A conservative well-balanced hybrid SPH scheme for the shallow-water - PowerPoint PPT Presentation

A conservative well-balanced hybrid SPH scheme for the shallow-water model A conservative well-balanced hybrid SPH scheme for the shallow-water model C. Berthon 1 , M. de Leffe 2 , V. Michel-Dansac 1 1 Laboratoire de Math´ ematiques Jean Leray, Universit´ e de Nantes 2 HydrOcean Wednesday, May 14th, 2014

A conservative well-balanced hybrid SPH scheme for the shallow-water model Contents 1 General introduction to the SPH method Brief history Core of the SPH method Discretization of the SPH equations Application to shallow-water equations 2 A well-balanced scheme Goals A change of variables Conservative fix Main result 3 Numerical results Inconsistency and non-well-balancedness Validation of the scheme 4 Conclusion and perspectives

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method 1 General introduction to the SPH method Brief history Core of the SPH method Discretization of the SPH equations Application to shallow-water equations 2 A well-balanced scheme Goals A change of variables Conservative fix Main result 3 Numerical results Inconsistency and non-well-balancedness Validation of the scheme 4 Conclusion and perspectives

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Brief history 1 General introduction to the SPH method Brief history Core of the SPH method Discretization of the SPH equations Application to shallow-water equations 2 A well-balanced scheme Goals A change of variables Conservative fix Main result 3 Numerical results Inconsistency and non-well-balancedness Validation of the scheme 4 Conclusion and perspectives

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Brief history History of the SPH method 1977: Monaghan, Gingold, Lucy - particle method for astrophysics, coined the term SPH: “smoothed particle hydrodynamics” 1994: Monaghan - SPH for free-surface hydrodynamics 1998: Vila - SPH formulation using Riemann problems recent & ongoing work: multi-fluid SPH variable mesh viscous terms

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Core of the SPH method 1 General introduction to the SPH method Brief history Core of the SPH method Discretization of the SPH equations Application to shallow-water equations 2 A well-balanced scheme Goals A change of variables Conservative fix Main result 3 Numerical results Inconsistency and non-well-balancedness Validation of the scheme 4 Conclusion and perspectives

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Core of the SPH method The regularizing kernel General kernel expression � | r | � W ( r, h ) = C θ h θ , with h θ cut-off function C θ normalization constant properties of this kernel: 1 bell-shaped even function � of class C ∞ W ( r, h ) dr = 1 4 R 2 compact support K � W ′ ( r, h ) dr = 0 5 3 bell parameters: R r (position) and h (width)

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Core of the SPH method The particle approximation f ( x ) = ( f ∗ δ )( x ), with f : R �→ R and δ the Dirac distribution � = f ( y ) δ ( x − y ) dy R Π h ( f )( x ) = ( f ∗ W )( x ) � = f ( y ) W ( x − y, h ) dy ≃ f ( x ) K � Π h ( f ′ )( x ) = f ′ ( y ) W ( x − y, h ) dy K � f ( y )( W ( x − y, h )) ′ dy = [ f ( y ) W ( x − y, h )] ∂K − K � f ( y ) W ′ ( x − y, h ) dy ≃ f ′ ( x ) = K

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Core of the SPH method The particle approximation Accuracy of the continuous approximation second-order accuracy requires properties 4 and 5 (Mas-Gallic - Raviart, 1987; Monaghan, 1992): � W ( r, h ) dr = 1, i.e. Π h (1) = 1 R � W ′ ( r, h ) dr = 0, i.e. Π h (1 ′ ) = 0 R

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Core of the SPH method Different kernels cut-off for the cubic spline kernel (Monaghan, 1998): 4 − 6 q 2 + 3 q 3  if 0 ≤ q < 1  (2 − q ) 3 θ ( q ) = if 1 ≤ q < 2 0 otherwise  cut-off for the Wendland kernel (Wendland, 1995): � (2 − q ) 4 (1 + 2 q ) if 0 ≤ q < 2 θ ( q ) = 0 otherwise

