A Grid-Free, Nonlinear Shallow-Water Model with Moving Boundary

Transcription

1 A Grid-Free, Nonlinear Shallow-Water Model with Moving Boundary X. Zhou Y.C. Hon* Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong K.F. Cheung Department of Ocean and Resources Engineering, University of Hawaii at Manoa, Honolulu, USA ABSTRACT: A meshless method based on the radial basis function is derived to solve the nonlinear, nondispersive shallow-water equations. The one-dimensional initboundary value problem has a fixed boundary at one end and a free boundary at the other. The formulation employs a Lagrangian-Eulerian scheme to track the movement of the free boundary and transform the problem to a time-independent domain. The radial basis function evaluates the spatial derivatives, while the Wilson-θ method integrates the development of the flow in time. The resulting model is applied to calculate the flow of floodwater resulting from dam collapse and the runup of a wave on a plane beach. Comparisons of the computed results with analytical and finite difference solutions demonstrate the accuracy and capability of this meshless model in engineering applications. AMS: 65N15, 65M30, 35R25 Key words: meshless, radial basis functions, shallow-water equations, moving boundary * Corresponding author The work described in this paper was partially supported by a grant from CityU (Project No ) 1

2 1. Introduction A wide variety of numerical solutions have been proposed for the nonlinear, shallow-water equations with moving boundaries. The finite difference method, owing to its flexibility in programming and ease of application, has been one of the commonly used approaches. Besides other complexities, which are generally amenable to computation, the main challenge lies in capturing the moving waterlines. Researchers have developed several strategies to treat this problem. The first approach is to adjust the boundary from grid to grid, or obtaining a local solution at the waterline by extrapolation (e.g., Titov et al. [1], Liu et al. [2], Zelt et al. [3], and Bellos et al. [4] ). Another approach is to transform the boundary value problem to a time-invariant domain using a Lagrangian-Eulerian scheme (Zhang et al. [5]). This approach, however, gives rise to several new nonlinear terms in the governing equations that render the numerical computation considerably more complicated. The present paper investigate the use of the newly developed radial basis function method to the solution of this moving-boundary problem. The idea is to form a basis function space V = {ϕ( x x j )}, where x x j denotes the radial distance between any point x and the given scattered data point x j. Powell [6] and Wu [7] discussed the application of this method for scattered data interpolation. This function space can also be used for approximating the solutions of partial differential and integral equations (e.g., Kansa [8] and Golberg and Chen [9]). Hon et al. further extended the applications of RBFs method to the numerical solutions of various ordinary and partial differential equations including general initial value problems [11], nonlinear Burgers equation with shock wave [12], shallow water equation for tide and currents simulation under irregular boundary [13], and free boundary problems like American option pricing [14][15]. The computations showed the definite advantages in using this truly mesh-free method for solving various initial and boundary values problems. Furthermore, this representation enjoys the benefit of spatial independence and data structure is flexible. The existence, uniqueness, and convergence proofs in applying the RBFs were given by Micchelli [16], Powell [17], Madych and Nelson [18] for scattered data interpolation and Wu [19], Franke and Schaback [20], and Wendland [21] for solving PDEs respectively. In these papers, two important features of the RBFs method had been observed: (1) it is a truly mesh-free algorithm; and (2) it is spatial dimension independent in the sense that the convergence order is of O(h d+1 ) (one of the estimates for MQ), where h is the density of the 2

