Central-Upwind Schemes for Shallow Water Models. Alexander Kurganov. Tulane University Mathematics Department

Transcription

1 Central-Upwind Schemes for Shallow Water Models Alexander Kurganov Tulane University Mathematics Department kurganov Supported by NSF and ONR

2 Shallow Water Equations Shallow water equations are derived by depth-averaging the Navier- Stokes equations in the case when the horizontal length scale is much greater than the vertical length scale, e.g. [Gerbeau, Perthame; ]. Shallow water equations are widely used to model water flows in Rivers Channels Lakes Coastal areas Shallow water equations are also used in oceanography and atmospheric sciences to model a variety of waves and phenomena including such natural disasters as tropical cyclones, tsunamis and floods caused by rainfalls, storm surges, dam breaks, etc. Shallow water models are extremely important in hydraulic engineering and coastal management.

3 z w=b+h h(x,t) B(x) 3

4 -D Saint-Venant System h t + q x = q t + ( hu + g h) x = ghb x This is a system of hyperbolic balance laws U t + F (U, B) x = S(U, B), U := (h, q) h: depth u: velocity q := hu: discharge B: bottom topography g: gravitational constant 4

5 Finite-Volume Methods -D System: U t + F (U) x = U j (t) x C j U(x, t) dx : cell averages over C j := (x j, x j+ ) This solution is approximated by a piecewise polynomial (conservative, high-order accurate, non-oscillatory) reconstruction: Ũ(x) = P j (x) for x C j Second-order schemes employ piecewise linear reconstructions: Ũ(x) = U j + (U x ) j (x x j ) for x C j 5

6 For example, (U x ) j = minmod θ U j U j x where the minmod function is defined as: minmod(z, z,...) :=, U j+ U j, θ U j+ U j x x min j {z j }, if z j > j, max j {z j }, if z j < j,, otherwise. θ [, ] The reconstructed point values at cell interfaces are: U j+ U + j+ := P j (x j+ ) = U j + x (U x) j := P j+ (x j+ ) = U j+ x (U x) j+ 6

7 x j x j x j+ x j+ The discontinuities appearing at the reconstruction step at the interface points {x j+ } propagate at finite speeds estimated by: a + j+ a j+ := max := min ( ( {λ N A(U ), j+ ) λ N A(U + ) } j+, ) ( ( {λ A(U ), j+ λ A(U ) + ) } j+ ), λ < λ <... < λ N : N eigenvalues of the Jacobian A(U) := F U

8 Central-Upwind Schemes Godunov-type central schemes with a built-in upwind nature [Kurganov, Tadmor; ] [Kurganov, Petrova;, ] [Kurganov, Noelle, Petrova; ] [Kurganov, Lin; 7] 8

9 -D Semi-Discrete Central-Upwind Scheme d H dt U j+ (t) H j(t) = j (t) x The central-upwind numerical flux is: H j+ = a + j+ F (U j+ ) a j+ F (U + j+ ) + a + j+ a j+ U + j+ U j+ d j+ a + j+ a j+ a + j+ a j+ The built-in anti-diffusion term is: d j+ = minmod U + j+ a + j+ U j+ a j+ U j+ U j+, a + j+ a j+ The intermediate values U j+ are: U j+ = a + j+ U + j+ a j+ U j+ a + j+ { F (U + } j+ ) F (U j+ ) a j+ 9

10 Remarks. d j+ corresponds to the central-upwind scheme from [Kurganov, Noelle, Petrova; ]. For the system of balance laws the central-upwind scheme is: where U t + F (U) x = S d H j+ (t) H dtūj(t) = j (t) x + S j (t) S j (t) x j+ S(x, t) dx x x j

11 -D Semi-Discrete Central-Upwind Scheme Rectangular Grid [Kurganov, Petrova; ] [Kurganov, Noelle, Petrova; ] [Kurganov, Tadmor; ] [Kurganov, Lin; 7] Triangular Grid [Kurganov, Petrova; 5]

12 Saint-Venant System Numerical Challenges h t + q x = q t + ( hu + g h) x = ghb x Steady-state solutions: q = Const, u + g(h + B) = Const Lake at rest steady-state solutions: u =, h + B = Const Dry (h = ) or near dry (h ) states

