Non-Hydrostatic Pressure Shallow Flows: GPU Implementation Using Finite-Volume and Finite-Difference Scheme.

Transcription

1 arxiv: v1 [math.na] 14 Jun 17 Non-Hydrostatic Pressure Shallow Flows: GPU Implementation Using Finite-Volume and Finite-Difference Scheme. C. Escalante 1 T. Morales de Luna and M.J. Castro 1 1 Departamento de Análisis Matemático Estadística e Investigación Operativa y Matemática Aplicada Universidad de Málaga Spain Departamento de Matemáticas Universidad de Córdoba Spain June Abstract We consider the depth-integrated non-hydrostatic system derived by Yamazaki et al. An efficient formally second-order well-balanced hybrid finite volume/difference numerical scheme is proposed. The scheme consists in a two-step algorithm. First the hyperbolic part of the system is discretized using a PVM path-conservative finitevolume method. Second the dispersive terms are solved by means of compact finite differences. A new methodology is also presented to handle wave breaking over complex bathymetries. This adapts well to GPU-architectures and guidelines about its GPU implementation are introduced. The method has been applied to idealized and challenging experimental test cases which shows the efficiency and accuracy of the method. address: escalante@uma.es; Corresponding author 1

2 Keywords: Non-hydrostatic Shallow-Water Finite-Difference Finite-Volume GPU Path Conservative Tsunami Simulation Dispersive effects Wave Breaking 1 Introduction When modelling and simulating geophysical flows the Nonlinear Shallow- Water equations hereinafter SWE is often a good choice as an approximation of the Navier-Stokes equations. Nevertheless SWE do not take into account effects associated with dispersive waves. In recent years effort has been made in the derivation of relatively simple mathematical models for shallow water flows that include long nonlinear water waves. As computational power increases Boussinesq Type Models ([1] [] [3] [4] [5] [6] [7] [8] [9]) become more accessible. This means that one can use more sophisticated models in order to better accurately describe reality despite the higher computational cost. Moreover in order to improve nonlinear dispersive properties of the model information on the vertical structure of the flow should be included. The Boussinesq-type wave equations have prevailed due to their computational efficiency. The main idea is to include non-hydrostatic effects due to the vertical acceleration of the fluid in the depth-averaging process of the equations. For instance one can assume that both non-linearity and frequency dispersion are weak and of the same order of magnitude. Since the early works of Peregrine [6] several improved and enhanced Boussinesq models have been proposed over the years: Madsen and Sørensen [4] Nwogu [5] Serre Green-Naghdi equations [3] and nonlinear and non-hydrostatic higher order Shallow-Water type models [1] [11] among many others. One may use different approaches to improve nonlinear dispersive properties of the models: to consider a Taylor expansion of the velocity potential in powers of the vertical coordinate and in terms of the depth-averaged velocity [4] or the particle velocity components (u w) at a chosen level [5]; to use a better flow resolution in the vertical direction with a multi-layer approach [1]; to include a non-hydrostatic effects in the depth-averaging process ([11] [1]). The development of non-hydrostatic models for coastal water waves has been the topic of many studies over the past 15 years. Non-hydrostatic models are capable of solving many relevant features of coastal water waves

3 such as dispersion non-linearity shoaling refraction diffraction and run-up (see [ ]). The approach used by Yamazaki in [11] has the advantage of including such non-hydrostatic effects while not adding excessive complexity to the model. This is an advantage from the practical point of view and we will use this technique in this paper. The paper is organized as follows. In Section the models is described. In Section 3 a numerical scheme is introduced based on a two-step algorithm. On the first step we solve the SWE in conservative form and on the second step we include the non-hydrostatic effects. In Section 4 breaking mechanism is discussed. The reader should keep in mind that detailed small-scale breaking driven physics are not described by the model. This means that one has to include some breaking mechanism in the depth-integrated equations as it is done by an ad-hoc submodel similar to [15]. The extension of the scheme to the D case is introduced in Section 5. Finally in Section 6 some numerical tests including comparisons with laboratory data are shown. Governing equations In [11] a D non-hydrostatic model was presented. The governing equations are derived from the incompressible Navier-Stokes equations. The equations are obtained by a process of depth averaging on the vertical direction z. Unlike it is done for SWE the pressure is not assumed hydrostatic. Following Stelling and Zijlema and Casulli [13] total pressure is decomposed into a sum of hydrostatic and non-hydrostatic pressures. In order to provide the dynamic free-surface boundary condition non-hydrostatic pressure is assumed to be zero at free surface level. In the process of depth averaging vertical velocity is supposed to have linear vertical profile. Moreover in the vertical momentum equation the vertical advective and dissipative terms which are small compared with their horizontal counterparts are neglected. The resulting x y and z momentum equations as well as the continuity equation described in [11] are 3

