meaningful asymptotic order of convergence use consistently refined grids, i.e. for N = degrees of freedom, the characteristic length h N = (L x L y )/N, For fair comparison between NCFV and CCFV need to derive equivalent meshes, based on the degrees of freedom N, Measure the grid irregularity, e.g. use proper grid metrics. CMWR 2010, Barcelona 5

24 FV discretization schemes on triangles: NCFV approach C P U t dxdy + C P (Fñ x + Gñ y ) dl = C P Ldxdy CMWR 2010, Barcelona 6

25 FV discretization schemes on triangles: NCFV approach C P U t dxdy + U P t C P + Φ PQ = Z ( U L PQ,n PQ ) + J PQ C P (Fñ x + Gñ y ) dl = C P Ldxdy { } Φ PQ + Φ P,out = Ldxdy where Q K P Q K T P PQ. ( ) U R PQ U L PQ, with J ( P Λ P 1 ) PQ =, PQ Φ PQ is the Roe numerical flux, evaluated at U L PQ and UR PQ reconstructed values. CMWR 2010, Barcelona 6

26 For MUSCL ( W) P has to be computed in each dual cell P Green-Gauss (G-G) linear reconstruction ( w i ) P = 1 C P 1 ) (w i,p + w i,q n PQ. 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 ), CMWR 2010, Barcelona 7

27 For MUSCL ( W) P has to be computed in each dual cell P Green-Gauss (G-G) linear reconstruction ( w i ) P = 1 C P 1 ) (w i,p + w i,q n PQ. 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 ) upw P = 2( w i ) P ( w i ) cent CMWR 2010, Barcelona 7

28 For MUSCL ( W) P has to be computed in each dual cell P Green-Gauss (G-G) linear reconstruction ( w i ) P = 1 C P 1 ) (w i,p + w i,q n PQ. 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 ) upw P = 2( w i ) P ( w i ) cent Monotonicity in the reconstruction is enforced by using van Albada-van Leer edge-based slope limiter. CMWR 2010, Barcelona 7

29 For MUSCL ( W) P has to be computed in each dual cell P Green-Gauss (G-G) linear reconstruction ( w i ) P = 1 C P 1 ) (w i,p + w i,q n PQ. 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 ) upw P = 2( w i ) P ( w i ) cent Monotonicity in the reconstruction is enforced by using van Albada-van Leer edge-based slope limiter. The same reconstruction is used to compute the gradient for B(x, y). CMWR 2010, Barcelona 7

30 FV discretization schemes on triangles: CCFV approach U p t T p = q K(p) Φ q + T p LdΩ. With the usual one point quadrature at M, Roe s solver is again utilized Φ q = Z ( U L ) ( ) p,n q + J LR U R q U L p. CMWR 2010, Barcelona 8

31 FV discretization schemes on triangles: CCFV approach U p t T p = q K(p) Φ q + T p LdΩ. With the usual one point quadrature at M, Roe s solver is again utilized Φ q = Z ( U L ) ( ) p,n q + J LR U R q U L p. Linear reconstruction for the CCFV scheme 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 ) CMWR 2010, Barcelona 8

32 FV discretization schemes on triangles: CCFV approach U p t T p = q K(p) Φ q + T p LdΩ. With the usual one point quadrature at M, Roe s solver is again utilized Φ q = Z ( U L ) ( ) p,n q + J LR U R q U L p. Linear reconstruction for the CCFV scheme 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) 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. CMWR 2010, Barcelona 8

33 CCFV approach: Gradient operators Three element (compact stencil) gradient w i,p = 1 1 ) (w Cp c i,q + w i,r n qr. 2 q,r K(p) r q CMWR 2010, Barcelona 9

34 CCFV approach: Gradient operators Three element (compact stencil) gradient w i,p = 1 1 ) (w Cp c i,q + w i,r n qr. 2 q,r K(p) r q CMWR 2010, Barcelona 9