13 Well-Balanced Schemes (preserving lake at rest steady states) [LeVeque; 998]: incorporating the source term into the Riemann solver [Shi Jin; ]: well-balanced source term averaging [Perthame, Simeoni; ]: kinetic scheme [Kurganov, Levy; ]: central-upwind scheme [Gallouët, Hérard, Seguin; 3]: Roe-type scheme [Audusse, Bouchut, Bristeau, Klein, Perthame; 4]: hydrostatic reconstruction [Russo; 5]: staggered central scheme [Xing, Shu; 5, 6]: WENO schemes [Noelle, Pankratz, Puppo, Natvig; 6]: high-order schemes [Lukácová-Medvidová, Noelle, Kraft; 7]: FVEG scheme [Berthon, Marche; 8]: relaxation schemes [Fjordholm, Mishra, Tadmor; 8, ]: energy stable schemes Abgrall, Audusse, Bristeau, Castro, Chertock, Dawson, Donat, Epshteyn, George, Karni, Klingenberg, Mohammadian, Parés, Ricchiuto,... 3

14 Well-Balanced Schemes (preserving moving steady states) [Noelle, Shu, Xing; 7, 9, ]: WENO schemes [Russo, Khe; 9, ]: staggered central schemes [Xing; 4]: discontinuous Galerkin method Positivity Preserving Schemes [Perthame, Simeoni; ]: kinetic scheme [Audusse, Bouchut, Bristeau, Klein, Perthame; 4]: hydrostatic reconstruction [Kurgnov, Petrova; 7]: central-upwind scheme with continuous piecewise linear bottom reconstruction [Berthon, Marche; 8]: relaxation schemes [Bollermann, Noelle, Lukáčová-Medvid ová; ]: special timequadrature for the fluxes [Bollermann, Chen, Kurganov, Noelle; 3]: well-balanced reconstruction of wet/dry fronts 4

15 Central-upwind schemes is simple, highly accurate and efficient tool that allows one to quite easily incorporate techniques required to numerically solve more realistic (and more complicated) Saint-Venant based shallow water models. Additional terms: friction, viscosity, Coriolis forces, temperature variations, third-order dispersive terms Robustness of central-upwind schemes made them attractive to practitioners. For example: Central-upwind schemes have been recently implemented at the National Center of Computational Hydroscience and Engineering at the University of Mississippi, where they became the core part of the GIS-based decision support systems for flood mitigation and management. These systems are currently being used by several US federal agencies. 5

16 Well-Balanced Central-Upwind Scheme [Kurganov, Levy; ] w = h + B: water surface h t + q x = q t + ( hu + g h) x = ghb x (h, q) (w, q) w t + q x = ( q t + q w B + g ) (w B) x = g(w B)B x Lake at rest steady states: u, w Const 6

17 At the lake at rest steady state: q =, w = Const = the flux is F = (q, q w B + g (w B) ) T = (, g (w B) ) T = the second component of the numerical flux is H () j+ = g ( ) w B(x j+ ), H () j = g = d H () dt q j+ (t) H () j (t) j(t) = x +S () j (t) ( ) w B(x j ) = g B(x j+ ) B(x j ) x (w B(x j+ )) + (w B(x j )) +S () j (t) = The well-balanced quadrature is S () j (t) = g B(x j+ ) B(x j ) x w j B(x j+ ) + B(x j ) 7

18 Well-Balanced Positivity Preserving Central-Upwind Scheme [Kurganov, Petrova; 7] Step : Piecewise linear reconstruction of the bottom B(x) B j+/ Bj B j+ x j / x j+/ x j+3/ 8

19 Step : Positivity preserving reconstruction of w w + j+/ w j+ w j w j+/ B j+ B j x j / x j+/ x j+3/ 9

20 h ± j+ = w ± j+ B j+ w B j / + j / w j w j+/ x j / B j x j B j+/ x j+/

21 B j / B j / = w j / + w + j / w j B j w j+/ w j B j w j+/ x j / x j B j+/ x j+/ x j / x j B j+/ x j+/

22 Step 3: Desingularization (u q h for small h) Simplest: u = q h, if h ε, if h < ε More sophisticated (smoother transition for small h): u = h q h + max(h, ε) or u = h q h 4 + max(h 4, ε) Remark: For consistency, one has to recompute the discharge: q = h u

