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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 1 1 Laboratoire de Mathématiques Jean Leray, Université de Nantes 2 Centre of Mathematics, Minho University 3 École Centrale de Nantes Monday, May 23rd, 2016

A well-balanced scheme for the shallow-water equations with topography and Manning friction Contents 1 Introduction 2 A well-balanced scheme 3 1D numerical experiments 4 Two-dimensional and high-order extensions 5 2D numerical experiments 6 Conclusion and perspectives

A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction 1 Introduction 2 A well-balanced scheme 3 1D numerical experiments 4 Two-dimensional and high-order extensions 5 2D numerical experiments 6 Conclusion and perspectives

A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction The shallow-water equations 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 η we can rewrite the equations as ∂ t W + ∂ x F ( W ) = S ( W ) η = 7/3 and g is the water surface gravitational constant h ( x, t ) u ( x, t ) k ≥ 0 is the so-called channel bottom Z ( x ) Manning coefficient: a higher k leads to a stronger x Manning friction 1 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction Steady state solutions 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 η 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 η 2 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction Introduction Objectives Objectives 1 derive a scheme that: is well-balanced for the shallow-water equations with friction and/or topography, i.e.: preservation of all steady states with k = 0 and Z � = cst preservation of all steady states with k � = 0 and Z = cst preservation of steady states with k � = 0 and Z � = cst preserves the non-negativity of the water height is able to deal with wet/dry transitions 2 provide two-dimensional and high-order extensions of this scheme, while keeping the above properties 3 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme 1 Introduction 2 A well-balanced scheme 3 1D numerical experiments 4 Two-dimensional and high-order extensions 5 2D numerical experiments 6 Conclusion and perspectives