35 CCFV approach: Gradient operators Three element (compact stencil) gradient w i,p = 1 1 ) (w Cp c i,q + w i,r n qr. 2 q,r K(p) r q Extended element (wide stencil) gradient w i,p = 1 C w p l,r K (p) r l 1 ) (w i,l + w i,r n lr 2 CMWR 2010, Barcelona 9

36 Typical behavior of the CCFV scheme at internal and boundary faces Keeping in mind that, we want to apply the same edge-based limiter as for the NCFV scheme CMWR 2010, Barcelona 10

37 Typical behavior of the CCFV scheme at internal and boundary faces Keeping in mind that, we want to apply the same edge-based limiter as for the NCFV scheme In an ideal unstructured grid, variables are extrapolated at M which will coincide with D (intersection point of face T q T p and pq). CMWR 2010, Barcelona 10

38 Typical behavior of the CCFV scheme at internal and boundary faces Keeping in mind that, we want to apply the same edge-based limiter as for the NCFV scheme In an ideal unstructured grid, variables are extrapolated at M which will coincide with D (intersection point of face T q T p and pq). There can be a "large" distance between M and D (also on boundary faces, where ghost cells are used). CMWR 2010, Barcelona 10

39 Topography source discretization (wet/wet case) C-property: (Bermudez and Vazquez, 1994) A numerical should resolve exactly steady-state cases where: u = v 0, B R B L = (h R h L ). CMWR 2010, Barcelona 11

40 Topography source discretization (wet/wet case) C-property: (Bermudez and Vazquez, 1994) A numerical should resolve exactly steady-state cases where: u = v 0, B R B L = (h R h L ). Upwind of the numerical source contribution to cell-i as Ψ = 1 2 ( P ( I Λ Λ 1) ) P 1 R CMWR 2010, Barcelona 11

41 Topography source discretization (wet/wet case) C-property: (Bermudez and Vazquez, 1994) A numerical should resolve exactly steady-state cases where: u = v 0, B R B L = (h R h L ). Upwind of the numerical source contribution to cell-i as Ψ = 1 2 ( P ( I Λ Λ 1) ) P 1 R Satisfies the discrete steady state if B R = B j and B L = B i (1st order scheme): Z ( U L,n ) + J ( U R U L) = ( 1 2 P ( I Λ Λ 1) ) P 1 R CMWR 2010, Barcelona 11

42 Topography source discretization (wet/wet case) C-property: (Bermudez and Vazquez, 1994) A numerical should resolve exactly steady-state cases where: u = v 0, B R B L = (h R h L ). Upwind of the numerical source contribution to cell-i as Ψ = 1 2 ( P ( I Λ Λ 1) ) P 1 R Satisfies the discrete steady state if B R = B j and B L = B i (1st order scheme): when Z ( U L,n ) + J ( U R U L) ( 1 = 2 P ( I Λ Λ 1) ) P 1 R 0 R = g hl + h R ( B R B L) n x 2 g hl + h R ( B R B L) n y 2 CMWR 2010, Barcelona 11

43 but in the 2nd order MUSCL discretization a correction term should be added to the source term Ψ (Hubbard and Garcia-Navarro, 2000), i.e. g hl + h i 2 g hl + h i 2 0 ( ) B L B i nx ( B L B i ) ny. CMWR 2010, Barcelona 12

44 but in the 2nd order MUSCL discretization a correction term should be added to the source term Ψ (Hubbard and Garcia-Navarro, 2000), i.e. g hl + h i 2 g hl + h i 2 0 ( ) B L B i nx ( B L B i ) ny. the above term vanishes if B L = B i, i.e. 1st order schemes CMWR 2010, Barcelona 12

45 but in the 2nd order MUSCL discretization a correction term should be added to the source term Ψ (Hubbard and Garcia-Navarro, 2000), i.e. g hl + h i 2 g hl + h i 2 0 ( ) B L B i nx ( B L B i ) ny. the above term vanishes if B L = B i, i.e. 1st order schemes for hydrostatic (steady) conditions inside each cell-i B L B i = ( h L h i ) CMWR 2010, Barcelona 12