4 h t + q = ( ) q q q t + div h hw t = p ( 1 + gh + 1 ) hp = (gh + p) H τ (1) u + W η W b = h where t is time and g is gravitational acceleration. u = (u v) contains the depth averaged velocities components in the x and y directions respectively. w is the depth averaged velocity component in the z direction. q = hu is the discharge vector in the x and y directions. W η and W b are the vertical velocities at the free-surface and bottom. p is the non-hydrostatic pressure at the bottom. The flow depth is h = η + H where η is the surface elevation measured from the still-water level H is the still water depth and τ is a friction law term (see Figure 1). Operators and div denote the gradient vector field and the divergence respectively in the (x y) direction. The vertical velocity at the bottom is evaluated from the boundary condition W b = u H. () Figure 1: Sketch of the domain for the fluid problem Due to the boundary condition () and the assumption of a linear profile of the vertical velocity W η W b h = w + u (H). h/ 4

5 Finally the last equation in system (1) is multiplied by h rewritten in the form so that it is h t + q = ( ) q q q t + div h hw t = p h q q (η h) + hw =. ( 1 + gh + 1 ) hp = (gh + p) H τ If we consider in system (3) the vertical velocity equation (hw) t + (hwu) x = p then system (3) matches with the one proposed in [1]. In this case the system verifies an exact energy balance. This property can not be guaranteed for the approach used by Yamazaki in [11] but it has the advantage of not adding an excessive complexity to the model. Nevertheless the numerical scheme proposed in this work can be easily extended to the model proposed in [1]. From the numerical point of view the results considered in this work do not present relevant differences when comparing both alternatives. 3 Numerical scheme System (3) in the one-dimensional case can be written in the compact form (3) U t + (F SW (U)) x G SW (U)H x = T NH (h h x H H x p p x ) τ hw t = p B(U U x H H x w) = where we introduce the notation ( ) h q U = F q SW (U) = q h + 1 G SW (U) = gh ( ) gh (4) 5

6 T NH (h h x H H x p p x ) = ( 1 (hp x + p(η h) x ) and for the friction term vector Manning empirical formula is used ( ) τ = τ = gh n u u τ h. 4/3 Finally where B(U U x H H x w) = hq x q (η h) x + hw U x = ( hx We describe now the numerical scheme used to discretize the 1D system (4). To do so we shall solve first the hyperbolic problem (SWE). Then in a second step non-hydrostatic terms will be take into account. The SWE written in vector conservative form is given by q x ). U t + (F SW (U)) x = G SW (U)H x. (5) The system is solved numerically by using a finite volume method. In particular an efficient second-order well-balanced PVM path-conservative finitevolume method [16] is applied. As usual we consider a set of finite volume cells I i = [x i 1/ x i+1/ ] with lengths x i and define U i (t) = 1 U(x t)dx x i I i the cell average of the function U(x t) on cell I i at time t. ) 6

7 Figure : Numerical scheme stencil. Up: finite volume mesh. Down: staggered mesh for finite differences. Regarding non-hydrostatic terms we consider a staggered-grid (see Figure ) formed by the points x i 1/ x i+1/ of the interfaces for each cell I i and denote the point values of the functions p and w on point x i+1/ at time t by p i+1/ (t) = p(x i+1/ t) w i+1/ (t) = w(x i+1/ t). Non-hydrostatic terms will be approximated by second order compact finitedifferences. The resulting ODE system is discretized using a TVD Runge- Kutta method [17]. For the sake of clarity only a first order discretization in time will be described. The source terms corresponding to friction terms are discretized semi-implicitly. Thus friction terms are neglected and only flux and source terms are considered. 3.1 Finite volume discretization for the SWE For the sake of simplicity we shall consider a constant cell length x. A first order path-conservative PVM scheme for system (4) reads as follows (see [16]): U i(t) = 1 ) (D i+1/ x (t) + D+i 1/ (t) where avoiding the time dependence D ± i+1/ (t) = D± i+1/ (U i(t) U i+1 (t) H i H i+1 ) = = 1 ( F(Ui+1 ) F(U i ) G i+1/ (H i+1 H i ) ) ± 1 ( ) Q i+1/ (U i+1 U i ) A 1 i+1/ G i+1/ (H i+1 H i ) (6) 7

8 where and ( G i+1/ = gh i+1/ ) ( ) 1 A i+1/ = u i+1/ + gh i+1/ u i+1/ is the Roe Matrix associated to the flux F(U) from the SWE being h i+1/ = h i + h i+1 u i+1/ = u i hi + u i+1 hi+1 hi +. h i+1 Q i+1/ is the viscosity matrix associated to the numerical method. For PVM schemes Q i+1/ is obtained by a polynomial evaluation of the Roe Matrix. In this work the viscosity matrix is defined as Q i+1/ = α Id + α 1 A i+1/ being where α = S R S L S L S R S R S L α 1 = S R S L S R S L ( S L = min u i+1/ gh i+1/ u i ) gh i ( S R = max u i+1/ + gh i+1/ u i+1 + ) gh i+1. Under this choice D ± i+1/ read as D ± i+1/ = 1 ( F(Ui+1 ) F(U i ) G i+1/ (H i+1 H i ) ) ± 1 ( ) ( ) α Id + α 1 A i+1/ (U i+1 U i ) A 1 i+1/ G i+1/ (H i+1 H i ). The scheme is a path-conservative extension of HLL scheme ([18]) Note that the above expression is not well defined for the resonant case when A i+1/ is not invertible. This problem can be avoided following the strategy described in [19] where A i+1/ is replaced by ( ) 1 A i+1/ =. gh i+1/ 8