3 collocation points and d is the spatial dimension. In other words, as the spatial dimension of the problem increases, the convergence order also increases and hence much fewer scattered collocation points will be needed to maintain the same accuracy. In particular, Golberg and Chen [10] showed that a three-dimensional Poisson equation could be solved with only 60 randomly distributed knots to the same degree of accuracy as a FEM solution with 71,000 linear elements. This paper describes the application of the radial basis function to provide a solution for the nonlinear, nondispersive shallow water equations in the Lagrangian-Eulerian form. The use of the radial basis function in principle saves the computational time for evaluating the approximation of the solution and its derivatives. Furthermore, since the radial basis functions are smooth, it can easily be applied to solve high order differential equations. The one-dimensional model is applied to calculate the flow of floodwater resulting from dam collapse and the runup of a solitary wave on a plane beach. The performance of the model is evaluated against the analytic solutions and finite difference solutions of the problems. 2. Mathematical Model The nonlinear, nondispersive shallow-water model is based on the assumptions of hydrostatic pressure and uniform velocity distribution in the depth direction. The governing equations in the Eulerian form are given by y t + [(h + y)u] ζ = 0, (1) u t + uu ζ + gy ζ = 0, (2) where y=wave amplitude, u=depth-average velocity, h=undisturbed water depth, g=acceleration of gravity, and the time t and the spatial variable ζ in subscript denote partial differentiation. In the case of moving boundary problems, the Lagrangian-Eulerian hybrid method [5] was introduced to exactly capture the moving waterlines by combining the Lagrangian description for the moving waterline with the Eulerian description for the interior flow field. Let X(t) denote the position of the mass at waterline, then dx dt = u(x(t), t) = U 0(t). (3) 3

4 Transforming equation (2) to the Lagrangian form, we get du 0 dt = gy(x(t), t) ζ, (4) where U 0 (t) represents the Lagrangian velocity of the mass at the waterline. Introducing a geometric transformation ζ = (1 + X/L)x + X and t = t (5) the variable domain with the original region, say, [-L, 0], can be converted to a fixed one. In equation (5), X(t) locates the moving waterline and L is the other open end of the computation region. By the above transformation, the time-varying region L ζ X(t) is transformed into a fixed region L x 0. And then the equations (1), (2) and (4) become y t c 1 U 0 y x + c 2 [(h + y)u] x = 0, (6) u t c 1 U 0 u x + c 2 (uu x + gy x ) = 0, (7) du 0 dt = c 2 gy(0, t) x, (8) respectively, where c 1 = c 1 (x, t) = 1 + x/a 1 + X/A, (9) c 2 = c 2 (t) = X/A. (10) 4

5 3. Radial Basis Function Algorithm The idea of the meshless RBFs method is to represent an unknown function f(x), x R d by the following linear combination M f(x, t) λ j (t)φ j (x), (11) j=1 where x j are M distinct points in R d, λ j are unknown coefficients, and φ j (x) = φ( x x j ) are called radial basis functions because the Euclidean norm x x j represents the radial distance of x from x j. Denote the radial distance r = x y, x, y R d. The commonly used RBFs φ(r) are: multiquadrics : (r 2 + c 2 ) β/2, β > 0, β (2N + 1) inverse multiquaqrics : (r 2 + c 2 ) β/2, β < 0, β (2N + 1) Gaussians : exp( c 2 r 2 ) thin plate splines : r β log r, β > 0, β 2N smoothing splines : r β, β > 0, β (2N + 1) where c is called a shape parameter. Based on the RBFs method, we Assume that the unknown solutions y and u in equations (6) and (7) are approximated by the RBFs interpolants as follows: M y(x, t) = λ j (t)φ j (x), (12) j=1 M u(x, t) = γ j (t)φ j (x). (13) j=1 Collocating (12) and (13) at the M points (x i, i = 1, 2,..., M) we obtain the following system of equations for the undetermined coefficients λ j and γ j : Φ 0 0 Φ Λ Γ + (c 2u c 1 U)Φ x c 2 gφ x c 2 (h + y)φ x + c 2 h x Φ Λ = 0, (14) (c 2 u c 1 U)Φ x Γ 0 where Φ = [φ j (x i )], i, j = 1, 2,..., M, (15) [ ] d Φ x = dx φ j(x), x = x i, i, j = 1, 2,..., M, (16) 5