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations 1 General introduction to the SPH method Brief history Core of the SPH method Discretization of the SPH equations Application to shallow-water equations 2 A well-balanced scheme Goals A change of variables Conservative fix Main result 3 Numerical results Inconsistency and non-well-balancedness Validation of the scheme 4 Conclusion and perspectives

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations The “mesh”

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations Discrete SPH equations A quadrature formula � � � f ( y ) dy ≃ ω ( x j ) f ( x j ) = ω j f j , where: R j ∈ Z j ∈ Z x j are the quadrature points, or particles ω j = ω ( x j ) are their volumes f j denotes f ( x j ) W ij = W ( x i − x j , h ) P : set of interacting particles x j close enough to particle x i

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations Discrete SPH equations Approximation of a function � Π h ( f )( x ) = f ( y ) W ( x − y, h ) dy K � Π h ( f ) i becomes = ω j f j W ij ≃ f i j ∈P Approximation of its derivative � f ( y ) W ′ ( x − y, h ) dy Π h ( f ′ )( x ) = K � ω j f j W ′ ij ≃ f ′ Π h ( f ′ ) i becomes = i j ∈P

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations Main issue: consistency Properties not verified in discrete form! the discrete analogues of 4 and 5 are generally not true: � � ω j W ′ ω j W ij � = 1 and ij � = 0 j ∈P j ∈P � loss of the consistency aim of the SPH methods: numerical resolution of PDE’s − → we need a suitable derivation operator

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations Main issue: consistency Weak formulation reinforce the derivation operator: Π h ( f ′ ) i − f i Π h (1 ′ ) i D h ( f ) i = � ω j ( f j − f i ) W ′ ij ≃ f ′ = i j ∈P � D h ( f ) i is exactly 0 for constant f yet another issue: this formulation is not conservative !

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations Main issue: consistency Conservativity the formulation D h ( f ) i will be conservative iff � ω i D h ( f ) i = 0 i ∈ Z ij , with W ′ odd � � � ω i ω j ( f j − f i ) W ′ ω i D h ( f ) i = i ∈ Z i ∈ Z j ∈ Z   � �  � = 0 ω j W ′ = − 2  ω i f i ij i ∈ Z j ∈ Z

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Discretization of the SPH equations Main issue: consistency Strong formulation � D ∗ h , adjoint of D h with respect to � f, g � h = ω i f i g i : i ∈ Z D ∗ h such that ∀ ( f, g ) , � D h ( f ) , g � = − � f, D ∗ h ( g ) � h � � D ∗ ω j ( g i + g j ) W ′ ij ≃ g ′ h ( g ) i = i j ∈P this strong formulation is conservative!

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Application to shallow-water equations 1 General introduction to the SPH method Brief history Core of the SPH method Discretization of the SPH equations Application to shallow-water equations 2 A well-balanced scheme Goals A change of variables Conservative fix Main result 3 Numerical results Inconsistency and non-well-balancedness Validation of the scheme 4 Conclusion and perspectives

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Application to shallow-water equations Hybridization SPH - Finite Volumes SPH Finite Volumes s j x i x j j i × ← → × × − → × F i + F j F ij � ω j ( F i + F j ) W ′ � ∂ x F i ≃ ∂ x F i ≃ s j F ij ij j ∈P j ∈ γ ( i ) ( F ij : any conservative FV flux) Hybrid formulation FV-SPH � ω j 2 F ij W ′ ∂ x F i ≃ ij j ∈P

A conservative well-balanced hybrid SPH scheme for the shallow-water model General introduction to the SPH method Application to shallow-water equations Summary SPH-FV approximation of a PDE consider a general PDE of the form ∂ t Φ + ∂ x F (Φ) = S (Φ) � 2 ω j F ij W ′ SPH approximation of ∂ x F (Φ): ij j ∈P F ij : any conservative FV flux from particle i to particle j choice to make to discretize ∂ t Φ and S (Φ) conservative flux discretization