9 For this particular choice the numerical scheme reads as where being D ± i+1/ = 1 ( (1 ± α1 )R i+1/ ± α (U i+1 U i (H i+1 H i ) e 1 ) ) (7) R i+1/ = F c (U i+1 ) F c (U i ) + T pi+1/ e 1 = F c (U i ) = q i qi T pi+1/ = h i ( ( ) 1 ) gh i+1/ (η i+1 η i ) the corresponding discretization of convective and pressure terms for the SWE. Second order in space is obtained following [] by combining a MUSCL reconstruction operator (see [1]) with the PVM scheme presented above. Remark 1 Concerning the well-balancing properties the numerical scheme considered in this work (first or second order) is well-balanced for the water at rest solution and are linearly L -stable under the usual CFL condition. 3. Finite difference discretization for the non-hydrostatic terms In this Subsection non-hydrostatic variables p w will be discretized using second order compact finite differences. In order to obtain point value approximations for the non-hydrostatic variables p i+1/ w i+1/ and skipping notation in time operator B(U U x H H x w) will be approximated for every point x i+1/ of the staggered-grid (Figure ) by B(U i+1/ U xi+1/ H i+1/ H xi+1/ w i+1/ ) = h i+1/ q xi+1/ q i+1/ ( ηxi+1/ h xi+1/ ) + hi+1/ w i+1/ (8) where we will use second order point value approximations of U U x H and H x on the staggered-grid. They will be computed from the approximations 9

10 of the average values on the cell I i I i+1 as follows: h i+1/ = h i+1 + h i h xi+1/ = h i+1 h i η xi+1/ = η i+1 η i x x q i+1/ = q i+1 + q i q xi+1/ = q i+1 q i. x In a similar way a second order point value approximation in the center of the cell will be used for T NH computed as ( ) T NH (h i h xi H i H xi p i p xi ) = 1 (h. (9) ip xi + p i (η xi h xi )) Here h xi = h i+1 h i 1 x η xi = η i+1 η i 1 x p i = p i+1/ + p i 1/ p xi = p i+1/ p i 1/ x are second order point value approximations in the middle of the cell I i which are a second order approximation of the averaged variables. 3.3 Final numerical scheme Assume given time steps t n and denote t n = k n tk and U i (t n ) = U n i p i+1/ (t n ) = p n i+1/ w i+1/(t n ) = wi+1/ n. The numerical scheme proposed can be summarized as follows: In a first stage SWE approximations are solved. Let us define U n+1/ i as the averaged values of U on cell I i at time t n for the SWE as detailed in the Subsection (3.1). In a second stage we consider the system U n+1 i = U n+1/ i + tt NH (h n+1 i h n+1 xi H i H xi p n+1 i p n+1 xi ) w n+1 i+1/ = wn i+1/ i+1/ h n+1 i+1/ + t pn+1 B(U n+1 i+1/ U n+1 xi+1/ H i+1/ H xi+1/ w n+1 i+1/ ) = (1) where B(U n+1 i+1/ U n+1 xi+1/ H i+1/ H xi+1/ w n+1 i+1/ ) is given by (8) and T NH (h n+1 i h n+1 xi H i H xi p n+1 i p n+1 xi ) by (9). System (1) implies the solution 1

11 of a tridiagonal linear system for the unknowns p n+1 i+1/. It is efficiently solved using the Thomas algorithm. Then the values q n+1/ i are corrected with T NH (h n+1 i h n+1 xi H i H xi p n+1 i p n+1 xi ). The scheme presented here is only first order in time. To get a second order in time discretization we perform a second order TVD Runge-Kutta approach (see [17]). Therefore the resulting scheme is second order accurate in space and time. Remark that the usual CFL restriction should be considered. 4 Breaking wave modelling and wet-dry treatment As pointed in [15] in shallow water complex events can be observed related to turbulent processes. One of these processes corresponds to the breaking of waves near the coast. The model presented here cannot describe this process without an additional term which allows the model dissipate the required amount of energy on such situations. In this Section a simple eddy viscosity approach similar to the one introduced in [15] is taken into account in the momentum equation U t + (F SW (U)) x G SW (U)H x =T NH (h h x H H x p p x ) + (R b (U U x )) x τ hw t = p (11) B(U U x H H x w) = where being ν the eddy viscosity ( ) R b (U U x ) = R R b (U U x ) b (U U x ) = νhu x ν = Bh q x B = 1 q x U 1 where U 1 = B 1 gh U = B gh 11