6 Λ = [λ j ] T, and Λ = [ dλ j dt ]T j = 1, 2,..., M, (17) Γ = [γ j ] T, and Γ = [ dγ j dt ]T j = 1, 2,..., M. (18) By representing the system of equations (14) in the matrix form: [P ][Ḃ] + [Q][B] = F, (19) where [B] = Λ, Γ [Ḃ] = Λ, Γ [P ] = Φ 0, [Q] = (c 2u c 1 U)Φ x 0 Φ c 2 gφ x and [F] is a vector, we get, at each time step t = t n, c 2 (h + y)φ x + c 2 h x Φ (c 2 u c 1 U)Φ x, [P ] n [Ḃ]n + [Q] n [B] n = [F ] n, (20) U n 0 = c 2 gy(0, t n ) x, (21) and Ẋ n = U n 0 (22) respectively from equations (19), (8) and (3). By Employing the Wilson-θ method to B n and similarly to U n 0 (3) to and [B] n = [B] n 1 + t [ (1 θ)[ḃ]n 1 + θ[ḃ]n], (23) and Xn, we further discretize respectively the equations (19), (8) and [[P ] n θ t[q] n ] [Ḃ]n = [F ] n [Q] n [ [B] n 1 + (1 θ) t[ḃ]n 1], (24) U n 0 = U0 n 1 + t [ ] n 1 (1 θ) U 0 θc 2 gy(0, t n ) x, (25) X n = X n 1 + t [ (1 θ)ẋn 1 + θu n 0 ], (26) where 0 θ 1. When θ equals to zero or one, the integration scheme given by equation (23) is explicit and may need an extra stability constraint for convergence. In the following computations, we choose θ to be 0.5 so that the time integration scheme is implicit and unconditional stable. Furthermore, the Newton s iterative method for solving the nonlinear equation (24) is adopted for faster convergence. 6

7 4. Numerical Computations Dam-Break Flood-Wave Propagation on Dry Bed In order to verify the efficiency and accuracy of the meshless RBFs method devised in the last section, we first apply the method to calculate the flow of floodwater resulting from dam collapse in a reservoir (refer Figure 1). The depth of the reservoir is assumed to be constant. The bottom of the channel bottom is horizontal and friction is neglected. The analytical solution is given by Stoker [22] as: u(ζ, t) = 2 ( ) ζ 3 t + c 0, ( c 0 t ζ 2c 0 t, t > 0), (27) y(ζ, t) = 1 ( ζ 2 0) 9g t + 2c h, ( c 0 t ζ 2c 0 t, t > 0), (28) where u denotes the velocity of the front flood-wave, y represents the depth of the floodwave, and c 0 = gh. The analytical solutions are valid when the dam collapses and the flood-wave propagates to the right-end of the reservoir. It can be observed from equations (27) and (28) that the propagation velocity of the front flood-wave, the water depth and the velocity of the water-wave at the dam site are having constant values of 2c 0, 4 9 h, and 2 3 c 0 respectively when y + h equals zero. For the purpose of comparison, the finite difference method(fdm) is also applied to obtain numerical approximations to the solutions y and u of equations (6) and (7), in which the Richtmyer two-step Lax-Wendroff scheme is used: u n+1/2 i+1/2 y n+1/2 i+1/2 = 1 2 (yn i + yn i+1 ) + t 2 x c 1U 0 (y n i+1 yn i ) t = 1 2 (un i + un i+1 ) + t 2 x c 1U 0 (u n i+1 un i ) t 2 x c 2 2 x c [ 2 (h + y n i+1 )u n i+1 (h + ] yn i )un i, (29) [ 1 2 (un i+1 )2 + gy n i (un i )2 gy n i ], (30) where c 1 U 0 and c 2 are evaluated at the grid points (i + 1/2, n), and y n+1 i = yi n + t x c 1U 0 (y n+1/2 i+1/2 yn+1/2 i 1/2 ) t [ ] x c 2 (h + y n+1/2 i+1/2 )un+1/2 i+1/2 (h + yn+1/2 i 1/2 )un+1/2 i 1/2, (31) u n+1 i = u n i + t x c 1U 0 (u n+1/2 i+1/2 un+1/2 i 1/2 ) t [ 1 x c 2 2 (un+1/2 i+1/2 )2 + gy n+1/2 i+1/2 1 ] 2 (un+1/2 i 1/2 )2 gy n+1/2 i 1/2, (32) where c 1 U 0 and c 2 are evaluated at the grid points (i, n). The comparison on the numerical RBFs and FDM results and the analytical solution are given in Figure 1. The values of the parameters used in the computations are as follow: 7