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme Structure of the scheme The HLL scheme to approximate solutions of λ L λ R ∂ t W + ∂ x F ( W ) = 0 , we choose the HLL scheme (Harten, Lax, W HLL van Leer (1983)), which uses W R W L the approximate Riemann solver � W , to the right: the consistency condition (as per Harten and Lax) holds if: � ∆ x/ 2 � ∆ x/ 2 � x � � x � 1 1 � W ∆ t ; W L , W R dx = W R ∆ t ; W L , W R dx ∆ x ∆ x − ∆ x/ 2 − ∆ x/ 2 � � which gives W HLL = λ R W R − λ L W L − F ( W R ) − F ( W L ) h HLL = q HLL λ R − λ L λ R − λ L note that h HLL > 0 for | λ L | and | λ R | large enough 4 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme Structure of the scheme Modification of the HLL scheme to approximate solutions of λ L 0 λ R ∂ t W + ∂ x F ( W ) = S ( W ) , we W ∗ W ∗ R L use the following approximate Riemann solver (assuming W L W R λ L < 0 < λ R ): � h ∗ � � h ∗ � � 3 unknowns to determine: W ∗ L and W ∗ R L = R = ; q ∗ q ∗ Harten-Lax consistency gives us λ R h ∗ R − λ L h ∗ L = ( λ R − λ L ) h HLL S ∆ x q ∗ = q HLL + (with S = S ( W L , W R ) approximating λ R − λ L the mean of S ( W ) , to be determined) 5 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The full scheme for a general source term Determination of h ∗ L and h ∗ R assume that W L and W R define a steady state, i.e. satisfy the following discrete version of ∂ x F ( W ) = S ( W ) : � 1 � � h 2 � + g q 2 = S ∆ x 0 h 2 λ L 0 0 λ R W ∗ W ∗ L R − → W R W L W R W L for the steady state to be preserved, we need R = h R and q ∗ = q 0 W ∗ L = W L and W ∗ R = W R , i.e. h ∗ L = h L , h ∗ as soon as W L and W R define a steady state 6 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The full scheme for a general source term Determination of h ∗ L and h ∗ R two unknowns � we need two equations we have λ R h ∗ R − λ L h ∗ L = ( λ R − λ L ) h HLL we choose α ( h ∗ R − h ∗ L ) = S ∆ x q 2 where α = − ¯ + g 2( h L + h R ) , with ¯ q to be determined h L h R � using both relations, we obtain   λ R S ∆ x  h ∗  L = h HLL −  α ( λ R − λ L )  λ L S ∆ x   h ∗ R = h HLL −  α ( λ R − λ L ) 7 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The full scheme for a general source term Correction to ensure non-negative h ∗ L and h ∗ R however, these expressions of h ∗ L and h ∗ R do not guarantee that the intermediate heights are non-negative: instead, we use (see Audusse, Chalons, Ung (2014))  �� � � � � λ R S ∆ x 1 − λ R   h ∗ L = min h HLL − , h HLL   α ( λ R − λ L ) λ L + �� � � � �  λ L S ∆ x 1 − λ L   h ∗  R = min h HLL − , h HLL α ( λ R − λ L ) λ R + note that this cutoff does not interfere with: the consistency condition λ R h ∗ R − λ L h ∗ L = ( λ R − λ L ) h HLL the well-balance property, since it is not activated when W L and W R define a steady state 8 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The full scheme for a general source term Summary using a two-state approximate Riemann solver with � h ∗ � � h ∗ � intermediate states W ∗ L and W ∗ R L = R = given by q ∗ q ∗  S ∆ x  q ∗ = q HLL +     λ R − λ L    �� � � � �  λ R S ∆ x 1 − λ R h ∗ L = min h HLL − , h HLL α ( λ R − λ L )  λ L  +  �� � � � �    λ L S ∆ x 1 − λ L   h ∗ R = min h HLL − , h HLL  α ( λ R − λ L ) λ R + yields a scheme that is consistent, non-negativity-preserving and well-balanced; we now need to find S and α (i.e. ¯ q ) according to the source term definition 9 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The cases of the topography and friction source terms The topography source term we now consider S ( W ) = S t ( W ) = − gh∂ x Z : discrete smooth steady states are governed by � 1 � � h 2 � + g = S t ∆ x q 2 0 h 2 � 1 � q 2 0 + g [ h + Z ] = 0 h 2 2 we can exhibit an expression of q 2 0 and thus obtain [ h ] 3 S t = − g 2 h L h R [ Z ] g ∆ x + h L + h R 2∆ x h L + h R but when Z L = Z R , we have S t � = O (∆ x ) � loss of consistency with S t (see for instance Berthon, Chalons (2015)) 10 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The cases of the topography and friction source terms The topography source term instead, we set, for some constant C ,  [ h ] 3 S t = − g 2 h L h R [ Z ] g   c ∆ x +   h L + h R 2∆ x h L + h R �  h R − h L if | h R − h L | ≤ C ∆ x   [ h ] c =  sgn( h R − h L ) C ∆ x otherwise . Theorem: Well-balance for the topography source term If W L and W R define a steady state, i.e. verify � 1 � q 2 0 + g [ h + Z ] = 0 , 2 h 2 then we have W ∗ L = W L and W ∗ R = W R . q = q ∗ this result holds for any ¯ q : we choose ¯ 11 / 33

A well-balanced scheme for the shallow-water equations with topography and Manning friction A well-balanced scheme The cases of the topography and friction source terms The friction source term we consider, in this case, S ( W ) = S f ( W ) = − kq | q | h − η the average of S f we choose is S f = − k ˆ q | h − η , with q | ˆ ˆ q the harmonic mean of q L and q R (note that ˆ q = q 0 at the equilibrium), and h − η a well-chosen discretization of h − η , depending on h L and h R , and ensuring the well-balance property we determine h − η using the same technique (with µ 0 = sgn( q 0 ) ): � 1 � � h 2 � + g q 2 = − kµ 0 q 2 0 h − η ∆ x 0 2 h � h η − 1 � � h η +2 � − q 2 η + 2 = − kµ 0 q 2 η − 1 + g 0 ∆ x 0 12 / 33