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A fully well-balanced scheme for the shallow-water model with topography and bottom friction A fully well-balanced scheme for the shallow-water model with topography and bottom friction C. Berthon 1 , V. Michel-Dansac 1 1 Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes Wednesday, December 10th, 2014

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Contents 1 Introduction 2 Steady states for the shallow-water model with friction 3 A fully well-balanced scheme Brief introduction to Godunov’s method Structure and properties of our scheme The full scheme 4 Numerical experiments 5 Conclusion and perspectives

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Introduction 1 Introduction 2 Steady states for the shallow-water model with friction 3 A fully well-balanced scheme Brief introduction to Godunov’s method Structure and properties of our scheme The full scheme 4 Numerical experiments 5 Conclusion and perspectives

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Introduction The Saint-Venant equations and their source terms  ∂ t h + ∂ x ( hu ) = 0  � � hu 2 + 1 − gh∂ x Z − kq | q | 2 gh 2 ∂ t ( hu ) + ∂ x =  h η where: h ( x, t ) > 0 is the water height u ( x, t ) is the water velocity q ( x, t ) is the water discharge, equal to hu Z ( x ) is the shape of the water bed η = 7/3 and g is the gravitational constant k is the so-called Manning coefficient: a higher k leads to a stronger bottom friction 1 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Introduction Steady states rewrite the shallow-water equations as ∂ t W + ∂ x F ( W ) = S ( W ) , with: � h   0 � hu � � W = , F ( W ) = hu 2 + 1 , S ( W ) = − gh∂ x Z − kq | q |   2 gh 2 hu h η Definition: Steady states W is a steady state iff ∂ t W = 0, i.e. ∂ x F ( W ) = S ( W ) 2 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Introduction Steady states 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 2 h + 1 � = − gh∂ x Z − kq 0 | q 0 | 0 2 gh 2 q = cst = q 0 and ∂ x h η (1) 3 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Introduction Objectives � q 2 � h + 1 = − gh∂ x Z − kq 0 | q 0 | 0 2 gh 2 ∂ x h η derive a fully well-balanced scheme for the shallow-water equations with friction and topography, i.e.: preservation of all steady states with friction and Z = cst preservation of the lake at rest steady state ( q = 0) preservation of all steady states with k = 0 and q � = 0 preservation of some steady states with k � = 0 and Z � = cst (not presented here) 4 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Steady states for the shallow-water model with friction 1 Introduction 2 Steady states for the shallow-water model with friction 3 A fully well-balanced scheme Brief introduction to Godunov’s method Structure and properties of our scheme The full scheme 4 Numerical experiments 5 Conclusion and perspectives

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Steady states for the shallow-water model with friction Obtaining the equations taking a flat bottom in (1), i.e. Z = cst, yields � q 2 h + 1 � = − kq 0 | q 0 | 0 2 gh 2 ∂ x , h η which we rewrite as: 1 h + g 2 ∂ x h 2 = − kq 0 | q 0 | − q 2 (2) 0 ∂ x h η for smooth solutions, we have q 2 g η − 1 ∂ x h η − 1 + η + 2 ∂ x h η +2 = − kq 0 | q 0 | 0 − (3) 5 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Steady states for the shallow-water model with friction Finding solutions integrating (3) between some x 0 and x yields − 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 (4) with h = h ( x ) and h 0 = h ( x 0 ) (4) is a nonlinear equation with unknown h for given x ; use Newton’s method to find h for any x , assuming q 0 < 0 6 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Steady states for the shallow-water model with friction Finding solutions zones and variations: analytical study solution shape: Newton’s method needed to solve (4) 7 / 36

A fully well-balanced scheme for the shallow-water model with topography and bottom friction Steady states for the shallow-water model with friction Other solutions? 1 use Rankine-Hugoniot relations to find admissible discontinuities linking two different increasing solutions, thus filling R by waves problem: we cannot fill ] − ∞ , x 0 [ 2 find admissible discontinuities linking any two different solutions, thus filling R same problem: we cannot fill ] − ∞ , x 0 [ 8 / 36