Solving the shallow-water equations by a pseudo-spectral method

Size: px

Start display at page:

Download "Solving the shallow-water equations by a pseudo-spectral method"

Hubert Griffith
5 years ago
Views:

1 Solving the shallow-water equations by a pseudo-spectral method Simon Chabot June 9th, 2015 Abstract In this short tutorial, you will use python to solve a non-linear system of equations using a pseudo-spectral method. The aim of this tutorial is that you use different tools provided by python libraries and get use to them. 1 Introduction to the set of equations The shallow water equations are a set of hyperbolic partial differential equations that describes the flow below a free surface, typically a water-wave, where the wave-length is much bigger than the water height. These equations are obtained from the Euler equations, that describes a ideal fluid 1. Let s consider an ideal and incompressible fluid. The fluid is bounded by a free surface and the seabed. We use a Cartesian coordinate system, such that the free surface y = 0 corresponds to the still water level. The horizontal independent variables are denoted by x = (x 1, x 2 ) and the upward vertical one by y. The free surface is given by y = η (x, t), and the bottom is defined by y = d (x, t). Finally, we denote by u the horizontal component of the fluid s velocity and by v the vertical one. These definitions are shown by the figure 1. Using these definitions, and taking the fluid density ρ = 1, the Euler s equations read : u + y v = 0 t u + (u ) u + (v y ) u = P (1) t v + (u ) v + (v y ) v = y P g 1 i.e. non-viscous 1 CC-by

Figure 1: domain definition where = ( x1, x2 ), and with the following boundary conditions 2 : v s = t η + u s η at y = η (x, t) (2) v b = t d + u b η at y = d (x, t) (3) These boundary conditions

2 Figure 1: domain definition where = ( x1, x2 ), and with the following boundary conditions 2 : v s = t η + u s η at y = η (x, t) (2) v b = t d + u b η at y = d (x, t) (3) These boundary conditions mean that the free surface and the seabed are impermeable. That is to say that material particles on the free surface (seabed) remain on the free surface (seabed). The difficulty in solving this problem is that you do not known where the free surface is. You have to solve a system, where the boundary is one of the unknown of the problem. You lake information 3. To close the problem, we can assume that the wave length is much bigger than the water height ; this is typically the case for tsunamis, roll-waves or flows in a canal. This implies that the movement of the flow is mainly horizontal, and therefore that you can average the equations (1) over the depth, from d to η to obtain two equations ; one for the water height h = η + d and one for the averaged horizontal velocity field ū. Where : ū (x, t) = 1 η(x,t) u (x, y, t) dy (4) η + d d(x,t) 2 ϕ s (x, t) ϕ (x, y = η, t) and ϕ b (x, t) ϕ (x, y = d, t) 3 five unknowns (the 3 components of the velocity, the pressure P, and the free surface η), and four equations. 2 CC-by

3 Then, neglecting the vertical component of the velocity v, you obtain the shallow water equations : t h + (hū) = 0 t ū + ( 1 + g (h d) ) (5) = 0 2ū2 Remark 1.1 The form (5) supposes that the flow is irrotational, i.e. ū = 0. If this assumption is false, the last equation becomes : ( t (hū) + hū ū + 1 ) 2 gh2 I = hg d 2 Three words on the pseudo-spectral method Let s take a generic PDE : t y + x f(y) = 0 y (x, t = 0) = y 0 (x) One can apply a spacial Fourier transform on this PDE. Then, in the Fourier space, it reads : t ŷ = ikf f (y)} (7) ŷ (k, t = 0) = ŷ 0 (k) where ˆϕ F ϕ} denotes the spatial Fourier transform of a quantity ϕ. Remark 2.1 We keep the notation F } when the treated term is non-linear and going back to the real space is needed for its computation. This method uses Fourier space to compute spatial derivative which become a simple product. The real space is used to compute the non-linear term. They cannot be computed in the Fourier space, because it would imply convolution operations which are to much expensive. Because the equation (7) tells us how varies ŷ in the time, ŷ can be numerically approximated by a Runge-Kutta method, for instance, at any time t. The solution in the real space is then recovered by applying the inverse Fourier transform F 1 }. 3 The python tasks Your task, should you choose to accept it, involves the completion of a python code that solves the shallow-water equations, with a moving seabed, using the pseudospectral method. You can start by downloading the code at this address : http: //huit.re/ljad-python-sw. You will find two files : (6) 3 CC-by

