An efficient splitting technique for two-layer shallow-water model - PowerPoint PPT Presentation

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS An efficient splitting technique for two-layer shallow-water model Christophe Berthon 1 , Françoise Foucher 1 and Tomas Morales 2 1-Laboratoire de mathématiques Jean Leray, CNRS and Université de Nantes, France 2-Dpto. de Matemáticas, Universidad de Córdoba, Spain HYP 2012 – Padova

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Outline 1. Introduction 2. One-layer system 3. Two-layer splitting system 4. Properties 5. Second-order scheme 6. Numerical results

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Two superposed layers on a non flat bottom h h 1 (t,x)+h 2 (t,x)+z(x) : surface ρ 1 > 0 h 1 (t,x) h 2 (t,x)+z(x) : interface u 1 (t,x) ρ 2 > ρ 1 h 2 (t,x) r = ρ 1 / ρ 2 z(x) : topography u 2 (t,x) x

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Equations We consider the following system of equations modelling the flow of two shallow water layers :  ∂ t h 1 + ∂ x ( h 1 u 1 ) = 0    1 + g ∂ t ( h 1 u 1 )+ ∂ x ( h 1 u 2 2 h 2  1 ) = − gh 1 ∂ x ( h 2 + z )  ∂ t h 2 + ∂ x ( h 2 u 2 ) = 0    2 + g ∂ t ( h 2 u 2 )+ ∂ x ( h 2 u 2 2 h 2  2 ) = − gh 2 ∂ x ( r h 1 + z )  This problem has been already adressed, we can cite C. Parés, M. Castro, J. Macías, F. Bouchut, T. Morales, T. Chacón, E. Fernández, J. García, E. Audusse, J. Sainte-Marie, . . .

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Motivations We are looking for a scheme which is expected to : • preserve h 1 ≥ 0, h 2 ≥ 0, • preserve the steady states at rest (well-balancing property) :  u 1 = u 2 = 0   h 1 + h 2 + z = cst  r h 1 + h 2 + z = cst  • be in agreement with real results especially if r approaches 1. Idea : apply what we did for the one-layer problem : C. Berthon, F. Foucher, Efficient well-balanced hydrostatic upwind schemes for shallow-water equations, JCP , 2012.

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS One-layer system � ∂ t h + ∂ x ( hu ) = 0 ∂ t ( hu )+ ∂ x ( hu 2 + g 2 h 2 ) = − hg ∂ x z We see that the steady states at rest are given by : � u = 0 h + z = cst In order to derive a well-balanced scheme, the idea is to introduce the free surface : H = h + z Then, using the associated fraction of water X = h H and writing : h 2 = h ( H − z ) = hH − hz = XH 2 − hz ,

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS New one-layer system we transform for weak solutions the initial system into : � ∂ t h + ∂ x ( XHu ) = 0 X ( Hu 2 + g 2 H 2 ) − g � � ∂ t ( hu )+ ∂ x = − gh ∂ x z 2 hz which can be written : ∂ t w + ∂ x ( Xf ( W )) = S ( H , h ) where  � � � � h H  w = and W = are state vectors, h ≥ 0 , H > 0 ,    hu Hu  � � � � Hu 0  f ( W ) = and S ( H , h ) =   Hu 2 + g g 2 H 2  2 ∂ x ( h ( H − h )) − gh ∂ x ( H − h ) 

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS New two-layer system We introduce : � H 1 = h 1 + h 2 + z and X 1 = h 1 H 1 H 2 = r h 1 + h 2 + z and X 2 = h 2 H 2 and state vectors : � � � � h j H j w j = and W j = , h j ≥ 0 , H j > 0 , j = 1 , 2 h j u j H j u j In order to transform the two-layers system, we write that � h 2 1 = X 1 H 1 ( H 1 − h 2 − z ) h 2 2 = X 2 H 2 ( H 2 − r h 1 − z )

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Two-layer splitting system Then we derive the new system :  ∂ t h 1 + ∂ x ( X 1 H 1 u 1 ) = 0    1 + g 1 ) − g � X 1 ( H 1 u 2 2 H 2 �  ∂ t ( h 1 u 1 )+ ∂ x 2 h 1 ( h 2 + z ) = − gh 1 ∂ x ( h 2 + z )  ∂ t h 2 + ∂ x ( X 2 H 2 u 2 ) = 0    2 + g 2 ) − g � X 2 ( H 2 u 2 2 H 2 �  ∂ t ( h 2 u 2 )+ ∂ x 2 h 2 ( r h 1 + z ) = − gh 2 ∂ x ( r h 1 + z )  Finally using that h 2 + z = H 1 − h 1 and r h 1 + z = H 2 − h 2 we turn the system into two similar systems with source terms ∂ t w j + ∂ x ( X j f ( W j )) = S ( H j , h j ) , j = 1 , 2

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Discretization of f ( W ) • Uniform mesh in space ∆ x = x i + 1 2 − x i − 1 2 • Time step ∆ t = t n + 1 − t n 2 ] at time t n : • Values in the cell [ x i − 1 2 , x i + 1 h n i , u n i , H n i , X n i , w n i , W n i , z i � f h ∆ x ( W n i , W n � i + 1 ) • f ∆ x ( W n i , W n i + 1 ) = computed by a numerical f hu ∆ x ( W n i , W n i + 1 ) scheme (HLLC, VFRoe, relaxation, ...) well-known for the homogeneous system ∂ t w + ∂ x f ( w ) = 0 : i − ∆ t w n + 1 = w n f ∆ x ( w n i , w n i + 1 ) − f ∆ x ( w n i − 1 , w n � � i ) i ∆ x