12 denote the flow speeds at the onset and termination of the wave-breaking process and B 1 B are calibration coefficients that should be calibrated through laboratory experiments (see [15]). Wave energy dissipation associated with breaking begins when q x U 1 and continues as long as q x U. The proposed definition of the viscosity ν requires a positive value of B. In order to satisfies that for negative values of B the viscosity ν is set to zero. An explicit discretization of (R b ) x leads to a severe restriction on the CFL number. This can be solved by considering an implicit discretization of the eddy viscosity term. The implicit discretization involves solving an extra tridiagonal linear system leading to a loss of efficiency. In this work we present to the best of our knowledge a new efficient treatment of the eddy viscosity term for depth averaged non-hydrostatic models. Let us define and p = p νu x (1) ( ) R b (U U x H) = R b (U U x H) R b (U U x H) = νu x H x. System (11) can be rewritten as U t + (F SW (U)) x G SW (U)H x =T NH (h h x H H x p p x ) + R b (U U x H) τ hw t = p + νu x B(U U x H H x w) =. (13) Terms νu x H x in the horizontal momentum equation and νu x in the vertical velocity equation are essentially first order derivatives of u and can be discretized explicitly. That gives us an efficient discretization of the eddy viscosity terms. A wet-dry treatment as described in [] in regions with emerging bottom is considered in the first step (SWE). In the second step no special treatment is required due to the rewriting of the last equation in (3) that has been multiplied by h. In presence of wet-dry fonts non-hydrostatic pressure vanishes. 1

13 Remark Reinterpretation of the eddy viscosity approach: Term νu x in the vertical velocity equation can be seen as a viscosity for the vertical velocity W. Indeed consider the term (ν z W ) z and integrate in the vertical direction. η b η (ν z W ) z dz = W z ν z dz = W z (ν η ν b ) = (ν η ν b )u x. b If we choose ν = (ν η ν b ) we get the term introduced in (13). 5 Numerical scheme in two dimensions We describe the numerical scheme used to discretize the D system (3). The computational domain is decomposed into subsets with a simple geometry called cells or finite volumes. We will use one common arrangement of the variables known as the Arakawa C-grid (see Figure (3)). This is an extension of the procedure used for the 1D case. Variables p and w will be computed at the intersections of the edges: p i+1/j+1/ (t) = p(x i+1/ y j+1/ t) w i+1/j+1/ (t) = w(x i+1/ y j+1/ t). Figure 3: Numerical scheme stencil 13

14 As in Section 3 we shall solve first the hyperbolic problem (SWE) and then correct it with the non-hydrostatic terms. The SWE are solved numerically by using a finite volume method. An efficient second-order well-balanced PVM path-conservative finite-volume method is applied following [16]. There second order in space is obtained following [] by combining a MUSCL reconstruction operator (see [1]) with the PVM scheme. Non-hydrostatic terms are approximated by second order compact finitedifferences. The resulting ODE system is discretized using a TVD Runge- Kutta method [17]. The source terms corresponding to friction terms are discretized semi-implicitly. Breaking terms are discretized following the ideas presented in Section 4. The final numerical scheme is U n+1 ij = U n+1/ ij + tt NH (h n+1 (h n+1 ) H (H) p n+1 (p n+1 )) ij w n+1 i+1/j+1/ = wn i+1/j+1/ + t pn+1 i+1/j+1/ h n+1 i+1/j+1/ B ( U n+1 (h n+1 ) ( Q n+1 ) H (H) w n+1) i+1/j+1/ =. (14) where we denote the vector of the state variables ( ) ( ) h q U = Q = 1 Q q and B T NH defined as in Section 3. B will be approximated for every point x i+1/j+1/ of the staggered-grid. To do that second order point value approximations of Ũ n+1 (h n+1 ) ( Q n+1 ) H (H) and w n+1 on the staggered-grid points will be computed from the approximations of the average values on the cell provided in the first SWE finite-volume step. In the same way a second order point value approximation in the center of the cell will be used for the approximation of T NH. System (14) leads to solve a linear system for the unknowns p n+1 i+1/j+1/. Linear system is solved using an iterative Jacobi method combined with a scheduled relaxation method following [3]. 14

15 Remark that the compactness of the numerical stencil and the easy parallelization of the Jacobi method adapts well to the implementation of the scheme on GPUs architectures. To define a convergence criteria we use U n(k+1) U n(k) < ɛ (15) where U n(k) denotes the k-th approximation of U n giving by the Jacobi algorithm and ɛ is a tolerance parameter. It is observed that Jacobi method converges in a few iterations for the problems tested here. A parallel implementation of the numerical scheme has been programmed on GPUs architecture following [4]. CUDA programming toolkit [5] has been used. To get a second order in time discretization we perform a second order TVD Runge-Kutta approach (see [17]). The details of the scheme can be found in the Appendix. 6 Numerical tests and results 6.1 Solitary wave propagation in a channel The propagation of a solitary wave over a long distance is a standard test of the stability and conservative properties of numerical schemes for Boussinesqtype equations ([1] [11] [15] [6] [14] [7]). A solitary wave propagates at constant speed and without change of shape over an horizontal bottom. An approximated expression of a solitary wave for system (3) is given by (see [7]) [ ] 3A gh η(x t) = A Sech (x ct)) u(x t) = η(x t) (16) 4H3 H where A is the amplitude and c = gh(a + H) is the wave propagation velocity. We simulate the propagation of a solitary wave over a constant depth H = 1. m with A =.1 m in a channel of length 5 m along the x direction. The domain is divided into 5 cells along the x axis. The final time is 5s. We set CF L =.4 and g = 1. m/s. Outflow boundary conditions are used and the initial condition is computed using (16). 15

