Uhaina Uhaina A brief ief intr introduction duction
Code Uhaina Actors EPOC, IMB, INRIA, I3M Issues Simulation of tsunami propagation at regional scale. Simulation wave shoaling and wave transformation in the surf and swash zones. Wave overtopping and coastal flooding during storm event or tsunami. Operational and open source community code. Realisation New numerical method for the resolution of the Green-Naghdi equations. Parallel implementation using the finite elements library AeroSol. 1 / 17
Architecture and developement tools Using Aerosol finite elements library Uhaina AeroSol Pre-processing Computation Post-processing Tools C++ CMake - CTest GForge Doxygen Git / SVN ci.inria Spack Parallel test framework. 2 / 17
Code Uhaina Inputs Mesh Bathymetry Initial conditions Boundary conditions. Output Submersion areas Elevation field Breaking zone etc... 3 / 17
Feature AeroSol Finite elements, both continuous and discontinuous. Unstructured and hybrid meshes to handle complex geometry. Efficient parallel computation. Specific feature Well-balanced formulation. Wet/Dry zones handling Positivity preserving limiter [Zhang et al. 2010] Shock limiter [Guermond et al. 2011] Efficient discretization of dispersive terms [Lannes et al. 2015] 4 / 17
Aerosol library Residual computation System inversion PaMPA Redistribution Time iterator Memory allocation Communications Spatial discretization Residual computation Mesh / connectivity Loop over elements Local operations Assembly of the matrix System inversion Operators Linear solver Geometry MUMPS Quadrature formulas UMFPACK Resolution Finite elements functions PaStiX Model PETSc Numerical flux Boundary condition 5 / 17
Well balanced shallow water equations Hyperbolic system ∂ W ∂ t + ∇ · � F ( W , z b ) = � S ( W , z b ) � η � � � q � W = F ( W , z b ) = , 2 g ( η 2 − 2 η z b ) uq + 1 q � � 0 � S ( W , z b ) = − g η ∂ z b ∂ x 6 / 17
Discontinuous Galerkin method Hyperbolic system ∂ W ∂ t + ∇ · � F ( W , z b ) = � S ( W , z b ) Domain Ω of R d Mesh M h such that Ω = ∪ e ∈M h Ω e P k polynomial with : deg ( P k ) ≤ k Approximation space � � V k v ∈ [ L 2 (Ω e )] d + 2 , v | e ∈ [ P k (Ω e )] d + 2 , h = ∀ Ω e ∈ M h 7 / 17
Well balanced shallow water equations Hyperbolic system ∂ W ∂ t + ∇ · � F ( W , z b ) = � S ( W , z b ) ∀ v h ∈ V k h W h approximation of W in V k h � � � � ∀ v h ∈ V k � ∂ t W h v h d Ω − F ( W h , z h ) · ∇ v h d Ω h Ω e Ω e Ω e ∈M h Ω e ∈M h � � v − h h b ( W − h , W b h , z − h , z b + h , n ) d σ σ σ ∈ Γ � � ] H ( W − h , W + h , z − h , z + + [ [ v h ] h , n ) d σ σ σ ∈ Σ i � � � = S ( W h , z h ) v h d Ω Ω e Ω e ∈M h 8 / 17
Positivity preserving limiter Zhang & Shu quadrature for P 1 and P 2 cases h min = min h ( x ) x ∈ S Zhang If h min ≤ ǫ : U ( x ) = Θ( U ( x ) − ¯ ˜ U ) + ¯ U , ∀ x ∈ S Gauss (1) With � � ǫ − ¯ h Θ = min 1 , (2) h min − ¯ h Zhang, X., Xia, Y., & Shu, C. (2012). Maximum-Principle-Satisfying and Positivity-Preserving High Order Discontinuous Galerkin Schemes for Conservation Laws on Triangular Meshes. Journal of Scientific Computing. 9 / 17
Shock limiter � � E = 1 hu 2 + g ( η 2 + z 2 ) 2 D = ∂ E ∂ t + ∇ · f ( E ) ≤ 0 D is always zero except in discontinuities. � � 1 . 0 , δ x | D | ν = βν max min ǫ N With : µ max = δ x max | λ | Neglecting face integral : � � ν ∇ v h · ∇ W h d Ω Ω e Ω e ∈M h Guermond et al (2011), Entropy viscosity method for nonlinear conservation laws. Journal of Computational Physics. 10 / 17
Shock limiter Entropy viscosity on simple breaking wave. 11 / 17
Test case : Well balanced property Well balanced property and dry/wet interface 12 / 17
Test case : Subcritical flow over a bump 1 . 0 0 . 8 0 . 6 h 0 . 4 0 . 2 Bathymetry Water level Exact solution 0 . 0 0 5 10 15 20 x Convergence with second and third order spatial discretization 13 / 17
Test case : Carrier and Greenspan periodic 0 . 40 0 . 35 0 . 30 0 . 25 h 0 . 20 0 . 15 0 . 10 Bathymetry 0 . 05 Water level Exact solution 0 . 00 0 5 10 15 20 25 30 35 x Moving Wet/Dry interface with second order spatial discretization 14 / 17
Test case : Synolakis Results for Synolakis test case at different time with third order spatial discretization. 15 / 17
Demo Simple demonstration of parallel computation 16 / 17
Thank you for your attention. 17 / 17
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