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Discretization of S ( H , h ) • We introduce upwind values on interfaces x i + 1 2 : � H n i , X n i si f h ( W n i , W n i + 1 ) > 0 2 = H i + 1 2 , X i + 1 H n i + 1 , X n i + 1 else H i + 1 f h ∆ x ( W n i , W n i + 1 ) 2 X i + 1 � � 2 � W n W n � i + 1 i x i − 1 x i + 1 2 2 • Then we define values h i + 1 2 by : 2 = H i + 1 h i + 1 2 X i + 1 2

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS One-layer scheme • We write the approximation : g 2 ∂ x h ( H − h ) − gh ∂ x ( H − h ) ≃ g � � 2 ( H i + 1 2 − h i + 1 2 ) − h i − 1 2 ( H i − 1 2 − h i − 1 2 ) h i + 1 2 − g � � 2 ( h i + 1 2 + h i − 1 2 ) ( H i + 1 2 − h i + 1 2 ) − ( H i − 1 2 − h i − 1 2 ) = g 2 ( h i + 1 2 − h i − 1 2 ) 2 H i − 1 2 H i + 1 = g 2 ( X i + 1 2 − X i − 1 2 ) 2 H i + 1 2 H i − 1 • We deduce the one-layer scheme : i − ∆ t w n + 1 = w n 2 f ∆ x ( W n i , W n 2 f ∆ x ( W n i − 1 , W n ∆ x ( X i + 1 i + 1 ) − X i − 1 i )) i � � ∆ t 0 + g 2 ( X i + 1 2 − X i − 1 2 ) ∆ x H i − 1 2 H i + 1 2

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS Two-layers scheme We recall the splitting two-layers system : ∂ t w j + ∂ x ( X j f ( W j )) = S ( H j , h j ) , j = 1 , 2 Now, we derive the following scheme to approximate this system, writing the previous one-layer scheme for each layer : j , i − ∆ t w n + 1 = w n 2 f ∆ x ( W n j , i , W n 2 f ∆ x ( W n j , i − 1 , W n ∆ x ( X j , i + 1 j , i + 1 ) − X j , i − 1 j , i )) j , i � � ∆ t 0 + g 2 ( X j , i + 1 2 − X j , i − 1 2 ) H j , i − 1 2 H j , i + 1 2 ∆ x

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS The scheme is well-balanced Let’s suppose u n j , i = 0, H n j , i = H j , where H j constant, j = 1 , 2.

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS The scheme is well-balanced Let’s suppose u n j , i = 0, H n j , i = H j , where H j constant, j = 1 , 2. • Since f ∆ x is consistent, we’ve got f ∆ x ( w , w ) = f ( w ) , so that � � 0 f ∆ x ( W n j , i , W n j , i + 1 ) = g 2 H 2 j

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS The scheme is well-balanced Let’s suppose u n j , i = 0, H n j , i = H j , where H j constant, j = 1 , 2. • Since f ∆ x is consistent, we’ve got f ∆ x ( w , w ) = f ( w ) , so that � � 0 f ∆ x ( W n j , i , W n j , i + 1 ) = g 2 H 2 j • Putting it in the scheme, we get : ( i ) h n + 1 = h n j , i and j , i j , i − ∆ t 2 ) g ( ii ) ( hu ) n + 1 = ( hu ) n 2 H 2 ∆ x ( X j , i + 1 2 − X j , i − 1 j , i j ∆ t + g 2 ( X j , i + 1 2 − X j , i − 1 2 ) ∆ x H j , i − 1 2 H j , i + 1 2

I NTRODUCTION O NE - LAYER SYSTEM T WO - LAYER SPLITTING SYSTEM P ROPERTIES S ECOND - ORDER SCHEME N UMERICAL RESULTS The scheme is well-balanced Let’s suppose u n j , i = 0, H n j , i = H j , where H j constant, j = 1 , 2. • Since f ∆ x is consistent, we’ve got f ∆ x ( w , w ) = f ( w ) , so that � � 0 f ∆ x ( W n j , i , W n j , i + 1 ) = g 2 H 2 j • Putting it in the scheme, we get : ( i ) h n + 1 = h n j , i and j , i j , i − ∆ t 2 ) g ( ii ) ( hu ) n + 1 = ( hu ) n 2 H 2 ∆ x ( X j , i + 1 2 − X j , i − 1 j , i j ∆ t + g 2 ( X j , i + 1 2 − X j , i − 1 2 ) ∆ x H j , i − 1 2 H j , i + 1 2 • Then from (i), we deduce : � H n + 1 = h n + 1 + h n + 1 + z i = h n 1 , i + h n 2 , i + z i = H n 1 , i = H 1 1 , i 1 , i 2 , i . H n + 1 = r h n + 1 + h n + 1 + z i = r h n 1 , i + h n 2 , i + z i = H n 2 , i = H 2 2 , i 1 , i 2 , i