46 but in the 2nd order MUSCL discretization a correction term should be added to the source term Ψ (Hubbard and Garcia-Navarro, 2000), i.e. g hl + h i 2 g hl + h i 2 0 ( ) B L B i nx ( B L B i ) ny. the above term vanishes if B L = B i, i.e. 1st order schemes for hydrostatic (steady) conditions inside each cell-i B L B i = ( h L h i ) But what about wet/dry fronts and flow over adverse slopes? CMWR 2010, Barcelona 12

47 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. In the MUSCL scheme for hydrostatic conditions we must have, at i cell CMWR 2010, Barcelona 13

48 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. In the MUSCL scheme for hydrostatic conditions we must have, at i cell B L B i = ( h L h i ) CMWR 2010, Barcelona 13

49 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. 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 CMWR 2010, Barcelona 13

50 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. 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 and similar for the j cell = ( B) j = ( h) j CMWR 2010, Barcelona 13

51 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. 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 and similar for the j cell = ( B) j = ( h) j 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 CMWR 2010, Barcelona 13

52 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. 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 and similar for the j cell = ( B) j = ( h) j 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 ) CMWR 2010, Barcelona 13

53 Topography source discretization (wet/dry case) Extended C-property, (Castro et al., 2005) A scheme is considered to be well-balanced if it can solve exactly steady-state solutions corresponding to flow at rest, regardless of including wet/dry transitions or not. 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 and similar for the j cell = ( B) j = ( h) j 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 ) For emerging bed situations, in the dry i node (cell), the bed value has to be redefined (Brufau et al., 2002) in R and in order to maintain hydrostatic conditions i.e. B = { (h R h L ), if h L > ε wd, h R ε wd and h R < (B R B L ), (B L B R ), otherwise. CMWR 2010, Barcelona 13

54 Solving at wet/dry front interface (continued) For water in motion over emerging slopes, (Castro et al., 2005) A further assumption is made for a flow in motion over a slope CMWR 2010, Barcelona 14

55 Solving at wet/dry front interface (continued) For water in motion over emerging slopes, (Castro et al., 2005) A further assumption is made for a flow in motion over a slope If h L > ε wd and h R ε wd and h L < (B R B L ), then set temporarily for the wet cell-i u L = v L = 0 Then calculate the well-balanced MUSCL numerical fluxes and sources. CMWR 2010, Barcelona 14

56 Solving at wet/dry front interface (continued) For water in motion over emerging slopes, (Castro et al., 2005) A further assumption is made for a flow in motion over a slope If h L > ε wd and h R ε wd and h L < (B R B L ), then set temporarily for the wet cell-i u L = v L = 0 Then calculate the well-balanced MUSCL numerical fluxes and sources. Finally in case of steep downhill slopes, h in wet cells may become negative. We apply the conservative approach of Brufau et. al. (2004) to control negative depths and conserve mass. CMWR 2010, Barcelona 14

57 Other implementation details: Dry cell identification: use a wet/dry tolerance parameter ε wd depending on grid characteristics (Ricchiuto and Bollerman, 2009). ε wd = ( hn L ref ) 2 CMWR 2010, Barcelona 15

58 Other implementation details: Dry cell identification: use a wet/dry tolerance parameter ε wd depending on grid characteristics (Ricchiuto and Bollerman, 2009). ε wd = ( hn L ref ) 2 Boundary conditions: use the theory of characteristics for: weak formulation for the NVFV scheme and ghost cells for CCFV scheme. CMWR 2010, Barcelona 15

59 Other implementation details: Dry cell identification: use a wet/dry tolerance parameter ε wd depending on grid characteristics (Ricchiuto and Bollerman, 2009). ε wd = ( hn L ref ) 2 Boundary conditions: use the theory of characteristics for: weak formulation for the NVFV scheme and ghost cells for CCFV scheme. Apply a four stage 2nd order Runge-Kutta scheme, due to its enhanced stability region, under the CFL condition t n = CFL min i ( R i ( u2 + v 2 + c ) i ). CMWR 2010, Barcelona 15