23 Positivity Preserving Property If an SSP ODE solver is used, then h n+ j = α j h j + α + j h + j + α j+ h j+ + α + j+ h + j+ where the coefficients α ± j± > provided an appropriate CFL condition is satisfied: -D: CFL number is / -D Cartesian mesh: CFL number is /4 -D triangular mesh: CFL number is /3 Remark: For high-order SSP methods, adaptive timestep control has to be implemented. 3

24 Example Small Perturbation of a Steady State The bottom topography is: B(x) = (x.3),.3 x.4.5 sin (5π(x.4)),.4 x.6 (x.7),.6 x.7, otherwise The initial data are: (w(x, ), u(x, )) = { ( + ε, ),. < x <. (, ), otherwise ε = 3 4

25 INITIAL DATA.5 INITIAL DATA ZOOM BOTTOM WATER SURFACE BOTTOM WATER SURFACE 5

26 x=/ NEW SCHEME OLD SCHEME. x=/

27 x=/8 BOTTOM NEW SCHEME OLD SCHEME x=/

28 Example Dry State and Discontinuous Bottom B(x) = {, x.5., x >.5 (w(x, ), u(x, )) = { (., ), x.5 (.846,.6359), x >.5 8

29 BOTTOM x=/ x=/64 BOTTOM x=/4 x=/

30 Example ShW with Friction and Discontinuous Bottom h t + q x = q t + (hu + g ) h x = ghb x κ(h)u, κ(h) =. + h B(x) =, x < cos (πx), x.4 cos (πx) +.5(cos(π(x.5)) + ),.4 x.5.5 cos 4 (πx) +.5(cos(π(x.5)) + ),.5 x.6.5 cos 4 (πx),.5 x <.5 sin(π(x )), < x.5, x >.5. (w(x, ), u(x, )) = { (.4, ), x < (B(x), ), x > (Dam break) 3

31 .5 t= t= t= t=.5.5 t=

32 .5 t= t= t= t=.5.5 t=

33 Central-Upwind Schemes for the -D Saint-Venant System Cartesian Grid: [Kurganov, Levy, ], [Kurganov, Petrova; 7] Triangular Grid: [Bryson, Epshteyn, Kurganov, Petrova; ] Polygonal Cell-Vertex Mesh: [Beljadid, Mohammadian, Kurganov; preprint] 33

34 Example Small Perturbation over an Exponential Hump The bottom consists of an elliptical shaped hump: B(x, y) =.8 exp( 5(x.9) 5(y.5) ) Initially, the water is at rest and its surface is flat everywhere except for.5 < x <.5: w(x, y, ) = { + ε,.5 < x <.5,, otherwise, u(x, y, ) v(x, y, ) We take ε = 4 34

35 t=.6 t= t=.9 t= t=. t=

36 t=.5 t= t=.8 t=

37 Example Small Perturbation over Submerged Flat Plateau ε ε ε B(x, y) = ε, r., ( ε)(. r),. r.,, otherwise, r := (x.5) + y w(x, y, ) = We take ε =. { + ε,. < x <.,, otherwise, u(x, y, ) v(x, y, ) 37

38

39 Example Small Perturbation Bending around an Island ε ε ε w(x, y, ) = B(x, y) =., r., (. r),. < r <.,, otherwise, + ε,.4 < x <.3, max (, B(x, y) ), otherwise, r := x + y u(x, y, ) v(x, y, ) We take ε =. 39

40

41 Shallow Water System with Friction Terms [Chertock, Cui, Kurganov, Wu; preprint] Friction from [Gerbeau, Perthame; ] h t + q x = q t + (hu + g ) h x = ghb x. + h u, Classical Manning friction [Manning; XIX century] n: Manning coefficient h t + q x = R q t + (hu + g ) h x = ghb x g n h /3 u u R = R(x, t): rain source 4

42 Steady-State Solutions Nontrivial steady states (for R ) q Const, h Const, B x Const correspond to the situation when the water flows over a slanted infinitely long surface with the constant slope. Assume that B x C, where C > and denote q q, then ( n ) q 3/ h h n =, q q, B x C, C where h n is the so-called normal depth. This steady state is stable in the super-critical case when h n < h c := ( ) /3 q g 4