8 The original spatial domain: [-10m, 0] Time domain: [0, 12s] The original water depth behind the dam: h=1m The total number of time steps: 600 (in RBFs) and 1200 (in FDM) The total number of points: 101 (in RBFs) and 201 (in FDM) The radial basis function used: r 3 In the numerical computation, the boundary conditions y(0, t) = h, u( 10, t) = 0 are imposed which leads to the following equations used in (14): M λ j (t)φ j (x M ) = h, j=1 and M γ j (t)φ j (x 1 ) = 0. j=1 Figure 1 and Figure 2 give respectively the time profiles of the water-surface and the floodflow, where the analytical solution at time t = 1.7s is marked with star for comparison. It can be observed that the numerical results are in excellent agreement with the analytical solution. Figure 3 demonstrates the propagation of the water-front with the stars indicating the analytical solution before the front of the flood-wave moves to the right-end of the reservoir. The numerical results show that the propagation velocity of the water-front is always equal to the constant 2c 0, which matches with the analytical solution. The numerical results on the water depth are shown in Figure 4, where the stars indicate the exact solution, indicate that the water depth remains constant before the front flood-wave extends to the right-end of the reservoir. In the numerical comparisons, the maximum relative error is less than the magnitude of We remark here that although the numerical results obtained by FDM as illustrated in Figures 1-3 are also in good agreement with the analytical solutions, the FDM method needs more time steps and denser grid density than the proposed meshless RBFs method for the same convergence and accuracy. 8

9 Wave Run-up on Sloping Plane Beach Secondly, we apply the RBFs method to successfully simulate the run-up of a long wave from an open ocean of depth h 0 on an uniform plane beach. The original water depth function is assumed to be: h = { αζ ( h 0 α ζ 0 ), h 0 ( x < h 0 α ), (33) where ζ represents the spatial horizontal coordinate and α denotes the slope of the plane beach. The wave is given as: ū(t) = Asin(ωt), (34) which is originally located L meters from the shoreline of the beach. domain is assumed to be [ L, 0]. The boundary conditions are: The fixed solution y(0, t) = αx(t), u( L, t) = ū(t). The initial conditions are taken to be: u(x, 0) = 0, y(x, 0) = 0. The parameters used in the computations are: h 0 = 1m, α = 0.5, A = 0.06, ω = g, L = 12m Time domain: [0, 10s] The total number of time steps: 500 The total number of the points used: 101 The radial basis function used: (r ) 1 2. The numerical simulation of the wave run-up and run-down at the slope of the beach is shown in Figure 5 and Figure 6. It can be observed from the figures that the RBFs method has successfully simulated the complicated wave runup phenomena, which is still considered to be a tedious numerical task by using the traditional finite difference or finite element methods. 9

10 5. Conclusions The meshless radial basis functions(rbfs) method combined with the Lagrangian-Eulerian hybrid scheme is devised in this paper to solve the nonlinear, nondispersive shallow-water equations with moving boundary. The advantages of the meshless method are demonstrated by applying the method to successfully obtain the numerical approximations of the solutions to the flood-flow in the dam-break problem and the wave run-up and run-down in the run-up problem. The resultant coefficient matrix of the system of equations resulted from the RBFs method is usually full and unsymmetric and hence leads to an ill-conditioning problem. The recently developed domain decomposition method for RBFs approximation by Hon et al. [23] has enabled the applications of the method to solve larger scale problems. This will be our future work in the simulation of a dimension wave run-up problem. 10