60 Other implementation details: Dry cell identification: use a wet/dry tolerance parameter ε wd depending on grid characteristics (Ricchiuto and Bollerman, 2009). ε wd = ( hn L ref ) 2 Boundary conditions: use the theory of characteristics for: weak formulation for the NVFV scheme and ghost cells for CCFV scheme. Apply a four stage 2nd order Runge-Kutta scheme, due to its enhanced stability region, under the CFL condition t n = CFL min i ( R i ( u2 + v 2 + c ) i ). Semi-implicit or implicit treatment for the friction source term within each cell following Brufau et al., 2004 and Serrano-Pacheco et al., CMWR 2010, Barcelona 15

61 NUMERICAL RESULTS and COMPARISONS 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) CMWR 2010, Barcelona 16

62 NUMERICAL RESULTS and COMPARISONS 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) A traveling vortex solution (with periodic boundary conditions) CMWR 2010, Barcelona 16

63 NUMERICAL RESULTS and COMPARISONS 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) A traveling vortex solution (with periodic boundary conditions) CMWR 2010, Barcelona 16

64 NUMERICAL RESULTS and COMPARISONS 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) A traveling vortex solution (with periodic boundary conditions) CMWR 2010, Barcelona 16

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70 CMWR 2010, Barcelona 17

71 A 2D potential (steady) solution with topography CMWR 2010, Barcelona 18

72 A 2D potential (steady) solution with topography CMWR 2010, Barcelona 18

73 A 2D potential (steady) solution with topography CMWR 2010, Barcelona 18

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76 CMWR 2010, Barcelona 19

77 Grid convergence including dry boundaries: Thacker s 2D solutions Known analytical periodic solutions. The motion is oscillatory. Friction is not included. Difficult test (wetting and drying) Used it to check convergence rates. CMWR 2010, Barcelona 20

78 Grid convergence including dry boundaries: Thacker s 2D solutions Known analytical periodic solutions. The motion is oscillatory. Friction is not included. Difficult test (wetting and drying) Used it to check convergence rates. Thacker s axisymmetric solution (wetting and drying phase) CMWR 2010, Barcelona 20

79 Grid convergence including dry boundaries: Thacker s 2D solutions Known analytical periodic solutions. The motion is oscillatory. Friction is not included. Difficult test (wetting and drying) Used it to check convergence rates. Thacker s axisymmetric solution (wetting and drying phase) CMWR 2010, Barcelona 20

80 Grid convergence including dry boundaries: Thacker s 2D solutions Known analytical periodic solutions. The motion is oscillatory. Friction is not included. Difficult test (wetting and drying) Used it to check convergence rates. Thacker s axisymmetric solution (wetting and drying phase) CMWR 2010, Barcelona 20

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86 CMWR 2010, Barcelona 22

87 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) CMWR 2010, Barcelona 23

88 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) CMWR 2010, Barcelona 23

89 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) CMWR 2010, Barcelona 23

90 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) Water depth evolution at t=800s, 1600s, 2400s CMWR 2010, Barcelona 23

91 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) Water depth evolution at t=800s, 1600s, 2400s CMWR 2010, Barcelona 23

92 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) Water depth evolution at t=800s, 1600s, 2400s CMWR 2010, Barcelona 23

93 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) Water depth evolution at t=800s, 1600s, 2400s CMWR 2010, Barcelona 23

94 The Maplasset dam-break test case (EDF grid, CFL=0.9, n m = 0.033) Water depth evolution at t=800s, 1600s, 2400s CMWR 2010, Barcelona 23

95 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. CMWR 2010, Barcelona 24

96 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. For the CCFV approach different convergence behavior is exhibited (with an order reduction) for grids where the center of the face does not coincide with the reconstruction location. CMWR 2010, Barcelona 24

97 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. For the CCFV approach different convergence behavior is exhibited (with an order reduction) for grids where the center of the face does not coincide with the reconstruction location. The proposed correction for the reconstruction values remedies the problem (for both the compact and wide stencil G-G gradient computations) CMWR 2010, Barcelona 24