43 The bottom setting in numerical examples: Left: -D steady state Right: a case of urban draining with obstacles like houses h(x) B(x) x 43

44 As before, we: replace the bottom with its piecewise linear approximation correct reconstruction of w = h + B to ensure that w > B regularize computed velocities to avoid division by h use well-balanced quadrature for the geometric source term A straightforward midpoint discretization of the friction term leads to the well-balanced positivity preserving central-upwind scheme 44

45 Modified Positivity Correction Instead of B j / B j / = w j / + w + j / w j B j w j+/ w j B j w j+/ x j / x j B j+/ x j+/ x j / x j B j+/ x j+/ 45

46 we do B j / w j / + w + j / w j B j w j+/ w j B j w j+/ x j / x j B j+/ x j+/ x j / x j B j+/ x j+/ 46

47 Example Small Perturbation of a Steady Flow Over a Slanted Surface (R ) {.hn, x.5, h(x, ) = h n + q(x, ) q, otherwise, t= t= t=.5 w B.5 w B.5 w B x x Supercritical case x.6.5 t= w B.6.5 t=.5 w B.6.5 t= w B x x Subcritical case x 47

48 Two-Dimensional Examples Example Rainfall-Runoff Over An Urban Area We consider a rainfall-runoff situation, which occurs over a -D surface containing houses as outlined in 48

49 The setting corresponds to the laboratory experiment reported in [Cea, Garrido, Puertas; ]. The surface structure is shown in The precise data was provided by Dr. Luis Cea 49

50 The experiment was built to mimic an urban area within the laboratory simulator of size m.5m. To model urban buildings, several blocks were placed onto the surface according to three different geometries:.5 X.5.5 5

51 .5 Y.5.5 5

52 .5 S Notice that across the walls of the houses the bottom topography is discontinuous and thus the bilinear interpolant B has very sharp gradients there 5

53 We set the almost dry initial conditions: h(x, y, ) 8, u(x, y, ) v(x, y, ) The rain of a constant intensity starts falling at time t = and stops at t = T s : R(x, y, t) =, t T s, otherwise We take T s =, 4 or 6 At the lower part of boundary, the total outlet discharge is recorded in the laboratory experiments at different time moments and then it is compared with the computed values of N x j= ( H y j,/ ) () 53

54 5 x 4 X, T s = x 4 Y, T s = x 4 S, T s = x 4 X, T s = x 4 Y, T s = x 4 S, T s = x 4 X, T s = x 4 Y, T s = x 4 S, T s =

55 Example Rainfall Runoff Over An Urban Area (REVISED) First, we remove the houses from the computational domain which becomes a punctured rectangle. Each of the holes is depicted in Ghost Cells Houses Edge Ghost Cells Houses Edge The house walls become the internal boundary, which is numerically treated using a solid wall ghost cell technique. 55

56 Second, we need to redistribute the rain water falling onto the roof so that it is placed inside the modified computational domain. In the laboratory experiment, the water falling on the house blocks streams down from the long (lower) edges and finally joins the surface water flow: Ghost Cells Houses Edge 56

57 In reality, the gutter system is commonly used and the rain water streams down from the rain pipes typically located at the house corners: Ghost Cells Houses Edge In both cases, the building-roof rainfall is uniformly distributed on the shaded cells near and outside the building edges. 57

58 The modified rain source can then be written as follows: + A h, in the shaded cells R(x, y, t) = A s, otherwise which is as before switched on only for t [, T s ] ( ) A h : the area of the house A s : the area of the shaded region near that house 58

59 5 x 4 X, T s = x 4 Y, T s = x 4 S, T s = x 4 X, T s = x 4 Y, T s = x 4 S, T s = x 4 X, T s = x 4 Y, T s = x 4 S, T s =

60 Water height snapshots for T s = 4 6

61 6

62 6

63 Shallow Water System with Coriolis Forces [Chertock, Dudzinski, Kurganov, Lukáčová-Medviďová; preprint] h t + (hu) x + (hv) y = (hu) t + (hu + g ) h + (huv) y = ghb x + fhv x (hv) t + (huv) x + (hv + g ) h = ghb y fhu y f: Coriolis parameter 63