16 .1 η/h.1 η/h.1 η/h.1 η/h.1 η/h x(m) Figure 4: Solitary wave propagation at T = s Figure 4 shows the evolution of the solitary wave at different times. As expected the wave s shape has not changed and propagates at constant speed (see Figure 5). 16

17 .1 η.5 x ct Figure 5: Comparison of analytical (red) and numerical (blue) surface at time T = 4 s Numerical simulations for different grids have been computed up to time t = 1. s in a channel of length 5 m. Table 1 shows the L 1 errors and numerical orders of accuracy obtained with CF L number.4. Since equation 16 is not an exact solution for system 3 we take as reference solution a numerical simulation at time t = 1. s for a very fine grid with 18 cells. Number of Cells L 1 error h L 1 order h L 1 error q L 1 order q 1.99E E E E E E E E E E E E Table 1: One-dimensional accuracy test. L 1 numerical errors and orders. 17

18 6. Head-on collision of two solitary waves The head-on collision of two equal solitary waves is again a common test for the Boussinesq-type models ([15] [6]). The collision of two solitary waves is equivalent to the reflection of one solitary wave by a vertical wall when viscosity is neglected. After the interaction of the two waves one should ideally recover the initial profiles. The collision of the two waves presents additional challenges to the model due to the sudden change of the nonlinear and frequency dispersion characteristics. We present here the interaction of two solitary waves propagating on a depth of H = 1 m with amplitude A =.1 m. The same computational scenario and same expression for the solitary wave (16) as in previous test is taken into account. Figure 6: Head-on collision of two solitary waves at T = s Figure 6 shows the collision of the two solitary waves at the midpoint of the domain. After the collision both maintain the initial amplitude and the same speed but in opposite directions. 18

19 6.3 Periodic waves breaking over a submerged bar The experiment of plunging breaking periodic waves over a submerged bar by Beji and Battjes [8] is considered here. The numerical test is performed in a one-dimensional channel with a trapezoidal obstacle submerged. Waves in the free surface are measured in seven point stations S S 1... S 6 ( See Figure 7). S S 1 S S 3 S 4 S 5 S m. m 1. m 1. m 1. m 1.6 m.4 m.3 m 6. m. m 3. m Figure 7: Periodic waves breaking over a submerged bar. topography and layout of the wave gauges Sketch of the The one-dimensional domain [ 5] is discretized with x =.5 m. and the bathymetry is defined in the Figure 7. u and η are set initially to. The boundary conditions are free outflow at x = 5 m and free surface is imposed at x = m using the data provide by the experiment at S. The CFL is set to.9 and g = 9.81 m/s. Figure 8 shows the time evolution of the free surface at points S 1... S 6. The comparison with experimental data emphasizes the need to consider a dispersive model to faithfully capture the shape of the waves near the continental slope. Both amplitude and frequency of the waves are captured on all wave gauges successfully. 19

20 . S 1. S S S S S Figure 8: Comparison of data time series (red) and numerical (blue) at wave gauges S 1 S S 3 S 4 S 5 S Solitary wave run-up on a plane beach Solitary wave run-up on a plane beach is one of the most intensively studied problems in long-wave modeling. Synolakis [9] carried out laboratory experiments for incident solitary waves of multiple relative amplitudes to study propagation breaking and run-up over a planar beach with a slope 1 : Many researchers have used this data to validate numerical models. With this test case we asses the ability of the model to describe shoreline motions and wave breaking when it occurs. Experimental data are available in [9] for surface elevation at different times. For this test the still water level is H = 1 m. The bathymetry of the problem is given by (see Figure 9)

21 .3 m 1. m 1. m m. m Figure 9: Sketch of the topography 1 + (x 19.85)/19.85 x H(x) = 1 x > A solitary wave of amplitude.3 m. is placed at point x = 5 m. given by (16). A Manning coefficient of n m =.38 was used in order to define the glass surface roughness used in the experiments. The computational domain is [ 1 4] and the numerical parameters used were x =.5 CF L =.9 and g = 1. m/s. Figure 1 shows snapshots at different times of a comparison of experimental and simulated data by using the breaking criteria described in Section 4 with B 1 =.15 and B =.5. Figure 11 shows the same that described previously although this time the breaking mechanism is not considered. In this case an overshoot value on the amplitude of the wave appears when the mesh is refined. The results are quite satisfactory in favour of the former. In addition good results are obtained at maximum run-up where breaking mechanism also played an important role. Note that no additional wetdry treatment on the second step of the scheme has been necessary. 1

22 .4 η.4 η.4 η.4 η x(m) Figure 1: Comparison of experiments data (red) and simulated ones (blue) at times T = s with a breaking criteria

23 .4 η.4 η.4 η.4 η x(m) Figure 11: Comparison of experiments data (red) and simulated ones (blue) at times T = s without a breaking criteria 6.5 Solitary wave propagation over reefs A test case on solitary wave over an idealized fringing reef examines the model s capability of handling nonlinear dispersive waves breaking waves and bore propagation. The test configurations include a fore reef a flat reef and an optional reef crest to represent fringing reefs commonly found in tropical environment. Figure 1 shows a sketch of the laboratory experiments carried out at the O.H. Hinsdale Wave Research Laboratory of Oregon State 3