98 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. For the CCFV approach different convergence behavior is exhibited (with an order reduction) for grids where the center of the face does not coincide with the reconstruction location. The proposed correction for the reconstruction values remedies the problem (for both the compact and wide stencil G-G gradient computations) For the wide stencil similar consistent convergence behavior to the NCFV is exhibited CMWR 2010, Barcelona 24

99 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. For the CCFV approach different convergence behavior is exhibited (with an order reduction) for grids where the center of the face does not coincide with the reconstruction location. The proposed correction for the reconstruction values remedies the problem (for both the compact and wide stencil G-G gradient computations) For the wide stencil similar consistent convergence behavior to the NCFV is exhibited Convergence to steady-state solutions is greatly improved. CMWR 2010, Barcelona 24

100 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. For the CCFV approach different convergence behavior is exhibited (with an order reduction) for grids where the center of the face does not coincide with the reconstruction location. The proposed correction for the reconstruction values remedies the problem (for both the compact and wide stencil G-G gradient computations) For the wide stencil similar consistent convergence behavior to the NCFV is exhibited Convergence to steady-state solutions is greatly improved. The effect of the grid s geometry at the boundary can lead to order reduction for the CCFV schemes (if the the center of the face does not coincide with the reconstruction location), even for good quality grids Results with the wide stencil CCFV scheme exhibited, in general, better accuracy. CMWR 2010, Barcelona 24

101 CONCLUSIONS The NCFV scheme exhibited identical convergence behavior on all grids. For the CCFV approach different convergence behavior is exhibited (with an order reduction) for grids where the center of the face does not coincide with the reconstruction location. The proposed correction for the reconstruction values remedies the problem (for both the compact and wide stencil G-G gradient computations) For the wide stencil similar consistent convergence behavior to the NCFV is exhibited Convergence to steady-state solutions is greatly improved. The effect of the grid s geometry at the boundary can lead to order reduction for the CCFV schemes (if the the center of the face does not coincide with the reconstruction location), even for good quality grids Results with the wide stencil CCFV scheme exhibited, in general, better accuracy. Both FV approaches to wetting and drying situations lead to order reduction but this drop is moderate and within acceptable limits Important differences observed between the wetting and drying phase, with the results in CMWR 2010, Barcelona 24

102 the drying phase being of reduced accuracy for the velocity computations CMWR 2010, Barcelona 25

103 the drying phase being of reduced accuracy for the velocity computations For the friction treatment an order reduction for the velocity components was observed proportional to the implicitness introduced to the numerical treatment CMWR 2010, Barcelona 25

104 the drying phase being of reduced accuracy for the velocity computations For the friction treatment an order reduction for the velocity components was observed proportional to the implicitness introduced to the numerical treatment For the Malpasset case, all formulations produced results close to field measurements, but with the NCFV scheme using half the degrees of freedom of the CCFV ones. CMWR 2010, Barcelona 25

105 the drying phase being of reduced accuracy for the velocity computations For the friction treatment an order reduction for the velocity components was observed proportional to the implicitness introduced to the numerical treatment For the Malpasset case, all formulations produced results close to field measurements, but with the NCFV scheme using half the degrees of freedom of the CCFV ones. References A.I. Delis, I.K. Nikolos and M.Kazolea, "Performance and comparison of cell-centered and node-centered unstructured finite volume discretizations for shallow water free surface flows", Archives of Computational Methods in Engineering, (accepted). I.K. Nikolos and A.I. Delis, "An unstructured nod-centered finite volume scheme for shallow water flows with wet/dry front over complex topography", Comput. Methods Appl. Mech. Engrg., 198:3723, THANK YOU FOR YOUR ATTENTION! CMWR 2010, Barcelona 25

106 2D steady-state with topography and friction Known analytical solution, Murillo et al., Start from water at rest, n m = 0.3 and impose sub and super-critical boundary conditions Compare explicit, semi-implicit and implicit friction term treatment (in terms of convergence rates) CMWR 2010, Barcelona 26

107 CMWR 2010, Barcelona 27

108 CMWR 2010, Barcelona 28