11 References [1] J.E. Zhang, T.Y. Wu, and T.Y. Hou, Coastal hydrodynamics of ocean waves on beach, Advances in Applied Mechanics, 37, , (2001). [2] V.V. Tito and C.E. Synolakis, Numerical modeling of tidal wave runup, Journal of Water, Port, Coastal, and Ocean Engineering, 124, , (1998). [3] P.L.-F. Liu, Y.-S. Cho, M.J. Briggs, U. Kanoglu, and C.E. Synolakis, Runup of solitary waves on a circular island, J. Fluid Mech., 302, , (1995). [4] J.A. Zelt and F. Raichlen, Overland flow from solitary waves, Journal of Water, Port, Coastal, and Ocean Engineering, 117, , (1991). [5] C.V. Bellos and J.G. Sakkas, 1-D dam-break flood-wave propagation on dry bed, Journal of Hydraulic Engineering, 113, , (1987). [6] M.J.D. Powell, Radial basis functions for multivariable interpolation: a review, in: Numerical Analysis, D. F. Griffiths and G. A. Watson, eds., Longman Scientific & Technical (Harlow), , [7] Z.M. Wu, Hermite-Bikhoff interpolation of scattered data by radial basis function, Approximation Theory and its Application, 8, 1-10, (1992). [8] E.J. Kansa, Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics - II. Solution to parabolic, hyperbolic and elliptic partial differential equations, Computers Math. Applic., 19, , (1990). [9] M.A. Golberg and C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Communications, 5, 57-61, (1994). [10] M.A. Golberg and C.S. Chen, Discrete projection methods for integral equations, Comput. Mech. Publ., Boston, MA, (1997). [11] Y.C. Hon and X.Z. Mao, A multiquadric interpolation method for solving initial value problems, Sci. Comput., 12, 51-55, (1997). [12] Y.C. Hon and X.Z. Mao, An efficient numerical scheme for Burgers equation, Appl. Math. Comput., 95, 37-50, (1998). 11

12 [13] Y.C. Hon, K.F. Cheung, X.Z. Mao, and E.J. Kansa, A multiquadric solution for shallow water equation, ASCE Journal of Hydraulic Engineering, 125, , (1999). [14] Y.C. Hon and X.Z. Mao, A radial basis function method for solving options pricing model, Journal of Financial Engineering, 8, 31-49, (1999). [15] Y.C. Hon and X. Zhou, A comparison on using various radial basis functions for options pricing, Int. J. Appl. Sci. Comput., 7, 29-47, (2000). [16] C.A. Micchelli, Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constructive Approximation, 2, 11-22, (1986). [17] M.J.D. Powell, The theory of radial basis function approximation in 1990, in: Advances in Numerical Analysis Vol. 2, W. Light, ed., Clarendon Press, Oxford, (1992), [18] W.R. Madych and S.A. Nelson, Multivariate interpolation and conditionally positive definite functions, Approx. Theory and its Appl. 4, 77-89, (1988). [19] Z. Wu, Solving PDE with radial basis functions, in: Advances in Computational Mathematics, Lecture Notes on Pure and Applied Mathematics, 202, Z. Chen, Y. Li, C.A. Micchelli, Y. Xu, and M. Dekker, GuangZhou, eds., (1998). [20] C. Franke and R. Schaback, Solving partial differential equations by collocation using radial basis functions, Appl. Math. Comput., 93, 73-82, (1998). [21] H. Wendland, Meshless Galerkin methods using radial basis functions, Math. Comp., 119, , (2001). [22] J.J. Stoker, Water waves, John Wiley & Sons, Inc., (1992). [23] X. Zhou, Y.C. Hon, and J.C. Li, Overlapping domain decomposition method by radial basis functions, Applied Numerical Mathematics, in press. 12

13 Depth of Water(m) t=3.3 t=4.9 t=6.5 t=8.1 t=9.7 t=11.3 t=1.7 t=0.1 Analytical Solutions o FDM Results RBFs Results Horizontal Position(m) * Figure 1. Water-surface profile in the dam break problem 13

14 t=0.1 t=1.7 t=3.3 t=4.9 t=6.5 t=8.1 t=9.7 Velocity of Water(m/s) t=11.3 Analytical Solutions * o FDM Results RBFs Results Horizontal Position(m) Figure 2. Velocity Distribution of the flood-flow field in the dam break problem 14

15 Wave Front Position(m) * Analytical Solutions o FDM Results RBFs Results Time(s) Figure 3. Propagation of the water-front in the dam break problem 15

16 Depth of the Water at the Location of the Dam(m) Time(s) Figure 4. History of water-depth at the dam-site in the dam break problem 16

17 t=4.8s 0.03 t=4.4s 0.02 Wave Elevation(m) t=3.8s t=5.4s t=5.8s Horizontal Position(m) Figure 5. Profile of wave run-up and run-down on the sloping plane beach 17

18 Vertical Amplitude of Run up and Run down(m) Time(s) Figure 6. Vertical Amplitude of the wave in the run-up problem 18