A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme for the shallow-water equations with topography and Manning friction C. Berthon 1 , S. Clain 2 , F. Foucher 1,3 , V. Michel-Dansac 4 1 Laboratoire de Mathématiques Jean Leray, Université de Nantes 2 Centre of Mathematics, Minho University 3 École Centrale de Nantes 4 INSA Toulouse Monday, September 25th, 2017
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Several kinds of destructive geophysical flows Dam failure (Malpasset, France, 1959) Tsunami (T¯ ohoku, Japan, 2011) Flood (La Faute sur Mer, France, 2010) Mudslide (Madeira, Portugal, 2010) 1 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations The shallow-water equations and their source terms ∂ t h + ∂ x ( hu ) = 0 � � hu 2 + 1 = − gh∂ x Z − kq | q | 2 gh 2 ∂ t ( hu ) + ∂ x (with q = hu ) h 7 � 3 � h � We can rewrite the equations as ∂ t W + ∂ x F ( W ) = S ( W ) , with W = . q Z ( x ) is the known water surface topography k is the Manning h ( x, t ) u ( x, t ) coefficient channel bottom Z ( x ) g is the gravitational constant x we label the water discharge q := hu 2 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Steady state solutions Definition: Steady state solutions W is a steady state solution iff ∂ t W = 0 , i.e. ∂ x F ( W ) = S ( W ) . Taking ∂ t W = 0 in the shallow-water equations leads to ∂ x q = 0 � q 2 � h + 1 = − gh∂ x Z − kq | q | 2 gh 2 ∂ x . h 7 � 3 The steady state solutions are therefore given by q = cst = q 0 � q 2 � h + 1 = − gh∂ x Z − kq 0 | q 0 | 0 2 gh 2 ∂ x . h 7 � 3 3 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Topography steady state not captured in 1D The initial condition is at rest; water is injected through the left boundary. 4 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Topography steady state not captured in 1D The classical HLL numerical scheme converges towards a numerical steady state which does not correspond to the physical one. 4 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Topography steady state not captured in 1D The classical HLL numerical scheme converges towards a numerical steady state which does not correspond to the physical one. 4 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations A real-life simulation: the 2011 T¯ ohoku tsunami. The water is close to a steady state at rest far from the tsunami. 5 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Objectives Our goal is to derive a numerical method for the shallow-water model with topography and Manning friction that exactly preserves its stationary solutions on every mesh. To that end, we seek a numerical scheme that: 1 is well-balanced for the shallow-water equations with topography and friction, i.e. it exactly preserves and captures the steady states without having to solve the governing nonlinear differential equation; 2 preserves the non-negativity of the water height; 3 can be easily extended for other source terms of the shallow-water equations (e.g. breadth). 6 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction and motivations Contents 1 Brief introduction to Godunov-type schemes 2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension 4 Numerical simulations 5 Conclusion and perspectives
A well-balanced scheme for the shallow-water equations with topography and Manning friction Brief introduction to Godunov-type schemes 1 Brief introduction to Godunov-type schemes 2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension 4 Numerical simulations 5 Conclusion and perspectives