24 University..5 m 1. m 17. m 5. m 3. m Figure 1: Sketch of the topography The uni-dimensional domain [ 45] is discretized with x =.45 m. The bathymetry is defined in the Figure 1. A solitary wave of amplitude.5 m. is placed at point x = 1 m. given by (16). A Manning coefficient of n m =.38 was used in order to define the glass surface roughness used in the experiments. Breaking mechanism is considered with B 1 =.15 and B =.5. Finally CF L =.9 and g = 1. m/s 4

25 η.4 η.4 η.4 η.4 η.4 η x(m) Figure 13: Comparison of experimental data (red points) and numerical (blue) at times T = s. Figure 13 shows snapshots at different times of a comparison of experimental and simulated data. The water rushes over the flat reef without producing a pronounced bore-shape. Simulation also captures the offshore component of the rarefaction falls exposing the reef edge below the initial water level. 6.6 Solitary wave on a conical island The goal of this D-numerical test is to compare numerical model results with laboratory measurements. Experiment is carried out at the Coastal and Hydraulic Laboratory Engineer Research and Development Center of the U.S. Army Corps of Engineers ([3]). The laboratory experiment consists in an idealized representation of Babi Island in the Flores Sea in Indonesia. 5

26 The produced data sets have been frequently used to validate run-up models ([31] [11]). A directional wave-maker is used to produce planar solitary waves of specified crest lengths and heights. Domain setup consists in a 5 3 m basin with a conical island situated near the center. The still water level is H =.3 m. The island had a base diameter of 7. m a top diameter of. m and it was.65 m high with a side slope 1 : 4. Wave gauges {W G 1 W G W G 3 W G 4 } are distributed around the island in order to measure the free surface elevation (see Figure 14). For the numerical simulation the computational domain is [ 5 3] [ 8] with x = cm and y = cm. Free outflow boundary conditions are imposed. As initial condition for η and u a solitary wave (16) of Amplitude A =.6 m centered at x = is given. The wave propagates until 3 s with CF L =.9 and g = 9.81 m/s. A Manning coefficient of n m =.15 is used and breaking mechanism with B 1 =.15 and B =.5 is considered. Numerical simulation shows two wave fronts splitting in front of the island and collide behind it (15). Comparison with measured and computed water level at gauges W G 1 W G W G 3 W G 4 shows good results as well as comparison between computed run-up and laboratory measurement. 6

27 Figure 14: Sketch of the topography 7

28 T = s T = 5 s T = 8 s T = 1 s Figure 15: Comparison of numerically calculated free surface at various times. 8

29 R(cm) Direction ( ) Figure 16: Maximum run-up measured (red) and simulated (blue) cm 15 cm cm 6 Figure 17: Maximum run-up measured (red) and simulated (blue) in polar coordinates 9

30 η(cm) η(cm) η(cm) WG 1 : x = 9.36 m y = 13.8 m WG : x = 1.36 m y = 13.8 m WG 3 : x = 1.96 m y = 11. m η(cm) WG 4 : x = m y = 13.8 m t(s) Figure 18: Comparison of data time series (red) and numerical (blue) at wave gauges W G 1 W G W G 3 W G Circular dam-break As D-test problem we consider a circular dam-break problem in the [ 5 5] [ 5 5] domain. The depth function is H(x y) = 1.5e x y and the 3

31 initial condition is h (x y) { Ui (x y) = h H(x y) if x + y (x y) =.5 H(x y) +.5 otherwise. The goal of this numerical test is to compare the execution times in seconds for the SWE and non-hydrostatic GPU codes for different mesh sizes. Simulations are carried out in the time interval [ 1]. CFL parameter is set to.9 and open boundary conditions are considered. Table shows execution times for both codes. Different parameters of ɛ { } were taken into account where ɛ was defined in (15). Figure19 shows a cross-section comparison of numerical simulations at different times for ɛ { }. In view of the figures non-hydrostatic code can achieve a good performance for ɛ = 1 3 with 17 millions of volumes per unit of times while the additional computation cost is only.4 times that of a SWE code. It can be stated thus that the scheme presented here is efficient and can model dispersive effects with a moderate computational cost. To our knowledge similar models and/or numerical schemes that intend to simulate dispersive effects in such frameworks are much more expensive from the computational point of view. Runtime (s) Number of Volumes SWE Non-Hydrostatic ɛ = 1 3 ɛ = 1 4 ɛ = Table : Execution times in sec for SWE and NH GPU implementations 31

32 in million for SWE and NH GPU implemen- Table 3: tations Volumes Iterations/Time ( 1 6 ) Number of Volumes SWE Non-Hydrostatic ɛ = 1 3 ɛ = 1 4 ɛ = Volumes Iterations Time 3

