A well-balanced scheme for the shallow-water equations with topography and bottom friction A well-balanced scheme for the shallow-water equations with topography and bottom friction C. Berthon 1 , S. Clain 2 , F. Foucher 1 , 3 , V. Michel-Dansac 1 1 Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes 2 Centre of Mathematics, Minho University 3 ´ Ecole Centrale de Nantes Friday, August 14th, 2015
A well-balanced scheme for the shallow-water equations with topography and bottom friction Contents 1 Introduction 2 A well-balanced scheme 3 Numerical experiments 4 Conclusion and perspectives
A well-balanced scheme for the shallow-water equations with topography and bottom friction Introduction 1 Introduction 2 A well-balanced scheme 3 Numerical experiments 4 Conclusion and perspectives
A well-balanced scheme for the shallow-water equations with topography and bottom 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 note we can rewrite the equations as ∂ t W + ∂ x F ( W ) = S ( W ) η = 7/3 and g is the gravitational constant k ≥ 0 is the so-called Manning coefficient: a higher k leads to a stronger bottom friction 1 / 26
A well-balanced scheme for the shallow-water equations with topography and bottom friction Introduction Steady states Steady states Definition: Steady states W is a steady state 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 states 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 / 26
A well-balanced scheme for the shallow-water equations with topography and bottom friction Introduction Steady states Smooth steady states for the friction source term assume a flat bottom ( Z = cst): the steady states are given by � q 2 � h + 1 = − kq 0 | q 0 | 0 2 gh 2 ∂ x h η assuming smooth steady states and integrating this steady equation between some x 0 ∈ R and x ∈ R yields the algebraic relation (with h = h ( x ) and h 0 = h ( x 0 )): − q 2 g � h η − 1 − h η − 1 � � h η +2 − h η +2 � 0 + + kq 0 | q 0 | ( x − x 0 ) = 0 0 0 η − 1 η + 2 but: � no global solution h ( x ) for all x ∈ R � for fixed x , we have 0, 1 or 2 solutions 3 / 26
A well-balanced scheme for the shallow-water equations with topography and bottom friction Introduction Steady states Smooth steady states for the friction source term zones and variations: analytical study with q 0 < 0 solution shape: Newton’s method 4 / 26
A well-balanced scheme for the shallow-water equations with topography and bottom friction Introduction Objectives Objectives 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 positivity of the water height is able to deal with wet/dry transitions 5 / 26
A well-balanced scheme for the shallow-water equations with topography and bottom friction A well-balanced scheme 1 Introduction 2 A well-balanced scheme 3 Numerical experiments 4 Conclusion and perspectives
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