A well-balanced scheme for the shallow-water equations with topography and Manning friction Brief introduction to Godunov-type schemes Setting Objective: Approximate the solution W ( x, t ) of the system ∂ t W + ∂ x F ( W ) = S ( W ) , with suitable initial and boundary conditions. We partition [ a, b ] in cells , of volume ∆ x and of evenly spaced centers x i , and we define: x i − 1 2 and x i + 1 2 , the boundaries of the cell i ; W n i , an approximation of W ( x, t ) , constant in the cell i and � ∆ x/ 2 1 W ( x, t n ) dx . at time t n , which is defined as W n i = ∆ x ∆ x/ 2 W n W ( x, t ) i x x x i − 1 x i + 1 x i 2 2 7 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Brief introduction to Godunov-type schemes Using an approximate Riemann solver As a consequence, at time t n , we have a succession of Riemann problems (Cauchy problems with discontinuous initial data) at the interfaces between cells: ∂ t W + ∂ x F ( W ) = S ( W ) � W n i if x < x i + 1 W ( x, t n ) = 2 W n i +1 if x > x i + 1 2 W n W n i i +1 x i x i + 1 x i +1 2 For S ( W ) � = 0 , the exact solution to these Riemann problems is unknown or costly to compute � we require an approximation. 8 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Brief introduction to Godunov-type schemes Using an approximate Riemann solver We choose to use an approximate Riemann solver, as follows. λ L λ R i + 1 i + 1 2 2 W n i + 1 2 W n W n i i +1 x i + 1 2 W n 2 is an approximation of the interaction between W n i and i + 1 W n i +1 (i.e. of the solution to the Riemann problem), possibly made of several constant states separated by discontinuities. λ L 2 and λ R 2 are approximations of the largest wave speeds i + 1 i + 1 of the system. 9 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Brief introduction to Godunov-type schemes Godunov-type scheme (approximate Riemann solver) t W ∆ ( x, t n +1 ) � �� � t n +1 W n W n i − 1 i + 1 2 2 λ L λ R i + 1 i − 1 2 2 W n W n W n i − 1 i +1 i t n x x i − 1 x i + 1 x i 2 2 We define the time update as follows: � x i + 1 1 W n +1 W ∆ ( x, t n +1 ) dx. 2 := i ∆ x x i − 1 2 Since W n 2 and W n 2 are made of constant states, the above i − 1 i + 1 integral is easy to compute. 10 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Derivation of a generic first-order well-balanced scheme 1 Brief introduction to Godunov-type schemes 2 Derivation of a generic first-order well-balanced scheme 3 Second-order extension 4 Numerical simulations 5 Conclusion and perspectives
A well-balanced scheme for the shallow-water equations with topography and Manning friction Derivation of a generic first-order well-balanced scheme The HLL approximate Riemann solver t To approximate solutions of λ L λ R ∂ t W + ∂ x F ( W ) = 0 , the HLL approximate W HLL Riemann solver (Harten, Lax, van Leer W L W R (1983)) may be chosen; it is denoted by W ∆ and displayed on the right. x − ∆ x/ 2 0 ∆ x/ 2 The consistency condition (as per Harten and Lax) holds if: � ∆ x/ 2 � ∆ x/ 2 1 1 W ∆ (∆ t, x ; W L , W R ) dx = W R (∆ t, x ; W L , W R ) dx, ∆ x ∆ x − ∆ x/ 2 − ∆ x/ 2 � h HLL � which gives W HLL = λ R W R − λ L W L − F ( W R ) − F ( W L ) = . q HLL λ R − λ L λ R − λ L Note that, if h L > 0 and h R > 0 , then h HLL > 0 for | λ L | and | λ R | large enough. 11 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Derivation of a generic first-order well-balanced scheme Modification of the HLL approximate Riemann solver The shallow-water equations with the topography and friction source terms read as follows: ∂ t h + ∂ x q = 0 , � q 2 � h + 1 + gh∂ x Z + k q | q | 2 gh 2 ∂ t q + ∂ x = 0 . h 7 � 3 12 / 39
A well-balanced scheme for the shallow-water equations with topography and Manning friction Derivation of a generic first-order well-balanced scheme Modification of the HLL approximate Riemann solver With Y ( t, x ) := x , we can add the equations ∂ t Z = 0 and ∂ t Y = 0 , which correspond to the fixed geometry of the problem: ∂ t h + ∂ x q = 0 , � q 2 � h + 1 + gh∂ x Z + k q | q | 2 gh 2 ∂ t q + ∂ x ∂ x Y = 0 , h 7 � 3 ∂ t Z = 0 , ∂ t Y = 0 . 12 / 39
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