33 .. η ǫ=1-5 η ǫ=1-4 η ǫ=1-3 -H η ǫ=1-5 η ǫ=1-4 η ǫ=1-3 -H Free surface cross-section at t =.5 s Free surface cross-section at t = 1. s u x ǫ=1-5 u x ǫ=1-4 u x ǫ= u x ǫ=1-5 u x ǫ=1-4 u x ǫ= u x cross-section at t =.5 s u x cross-section at t = 1. s Figure 19: Cross-section of numerical simulations at times T =.5 s (left) and T = 1. s (right) for ɛ { }. 7 Conclusions In this work a non-hydrostatic model has been considered in order to incorporate dispersive effects in the propagation of waves in a homogeneous inviscid and incompressible fluid. The numerical scheme employed combines a finite volume path-conservative for the underlying hyperbolic system and finite differences for discretization of non-hydrostatic terms. Furthermore it is second order accurate and it is 33

34 well-balanced for the water at rest solution and linearly L -stable under the usual CFL condition. A wet-dry treatment presented in [] for the SWE is adopted. Moreover no numerical truncation for the non-hydrostatic pressure is needed at wet-dry areas where non-hydrostatic pressure vanishes. This is due to the writing of the equations proposed in (3). To the best of our knowledge this is an improvement on non-hydrostatic numerical schemes where usually nonhydrostatic pressure is truncated to zero up to a threshold value. For such models it is necessary to consider some dissipative mechanism for breaking waves in order to accurately model waves near the coastal areas. Discretization of the viscosity term needs to solve an extra elliptic problem which results in additional computational cost. We have proposed a reinterpretation of the viscosity term which results in a new simple and efficient way to solve the problem. A GPU implementation of the D model is carried out. From a computational point of view non-hydrostatic code presents good computational times respect to the SWE GPU times. A numerical test was carried out in order to illustrate such claim. For a tolerance of ɛ = 1 3 for the iterative method that solves the linear system non-hydrostatic wall-clock times are no higher than.4 SWE times for refined meshes. Numerical simulations show that the approach presented here correctly solves the propagation of solitary waves preserving their shape for large integration times accurately. Comparison with experimental data is also presented. Experimental data justifies the need to incorporate dispersive effects to capture faithfully waves in the vicinity of the continental shelf. Moreover complex processes such as run-up shoaling wet-dry areas are simulated successfully for the proposed 1D and D tests. The numerical scheme presented in this work provides thus an efficient and accurate approach to model dispersive effects in the propagation of waves near coastal areas. A D numerical scheme We consider as in 3 the system: 34

35 U t + (F 1SW (U)) x + (F SW (U)) y = G 1SW (U)H x + G SW (U)H y + T NH (h h H H p p) hw t = p B(U h ( Q) H H w) = (17) where we denote the vector of the state variables and the corresponding flows ( ) ( ) h q U = Q = 1 Q q F 1SW (U) = q 1 q 1 h + 1 gh q 1 q h F SW (U) = q q 1 q h q h + 1 gh. The sources terms are given by G 1SW (U) = gh G SW (U) = gh and the friction term vector where a Manning empirical formula is used is 35

36 given by τ = n u 1 + u ghu 1 h 4/3. n u 1 + u ghu h 4/3 Finally non-hydrostatic terms are T NH (h h H H p p) = T Hor (h h x H H x p p x ) T V er (h h y H H y p p y ) being T Hor T V er the horizontal and vertical non-hydrostatic contributions respectively: T Hor (h h x H H x p p x ) = 1 (hp x + p((η h) x )) T V er (h h y H H y p p y ) = 1 (hp y + p((η h) y )) and he free divergence equation is B(U h ( Q) H H w) = h ( Q) Q (η h) + hw. We describe now the numerical scheme used to discretize the D system (17). The D-SWE are written in vector conservative form U t + (F 1SW (U)) x + (F SW (U)) y = G 1SW (U)H x + G SW (U)H y. (18) 36

37 To discretize (18) the computational domain is decomposed into subsets with a simple geometry called cells or finite volumes. Here we consider rectangular structured meshes: V ij = [x i 1/ x j 1/ ] [y i 1/ y j 1/ ] R i Nx j N y. Given a finite volume V ij V ij will represent its area and U ij (t) the constant approximation to the average of the solution in the cell V ij at time t provided by the numerical scheme: U ij (t) = 1 V ij V ij U(x t) dx. Regarding non-hydrostatic terms we will use one common arrangement of the variables known as the Arakawa C-grid (see Figure 3). This is an extension of the procedure used for the 1D case. Variables p and w will be computed at the intersections of the edges: p i+1/j+1/ (t) = p(x i+1/ y j+1/ t) w i+1/j+1/ (t) = w(x i+1/ y j+1/ t) and non-hydrostatic terms will be approximated by second order compact finite-differences. The resulting ODE system is discretized using a TVD Runge-Kutta method [17]. For the sake of clarity only a first order discretization in time will be described. The source terms corresponding to friction terms are discretized semi-implicitly. Thus friction terms are neglected and only flux and source terms are considered. A.1 Finite volume scheme For the finite-volume scheme we will follow the ideas given in [3] for the two-dimensional problem. In particular we use the D extension of the PVM scheme described in Section 3 (see [4]). A. Finite differences scheme In this subsection non-hydrostatic variables p and w will be discretized using second order compact finite differences. Following the same procedure as for 37