4 main.py : containing the code you will have to complete. conditions.py : containing the functions giving the initial condition and the seabed. The goal is to approximate the solution of the shallow-water equations, in 1D, by a spectral method : t h + x (hū) = 0 t ū + x ( 1 2ū2 + g (h d) ) = 0 (8) Here are your tasks : 1. You can start by having a look at the conditions.py file. You can launch it to see how the initial conditions are and how the seabed will evolve in time. You have nothing to change in this file for the moment. 2. Open the main.py. Basically, each time you see #XXX complete me at the end of a line, you have something to do. The first is at line 59. x = # XXX complete me You have to discretize the real space. The real space is periodic and must be zero-centered, its size is given by the variable length and you want N points. It is highly recommended to use the dedicated function linespace(start, stop, num=none, endpoint=false, retstep=false) for this purpose. Here some example of calls of this function : 0 >>> linspace (2.0, 3.0, num =5) array ([ 2., 2.25, 2.5, 2.75, 3. ]) 2 >>> linspace (2.0, 3.0, num =5, endpoint = False ) array ([ 2., 2.2, 2.4, 2.6, 2.8]) 4 >>> linspace (2.0, 3.0, num =5, retstep = True ) ( array ([ 2., 2.25, 2.5, 2.75, 3. ]), 0.25) 3. You now have to discretize the k vector of wavenumbers. The fundamental frequency is : k = 2π (9) L where L is the length of your domain. From k, we build the different harmonics as follow : p k if 0 p < N 2 k = 0 if p = N 2 = N Nyquist (10) (p N) k if p > N 2 Write the necessary python code to do this discretization (line 64). 4 CC-by

5 4. Complete, line 66, the function that compute the n th spatial derivative of a signal in the Fourier space. 66 def diff ( sig_hat, nth =1) : r """ compute the nth derivative of sig_ hat with respect to x, in Fourier 68 space """ 70 return # XXX complete me 5. Complete, line 72, the function yhat2hq(y_hat), that from the ŷ vector, in the Fourier space, returns h, u in the real space. Where the ŷ vector is built as follow : (ĥ ) ŷ = (11) û 6. Complete, line 79, the sw_rhs(y_hat) function that gives the right-hand-side member of the equation (8), in the Fourier space (of course). def sw_rhs (t, y_hat ): 80 r """ The RHS of Saint Venant equation 82 - y_ hat : the vector of dynamic variable, in Fourier space 84 Returns : - y_ hat_ t : the value of the time derivative of y_ hat 86 """ 88 # we recover h, u from y_ hat [0] and y_ hat [1] h, u = yhat2hu ( y_hat. reshape ((2, N))) 90 bottom = get_bottom (x, t=t) 92 h_ hat_ t = # XXX complete me u_ hat_ t = # XXX complete me 94 return array ([ h_hat_t, u_hat_t ]). flatten () 7. Have a look to the documentation of the ode object (here) and complete the last XXX complete me areas. 8. Cross the fingers, run the code, admire the result. 5 CC-by

6 4 More tasks if you are fast 1. Add an anti-aliasing treatment. Nonlinear computations come with aliasing errors. A zero-padding procedure should be added to avoid this phenomena. More information in [1] 2. Adapt the code to solve an other equation. For instance the Korteweg de Vries (KdV) equation : t η + c 0 x η + αη x η + β xxx η = 0 H (12) η(x, t = 0) = cosh 2 [ κ (x ct)] 2d where α = 2 3 g, β = d2 c 0, c d 6 0 = 3H gd, κ = d and c = c 0 ( 1 + H 2d). You can find at a proposal of solution. The main.py shows a solution of the different tasks that were proposed. An implementation of the anti-aliasing treatment by zero-padding is also suggested. Finally, the file kdv.py, based on the same code, solves the KdV equation. References [1] Holmås H., Clamond D. and Langtangen H.-P A pseudospectral Fourier method for a 1D incompressible two-fluid model. Int. J. Num. Meth. Fluids. 58, [2] Brice Eichwald, PhD thesis, Intégrateur exponentiel modifié pour la simulation des vagues non linéaires. University of Nice Sophia Antipolis [3] Simon Chabot, Master thesis, Modeling and numerical simulation of nonlinear water-waves. University of Nice Sophia Antipolis 6 CC-by

Homework 4A Due November 7th IN CLASS

CS207, Fall 2014 Systems Development for Computational Science Cris Cecka, Ray Jones Homework 4A Due November 7th IN CLASS Previously, we ve developed a quite robust Graph class to let us use Node and