64 Steady-State Solutions h t + (hu) x + (hv) y = (hu) t + (hu + g ) h + (huv) y = ghb x + fhv x (hv) t + (huv) x + (hv + g ) h = ghb y fhu y Lake at rest : u, v, h + B Const Geostrophic equlibria ( jets in the rotational frame ): Here, u, v y, h y, B y, K Const v, u x, h x, B x, L Const K := g(h + B V ) and L := g(h + B + U) are the potential energies defined through the primitives of the Coriolis force (U, V ) T : V x := f g v and U y := f g u 64

65 -D Shallow Water System with Coriolis Forces h t + (hu) x = (hu) t + (hu + g ) h (hv) t + (huv) x = fhu x = ghb x + fhv -D geostrophic equlibria ( jets in the rotational frame ): u, K Const K := g(h + B V ), V x := f g v 65

66 Special Piecewise Linear Reconstruction Way to design a well-balanced scheme (KEY IDEA): Reconstruct equilibrium variables! -D case: u, v, K -D case: u, v, K, L 66

67 -D Reconstruction First, compute the point values of the velocities at x j : u j = (hu) j h j, v j = (hv) j h j The values V (x j+ ) are obtained using the midpoint quadrature: V j+ = f g j m= v m x The corresponding values V (x j ) are V j = ) (V j + V j+ Thus, K(x j ) are computed as K j = g(h j + B j V j ) 67

68 Equipped with {u j }, {v j } and {K j }, we construct piecewise linear approximants ũ, ṽ and K: ũ(x) = u j + (u x ) j (x x j ), ṽ(x) = v j + (v x ) j (x x j ), K(x) = K j + (K x ) j (x x j ), x C j x C j x C j The corresponding reconstruction h then reads: h(x) = K(x) g + x [ (Vj ) B j (xj+ x) + ( ) ] V j+ B j+ (x xj ) (Recall that K := g(h + B V ) h = K g + V B) 68

69 Example -D Geostrophic Adjustment Simulation B, f =, g = h(x, y, ) = + 4 [ tanh ( (.5x +.4y ))] u(x, y, ), v(x, y, ) The initial perturbation generates two circular shock waves propagating outwards with a clockwise rotating elevation staying behind the shocks. As time evolves, the solution converges to a nontrivial geostrophic steady state, which is accurately captured by the well-balanced centralupwind scheme. 69

70 7

71 7

72 7

73 Two-Layer Shallow Water Equations (h ) t + (q ) x = (q ) t + ( h u + g h (h ) t + (q ) x = (q ) t + ( h u + g h ) ) x = gh B x gh (h ) x x = gh B x gh (ĥ) x u, u : velocities q := h u, q := h u : discharges g: gravitational constant r := ρ ρ : constant density ratio ĥ := rh. 73

74 (h ) t + (q ) x = (q ) t + ( h u + g h (h ) t + (q ) x = (q ) t + ( h u + g h ) ) x = gh B x gh (h ) x x = gh B x gh (ĥ) x Difficulties: Nonlinear hyperbolic system = shock waves rarefaction waves contact waves (if B is discontinuous) Geometric source terms (nonflat bottom topography) Nonconservative products (interlayer exchange) 74

75 [Kurganov, Petrova; 9] (h ) t + (q ) x = (q ) t + ( h u + g h (h ) t + (q ) x = (q ) t + ( h u + g h ) ) x = gh B x gh (h ) x x = gh B x gh (ĥ) x (h, q, h, q ) (h, q, w := h + B, q ) (h ) t + (q ) x = ( q (q ) t + + gεh h w t + (q ) x = ) x = gε(h ) x (q ) t + ( q w B + g w g rh gb ε ) x = g εb x gε(ĥ) x ε = h + h + B = h + w, ε := ĥ + w 75

76 The new system is preferable for numerical computations thanks to the following two reasons: (i) the lake at rest steady state is given by U (C,, C, ) T C, C constants with (ii) the coefficients in the nonconservative products gε(h ) x and gε(ĥ) x are proportional to ε, which vanishes at lake at rest steady states, and, what is even more important, in most oceanographic applications remains very small. Note that when r, the variable ε is also expected to be small in typical oceanographic applications 76

