Development of high-order well-balanced schemes for geophysical flows Development of high-order well-balanced schemes for geophysical flows Victor Michel-Dansac Thursday, September 29th, 2016 PhD advisors: Christophe Berthon and Françoise Foucher PhD reviewers: Manuel J. Castro-Díaz and Jean-Paul Vila Defense examiners: Christophe Chalons, Stéphane Clain and Fabien Marche
Development of high-order well-balanced schemes for geophysical flows Contents 1 Introduction and motivations 2 A well-balanced scheme 3 1D numerical experiments 4 Two-dimensional and high-order extensions 5 2D numerical experiments 6 Conclusion and perspectives
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations 1 Introduction and motivations 2 A well-balanced scheme 3 1D numerical experiments 4 Two-dimensional and high-order extensions 5 2D numerical experiments 6 Conclusion and perspectives
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations Geophysical flows 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 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations 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 � h � We can rewrite the equations as ∂ t W + ∂ x F ( W ) = S ( W ) , with W = . q 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 Manning coefficient: Z ( x ) a higher k leads to a stronger Manning x friction 2 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations 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 | 2 gh 2 0 ∂ x . h η 3 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations Steady state solutions 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 state solutions and integrating this relation between some x 0 ∈ R and x ∈ R yields (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 next step : study of the above nonlinear equation, denoted by χ ( h ; x, h 0 , x 0 , q 0 ) = 0 4 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations Steady state solutions Steady states for the friction source term 1 We show that ∂χ ∂h ( h ; x, h 0 , x 0 , q 0 ) < 0 if and only if h < h c , where � q 2 � 1 � 3 0 h c = . g As a consequence, χ ( h ) is: decreasing for h < h c ; increasing for h > h c . 2 In the context of a steady state solution, the Froude number is defined, using the sound speed c = √ gh , by: q 2 Fr ( h ) = u 0 c = � gh 3 . Therefore, Fr ( h c ) = 1 and the steady state is: supercritical if h < h c ; subcritical if h > h c . 5 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations Steady state solutions Steady states for the friction source term, assuming q 0 < 0 h < h c h > h c supercritical subcritical solution solution h c Sketches of χ ( h ; x ) for h ∈ [0 , 0 . 41] , for different values of x , and for h c = 0 . 25 . We are interested in the solutions of χ ( h ) = 0 . 6 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations Steady state solutions Steady states for the friction source term, assuming q 0 < 0 h(x) blue curve: subcritical solution; red curve: supercritical 7 / 46
Development of high-order well-balanced schemes for geophysical flows Introduction and motivations 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, where the friction source term is stiff. 2 Provide two-dimensional and high-order extensions of this scheme, while keeping the above properties. 8 / 46
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme 1 Introduction and motivations 2 A well-balanced scheme 3 1D numerical experiments 4 Two-dimensional and high-order extensions 5 2D numerical experiments 6 Conclusion and perspectives
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme Non-exhaustive state of the art Non-exhaustive state of the art Well-balanced schemes for the shallow-water equations introduction of the well-balance property: Vazquez (1994), Greenberg-LeRoux (1996) Bermudez- preservation of the lake at rest: Audusse et al. (2004), Berthon-Foucher (2012), Audusse et al. (2015) 1D fully well-balanced schemes: Gosse (2000), Castro et al. (2007), Fjordholm et al. (2011), Xing et al. (2011), Berthon-Chalons (2016) 1D high-order schemes that preserve steady states: Castro et al. (2006), Castro Díaz et al. (2013) 2D schemes preserving the lake at rest on unstructured meshes: Duran et al. (2013), Clain-Figueiredo (2014) for the friction source term: Liang-Marche (2009), Chertock et al. (2015) 9 / 46
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme Structure of the scheme The HLL scheme t To approximate solutions of λ L λ R ∂ t W + ∂ x F ( W ) = 0 , the HLL scheme W HLL (Harten, Lax, van Leer (1983)) may be W R W L chosen; it uses the approximate Riemann solver � W , 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. 10 / 46
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme Structure of the scheme Modification of the HLL scheme With Y ( t, x ) = x , we rewrite the shallow-water equations with a generic source term S as follows: ∂ t h + ∂ x q = 0 , � q 2 � h + 1 2 gh 2 ∂ t q + ∂ x − S∂ x Y = 0 , ∂ t Y = 0 . The equation ∂ t Y = 0 induces a stationary wave associated to the source term; we also note that q is a Riemann invariant for this wave. λ L 0 λ R To approximate solutions of W ∗ W ∗ ∂ t W + ∂ x F ( W ) = S ( W ) , we thus use L R the approximate Riemann solver W L W R displayed on the right (assuming λ L < 0 < λ R ). 11 / 46
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme Structure of the scheme Modification of the HLL scheme We now wish to apply the Harten-Lax consistency condition to ∂ t W + ∂ x F ( W ) = S ( W ) . Recall this condition: � ∆ 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 � ∆ x/ 2 1 � first step : compute W (∆ t, x ) dx (straightforward) ∆ x − ∆ x/ 2 � ∆ x/ 2 1 W (∆ t, x ) dx = W L + W R ∆ t ∆ t � ∆ x ( W R − W ∗ ∆ x ( W L − W ∗ − λ R R ) + λ L L ) ∆ x 2 − ∆ x/ 2 � ∆ x/ 2 1 second step : compute W R (∆ t, x ) dx ∆ x − ∆ x/ 2 12 / 46
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme Structure of the scheme Modification of the HLL scheme � ∆ t � ∆ x/ 2 1 1 ( ∂ t W R + ∂ x F ( W R ) ) dx dt = 0 ∆ t ∆ x 0 − ∆ x/ 2 13 / 46
Development of high-order well-balanced schemes for geophysical flows A well-balanced scheme Structure of the scheme Modification of the HLL scheme � ∆ t � ∆ x/ 2 1 1 ( ∂ t W R + ∂ x F ( W R ) ) dx dt = 0 ∆ t ∆ x 0 − ∆ x/ 2 �� ∆ x/ 2 � � ∆ x/ 2 0 = 1 1 W R (∆ t, x ) dx − W R (0 , x ) dx + ∆ t ∆ x − ∆ x/ 2 − ∆ x/ 2 �� ∆ t � � � � ∆ t � � 1 1 t, − ∆ x t, ∆ x F ( W R ) dt − F ( W R ) dt ∆ t ∆ x 2 2 0 0 t λ L 0 λ R ∆ t W ∗ W ∗ L R W L W R x 0 − ∆ x/ 2 0 ∆ x/ 2 13 / 46
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