38 the 1D equations. Let us define the North and South approximations in the middle of the horizontal edges for the volume V ij of T Hor NH by T Hor N(ij)(h h x H H x p p x ) = 1 h p i+1/j+1/ p i 1/j+1/ ij x 1 p i+1/j+1/ + p i 1/j+1/ T Hor S(ij)(h h x H H x p p x ) = 1 h ij 1 p i+1/j 1/ + p i 1/j 1/ η i+1j h i+1j (η i 1j h i 1j ) x p i+1/j 1/ p i 1/j 1/ x η i+1j h i+1j (η i 1j h i 1j ) x respectively. Same ideas for the East and West approximations in the middle of the vertical edges for the volume V ij of T V NH: er T V E(ij)(h er h y H H y p p y ) = 1 h p i+1/j+1/ p i+1/j 1/ ij y 1 p i+1/j+1/ + p i+1/j 1/ T V er W (ij)(h h y H H y p p y ) = 1 h ij 1 p i 1/j+1/ + p i 1/j 1/ η ij+1 h ij+1 (η ij 1 h ij 1 ) y p i 1/j+1/ p i 1/j 1/ y η ij+1 h ij+1 (η ij 1 h ij 1 ). y Note that if we approximate T NH (h (h) H (H) p (p)) ij 1 1 ( ) T Hor N(ij) + T Hor S(ij) ( T V er E(ij) + T V er W (ij) (19) ) then we have a second order approximation of T NH (h (h) H (H) p (p)) at the center of the volume V ij. 38

39 Likewise B(U (h) ( Q) H (H) w) will be discretized for every point (x j+1/ y i+1/ ) for the staggered mesh by B(U (h) ( Q) H (H) w) i+1/j+1/ h i+1/j+1/ ( Q) i+1/j+1/ Q i+1/j+1/ (η h) i+1/j+1/ + h i+1/j+1/ w i+1/j+1/ () being h i+1/j+1/ = 1 4 (h ij + h i+1j + h i+1j+1 + h ij+1 ) (1) ( Q) i+1/j+1/ = q 1E q 1W x + q N q S y Q i+1/j+1/ = q 1E + q 1W q N + q S () (η h) i+1/j+1/ = (η h) E (η h) W (η h) N (η h) S (3) where q 1E q 1W q N q S and (η h) E (η h) W (η h) N (η h) S are second order approximations of q 1 q and (η h) respectively in the middle of the edges (see Figure(3)). Expressions for this approximations will be introduced in the next section. Final Numerical Scheme Let be given time steps t n note t n = k n tk and U ij (t n ) = U n ij p i+1/ (t n ) = p n i+1/ w i+1/(t n ) = wi+1/ n. The numerical scheme proposed consists: 39

40 On a first stage SWE approximation is carried out. Let us define U n+1/ ij as the averaged values of U on cell I i at time t n for the SWE as detailed in the subsection (A.1). On a second stage we consider the system U n+1 ij = U n+1/ ij + tt NH (h n+1 h n+1 H H p n+1 p n+1 ) ij where: w n+1 i+1/j+1/ = wn i+1/j+1/ + t pn+1 i+1/j+1/ h n+1 i+1/j+1/ B (Ũ n+1 h n+1 ( Q ) n+1 ) H H w n+1 = i+1/j+1/ is defined by (19) is defined by (1) and T NH (h n+1 h n+1 H H p n+1 p n+1 ) ij h i+1/j+1/ (4) is defined by () being B(Ũ n+1 h n+1 ( Q n+1 ) H H w n+1 ) i+1/j+1/ q n+1 1E = 1 ( q n+1 xi+1j+1 + qn+1 xi+1j + 1 Hor tt S(i+1j+1)(h n+1 h n+1 y H H y p n+1 p n+1 y ) + 1 Hor tt N(i+1j)(h n+1 h n+1 y H H y p n+1 p n+1 y ) ) q n+1 1W = 1 ( q n+1 xij+1 + ) qn+1 xij + 1 Hor tt S(ij+1)(h n+1 h n+1 y H H y p n+1 p n+1 y ) + 1 Hor tt N(ij)(h n+1 h n+1 y H H y p n+1 p n+1 y ) 4

41 q n+1 N = 1 ( q n+1 yi+1j+1 + ) qn+1 xij V er tt W (i+1j+1)(h n+1 h n+1 y H H y p n+1 p n+1 y ) + 1 V er tt E(ij+1)(h n+1 h n+1 y H H y p n+1 p n+1 y ) q n+1 S = 1 ( q n+1 yi+1j + ) qn+1 xij + 1 V er tt W (i+1j)(h n+1 h n+1 y H H y p n+1 p n+1 y ) + 1 V er tt E(ij)(h n+1 h n+1 y H H y p n+1 p n+1 y ) (η h) n+1 E = η i+1j+1 h i+1j+1 + (η i+1j h i+1j ) (η h) n+1 W = η ij+1 h ij+1 + (η ij h ij ) (η h) n+1 N = η i+1j+1 h i+1j+1 + (η ij+1 h ij+1 ) (η h) n+1 S = η i+1j h i+1j + (η ij h ij ). Acknowledgements This research has been supported by the Spanish Government through the Research projects MTM C-1-R MTM C--R. 41

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