77 As in the single-layer case, we: replace the bottom with its piecewise linear approximation correct reconstruction of w to ensure that w > B regularize computed velocities to avoid division by h i use well-balanced quadrature for the geometric source term The e-values are given implicitly by We either: (λ u λ + u gh )(λ u λ + u gh ) = g ĥh use the first-order approximation of the e-values (when r and u u ) use their upper/lower bounds to enforce stability (if the system is still in the hyperbolic regime) 77

78 Example Interface Propagation (h, q, h, q )(x, ) = { (.5,.5,.5,.5), x <.3 (.45,.5,.55,.375), x >.3 B, r =.98 78

79 h h.5.49 x=/ x=/ x=/.5.49 x=/4 x=/8 x=/ x 4 ε x 4 ε x=/ x=/ x=/ x=/4 x=/8 x=/

80 (h, q, h, q )(x, ) = { (.8,,., ), x <, (.,,.8, ), x >, B, r =.98 h x 3 ε.5 x=/5 x=/ x=/5 5 x=/5 x=/ x=/

81 Example Lock Exchange Problem (h, q, h, q )(x, ) = { ( B(x),,, ), x < (,, B(x), ), x > B(x) = e x, r =.98 The computational domain is [ 3, 3] and the boundary conditions are q = q at each end of the interval 8

82 Initial Condition Steady State.5 SURFACE INTERFACE BOTTOM.5 SURFACE INTERFACE BOTTOM

83 Example Internal Dam Break (h, q, h, q )(x, ) = { (.95,,.95 B(x), ), x < (.5,,.5 B(x), ), x > B(x) =.5e x.5, r =

84 Initial Condition Steady State.5 SURFACE INTERFACE BOTTOM.5 SURFACE INTERFACE BOTTOM

85 Example -D Interface Propagation where (h, q, p, h, q, p )(x, y, ) { (.5,.5,.5,.5,.5,.5), if (x, y) Ω = (.45,.5,.5,.55,.375,.375), otherwise Ω = {x <.5, y < } { (x +.5) + (y +.5) <.5 } {x <, y <.5}. The initial location of the interface is shown by the dashed line. B(x, y) 85

86 ε h ε h

87 ε h h along the diagonal y=x x= y=/ x= y=/4 x= y=/

88 Savage-Hutter Type Model of Submarine Landslides and Generated Tsunami Waves [Fernández-Nieto, Bouchut, Bresch, Castro, Mangene; 8] z water surface upper fluid layer density ρ fluid-granular mixture density ρ h cos θ h cos θ θ B x 88

89 (h ) t + (q cos θ) x = (q ) t + ( h u cos θ + g h cos3 θ ) x = gh cos θb x g h cos θ (cos θ) x gh cos θ ( h cos θ ) x (h ) t + (q cos θ) x = ( (q ) t + h u cos θ + g h cos3 θ ) x = gh cos θb x g h cos θ (cos θ) x rgh cos θ ( h cos θ ) x + T cos θ B(x): non-erodible bottom θ(x): the angle of B(x) from horizontal T = T (h, u ; δ ): the Coulomb friction term that describes the friction between the lower layer and the bottom δ : angle of repose (parameter of the fluid-granular mixture) 89

90 [Kurganov, Miller; 4] Lake-at-rest steady-state solution (flat water surface in the absence of a lower-layer): h cos θ + B const, h, q q z water surface = const h cos θ B θ x 9

91 (h, h ) (w i := h i cos θ, ε := w + w + B) ε t + ( (q + q ) cos 3 θ ) x = (q + q ) cos θ (cos θ) x ( q (q ) t + cos 3 θ ε (w + B) + g ε ) ε(w + B) = g cos θ cos θ ε (w + B) x x g ε ε(w + B) cos (cos θ) θ x (w ) t + ( q cos 3 θ ) x = q cos θ (cos θ) x ( q (q ) t + cos 3 θ + g ( r)w + rw ε w cos θ + rg cos θ ε(w ) x g ) x ( r)w + rεw cos θ ( r)g = cos θ w B x (cos θ) x + T cos θ 9

