Numerical dispersion and Linearized Saint-Venant Equations M. Ersoy - PowerPoint PPT Presentation

Numerical dispersion and Linearized Saint-Venant Equations M. Ersoy Basque Center for Applied Mathematics 11 November 2010

Outline of the talk Outline of the talk 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 2 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 3 / 32

Motivation Even if an equation is nondispersive, any discrete model of it will be dispersive[Tref] L-.N. Trefethen Group velocity in finite difference schemes. SIAM Review , 24(1), p. 113–136, 1982. M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 4 / 32

A simple example : The Saint-Venant equations, approximation of the gravity waves The Saint-Venant equations are the equations obtained by vertical averaging of the Navier-Stokes system and are widely used for geophysical fluids, river, lakes, . . . The most numerical schemes introduce spurious modes ; the most dangerous modes are the stationnary ones : ◮ discrete solution may not be unique ◮ lead to oscillating solutions A fourier analysis is necessary to understand the behavior of discrete modes. M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 5 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 6 / 32

Derivation it models the shallow water physical configuration where the movements are principally horizontal and ∂ z u = 0 . the fluid is assumed incompressible, i.e. ρ = cte the pressure is hydrostatic, i.e. ∂ z P = − ρg the characteristic length L and the height H are such that H ≪ L Under these assumptions, a vertical averaging of the Navier-Stokes equations gives : � ∂ t h + H∂ x u = 0 ∂ t u + g∂ x h = 0 M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 7 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 8 / 32

Let us consider the preceding system : � ∂ t u + g∂ x h = 0 ∂ t h + H∂ x u = 0 As the system is linear, we seek for a solution : � h ˜ h e i ( kx + wt ) = u e i ( kx + wt ) u = ˜ where ˜ h , ˜ u : the amplitude kx + wt : the phase with and where ◮ k = 2 π/λ : the wave number where λ : the wavelength ◮ w = 2 π/T : the frequence where T : the periode M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 9 / 32

Substituting u and h in the previous equations, we get : � w � � ˜ � u gk = O R 2 . ˜ Hk w h A non identically zero solution is provided when the determinant of this matrix is zero, then we have the following relation : � w = ± gHk. We deduce the phase velocity : v = w � k = ± gH the group velocity : v g = ∂w � ∂k = ± gH As, v = v g , the equations are evidently non dispersive. Even if an equation is nondispersive . . . [Tref] M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 10 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 11 / 32

. . . any discrete model of it will be dispersive [Tref] It means that for several numerical schemes, unfortunately, the previous relations are not respected and introduce non physical mode, called spurious mode, which have consequences on the behavior of the solution. M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 12 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 13 / 32

The equations � ∂ t u + g∂ x h = 0 ∂ t h + H∂ x u = 0 are approximated by a cell-centered finite difference scheme where unknowns u j ( t ) and h j ( t ) are the approximation of u ( t, x j ) and h ( t, x j ) :  ∂ t u j + g h j +1 − h j − 1  = 0  2∆ x ∂ t h j + H u j +1 − u j − 1  = 0  2∆ x x j = j ∆ x M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 14 / 32

Cell centered finite difference scheme Substituting u j and h j , � h j ˜ h e i ( kx j + wt ) = u e i ( kx j + wt ) u j = ˜ in the previous discrete equations, we get :  he ik ∆ x − e − ik ∆ x u + g ˜  iw ˜ = 0  2∆ x ue ik ∆ x − e − ik ∆ x iw ˜  h + H ˜ = 0  2∆ x or equivalently   g sin( k ∆ x ) � ˜ w � u ∆ x   = O R 2 . ˜  H sin( k ∆ x )  h w ∆ x M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 15 / 32

Numerical dispersion We obvisously get the following frequency w = v sin( k ∆ x ) ∆ x where v is the phase velocity of the continuous model. We deduce then that : the phase velocity for the discrete model is non constant, that is : v ∗ ( k ) = w ( k ) /k = v sin( k ∆ x ) k ∆ x the group velocity is g ( k ) = ∂w v ∗ ∂k = v cos( k ∆ x ) M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 16 / 32

v v 0 − v v^* v^*_g v v_g π π 0 k ∆ x k ∆ x The phase speed is zero when k ∆ x = π The group speed is negative on the interval k ∆ x ∈ [ π/ 2 , π ] Consequently → the energy is propagated in the opposite direction M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 17 / 32

For k ∆ x = π , we have w = 0 u e ikj ∆ x and ◮ u j = ˜ ◮ h j = ˜ h e ikj ∆ x . Solution is stationnary and do not propagate ! We have : ◮ v ∗ = 0 ◮ v ∗ g = − v and solution oscillates at each nodes. Moreover, it is easy to check that this solutions belong to the kernel of the discrete gradient. Consequently → it does not allow to get the uniqness of the discrete solution. This mode is called spurious mode. For k = 0 , we have again w = 0 but, in this case : ◮ v ∗ = v ◮ v ∗ g = v g This mode is called hydrostatic mode. M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 18 / 32

Whenever, spurious mode exists, the solution belong to the kernel of the discrete gradient with u j = 0 , ∀ j , that is : h j +1 = h j − 1 , whence we can rewrite as follows : ( h 1 , h 2 , h 3 , h 4 , . . . ) = h 1 (1 , 0 , 1 , 0 , . . . ) + h 2 (0 , 1 , 0 , 1 , . . . ) � �� � � �� � d 1 d 2 Furthemore, we recover the hydrostatic mode for d 1 + d 2 , i.e. h j = h j +1 , ∀ j . M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 19 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 20 / 32

Now, we consider the following discretisation :  ∂ t u j + g h j +1 / 2 − h j − 1 / 2  = 0  2∆ x ∂ t h j +1 / 2 + H u j +1 − u j  = 0  ∆ x M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 21 / 32

v v v^* 0 v^*_g v v_g π π 0 k ∆ x k ∆ x Following the previous computation, we get : ∀ kh ∈ [0 , π ] , w = v sin( k ∆ x , v ∗ = v sin( k ∆ x 2 ) 2 ) g = v cos( k ∆ x , c ∗ ) . ∆ x k ∆ x 2 2 2 M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 22 / 32

Contrary to the previous scheme, for k ∆ x = π , the phase speed is not zero and the energy propagates in the right direction. Consequently → the upwinding of the unknowns on the mesh avoid the apparition of the spurious mode. In this case, the dimension of the kernel of the discrete gradient is 1 , i.e. it contains only the hydrostatic mode. M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 23 / 32

Outline Outline 1 Introduction 2 The Saint-Venant equations 3 Dispersion relations for the Saint-Venant equations 4 Numerical approximation Cell-centered finite difference scheme “Upwinded “ finite difference scheme Finite Element method 5 Perspectives M. Ersoy (BCAM) Numerical dispersion and LSVEs 11 November 2010 24 / 32