92 (ε, q, w, q ) T : equilibrium variables Lake-at-rest steady-state solution: z ε const, w, q q ε = water surface upper fluid layer density ρ w fluid-granular mixture density ρ w θ B x 9

93 As for the two-layer shallow water equations, we: replace the bottom with its piecewise linear approximation correct reconstruction of w to ensure that w > B regularize computed velocities to avoid division by w i use well-balanced quadrature for the geometric source term use upper/lower bounds of e-values to enforce stability 93

94 Coulomb Friction To include the friction term, we apply an operator splitting We first obtain (q ) j and then update q as follows: with = sgn{(q ) j} where (q ) j (t + t) = σ j + (ŵ ) j+ + (ŵ ) j (q ) j + T j t, if (q ) j > σ j t/ cos θ j,, otherwise, (û ) j sin θ j+ sin θ j x (ŵ ) σj j+ + (ŵ ) := g( r) j (cos θ j ) tan δ (w ) j (q ) j(cos θ j ) (û ) j := ((w ) j) 4 + max ( ((w ) j) 4, β ) tan δ, (ŵ ) j+ := ( (w ) j (cos θ j ) + (w ) ) j+ (cos θ j+ ) 94

95 Remark Desingularization is critical to the success of the friction procedure since the locations where (w ) is small is exactly where one would expect the friction cause the lower layer to stop. Without desingularization, the velocities (û ) at these locations may become very large, and as a result the momentum may never fall below the critical threshold σ t/ cos θ. 95

96 Example Small Perturbation of Steady State (ε, q, w, q ) (x, ) = B(x) = (,, η (x), ), if.9 < x <.95 (η,,, ), if < x <. (,,, ), otherwise.5(cos(5π(x )) + ), if.8 < x <., otherwise r =., g =, δ = 5 96

97 Case I η = 3, η (x) η

98 t=. x Reformulated System, Well Balanced Original System, not Well Balanced t=.6 x Reformulated System, Well Balanced Original System, not Well Balanced

99 t=.8 x Reformulated System, Well Balanced Original System, not Well Balanced t= x Reformulated System, Well Balanced Original System, not Well Balanced

100 t=.4 x Reformulated System, Well Balanced Original System, not Well Balanced t=3 x Reformulated System, Well Balanced Original System, not Well Balanced

101 Case II η =, η (x) =.9 B(x) η η..8.

102 t=. x Reformulated System, Well Balanced Original System, not Well Balanced t=.6 x Reformulated System, Well Balanced Original System, not Well Balanced

103 t= x Reformulated System, Well Balanced Original System, not Well Balanced t=3 x Reformulated System, Well Balanced Original System, not Well Balanced

104 Example Surface Wave Generation (ε, q, w, q ) (x, ) = (,, cos(arctan(.)), ), if 7 < x < 8 (,,, ), otherwise B(x) =.x.7 r =., g = 9.8, δ = 5 4

105 .5 t= t=

106 .5 t= t=

107 .5 t= t=

108 .5.5 x=/6 x=/8 x=/4 x=/ x=/ Bottom t= Zoom of the interface between the upper and lower layers 8

109 Example Large-Scale Wave Generation (ε, q, w, q ) (x, ) = (,, max{ B(x).8 4 (x ), }, ) B(x) = 49 e 6.49 ( x/), x 5 + e.55 (x/ ), x > r =., g = 9.8, δ = 9

110 5 t= x 4 5 t= x 4

111 t= Water Surface (ε) Bottom (B) x 4 t= Water Surface (ε) Bottom (B) x 4

112 t= Water Surface (ε) Bottom (B) x 4 t= Water Surface (ε) Bottom (B) x 4

113 t= Water Surface (ε) Bottom (B) x 4 t= Water Surface (ε) Bottom (B) x 4 3

114 Example Tsunami-Like Wave Generation (ε, q, w, q ) (x, ) = (,, max { B(x), }, ), if.5 < x < 3.5 (,,, ), otherwise B(x) = 4 [ ( x) sgn( x) ( ( x c + ) /c)] r =., g = 9.8, δ = 35 4

115 t=

116 t= x 4 Water Surface (ε) t=5 x 4 Water Surface (ε)

117 t= x 4 Water Surface (ε) t= x 4 Water Surface (ε)