Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Schwarz waveform relaxation algorithms : theory and applications Laurence HALPERN LAGA - Universit´ e Paris 13 DD17. July 2006 1 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Outline Introduction 1 The SWR algorithm for advection diffusion equation 2 Description Numerical experiments Back to the theoretical problem The two-dimensional wave equation 3 Dirichlet transmission conditions Optimized algorithms with overlap Conclusion und perspectives 4 2 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Coupling process Issues ♦ For a given problem, split the domain : domain decomposition. 3 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Coupling process Issues ♦ For a given problem, split the domain : domain decomposition. ♦ For a given problem, different numerical methods in different zones : FEM/FD, SM/FEM, AMR. 3 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Coupling process Issues ♦ For a given problem, split the domain : domain decomposition. ♦ For a given problem, different numerical methods in different zones : FEM/FD, SM/FEM, AMR. ♦ Couple two different models in different zones. 3 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Coupling process Issues ♦ For a given problem, split the domain : domain decomposition. ♦ For a given problem, different numerical methods in different zones : FEM/FD, SM/FEM, AMR. ♦ Couple two different models in different zones. ♦ Furthermore the codes can be on distant sites. 3 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives DDM for evolution problems Usual methods ⋄ Explicit + interpolation − > exchange of information every time step − > time consuming, possibly unstable for hyperbolic problems. 4 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives DDM for evolution problems Usual methods ⋄ Explicit + interpolation − > exchange of information every time step − > time consuming, possibly unstable for hyperbolic problems. ⋄ Implicit − > uniform time step. 4 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives DDM for evolution problems The goals ⋄ Different time and space steps in different subdomains, ⋄ Different models in different subdomains, ⋄ Different computing sites, ⋄ Easy to use, fast and cheap. 4 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives DDM for evolution problems The goals ⋄ Different time and space steps in different subdomains, ⋄ Different models in different subdomains, ⋄ Different computing sites, ⋄ Easy to use, fast and cheap. The means ⋄ Work on the PDE level, globally in time, ⋄ Split the space domain, ⋄ Use time windows, ⋄ Use the physical transmission conditions, transmit with improved (optimal/optimized) transmission conditions. ⋄ Then discretize separately.. 4 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives DDM for evolution problems The goals ⋄ Different time and space steps in different subdomains, ⋄ Different models in different subdomains, ⋄ Different computing sites, ⋄ Easy to use, fast and cheap. The means ⋄ Work on the PDE level, globally in time, ⋄ Split the space domain, ⋄ Use time windows, ⋄ Use the physical transmission conditions, transmit with improved (optimal/optimized) transmission conditions. ⋄ Then discretize separately.. Optimized Schwarz Waveform Relaxation 4 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Outline Introduction 1 The SWR algorithm for advection diffusion equation 2 Description Numerical experiments Back to the theoretical problem The two-dimensional wave equation 3 Dirichlet transmission conditions Optimized algorithms with overlap Conclusion und perspectives 4 5 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives Outline Introduction 1 The SWR algorithm for advection diffusion equation 2 Description Numerical experiments Back to the theoretical problem The two-dimensional wave equation 3 Dirichlet transmission conditions Optimized algorithms with overlap Conclusion und perspectives 4 6 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives The Schwarz algorithm L u := ∂ t u + a ∂ x u + ( b · ∇ ) u − ν ∆ u + cu in Ω × (0 , T ) ν > 0. t Ω 1 Γ 21 Γ 12 Ω 2 8 L u k +1 = f in Ω 1 × (0 , T ) 1 < u k +1 ( · , 0) = in Ω 1 u 0 1 B 1 u k +1 B 1 u k : = on Γ 12 × (0 , T ) 2 1 L u k +1 8 = f in Ω 2 × (0 , T ) 2 < u k +1 ( · , 0) = u 0 in Ω 2 2 B 2 u k +1 B 2 u k = on Γ 21 × (0 , T ) : 1 2 7 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives How to choose the transmission operators ? Transmission conditions B 1 u k +1 = B 1 u k B 2 u k +1 = B 2 u k 2 on Γ 12 × (0 , T ) , 1 on Γ 21 × (0 , T ) 1 2 8 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives How to choose the transmission operators ? Transmission conditions B 1 u k +1 B 2 u k +1 = B 1 u k = B 2 u k 2 on Γ 12 × (0 , T ) , 1 on Γ 21 × (0 , T ) 1 2 Classical Schwarz B j ≡ I AND overlap. 8 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives How to choose the transmission operators ? Classical Schwarz B j ≡ I AND overlap. 1D Numerical experiment a = 1 , ν = 0 . 2 , Ω = (0 , 6) , T = 2 . 5 , L = 0 . 08 . u 1 1 ( · , T ) , u 2 u 3 1 ( · , T ) , u 4 u 5 1 ( · , T ) , u 6 2 ( · , T ) 2 ( · , T ) 2 ( · , T ) 0.5 0.5 0.5 u u u 1 3 5 u 1 u 1 u 1 u 2 2 u 2 4 u 2 6 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 8 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives How to choose the transmission operators ? Transmission conditions B 1 u k +1 = B 1 u k B 2 u k +1 = B 2 u k 2 on Γ 12 × (0 , T ) , 1 on Γ 21 × (0 , T ) 1 2 Classical Schwarz B j ≡ I AND overlap. Optimized Schwarz Waveform relaxation B j ≡ absorbing boundary operator+optimization WITH OR WITHOUT overlap 8 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives How to choose the transmission operators ? Comparison u 1 1 ( · , T ) , u 2 u 3 1 ( · , T ) , u 4 u 5 1 ( · , T ) , u 6 2 ( · , T ) 2 ( · , T ) 2 ( · , T ) 0.5 0.5 0.5 u u u u 1 1 u 1 3 u 1 5 u 2 2 u 2 4 u 2 6 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 u 1 1 ( · , T ) , u 2 u 3 1 ( · , T ) , u 4 u 5 1 ( · , T ) , u 6 2 ( · , T ) 2 ( · , T ) 2 ( · , T ) 0.5 0.5 0.5 u u u 1 3 5 u 1 u 1 u 1 u 2 2 u 2 4 u 2 6 0.4 0.4 0.4 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6 8 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives The optimal SWR algorithm Ω 1 = ( −∞ , L ) × R n , Ω 2 = (0 , ∞ ) × R n . B j ≡ ∂ x + S j ( ∂ t , ∂ y ) a > 0, Fourier transform t ↔ ω, y ↔ κ S 1 ( i ω, i κ ) = δ 1 / 2 − a , S 2 ( i ω, i κ ) = δ 1 / 2 + a . 2 ν 2 ν δ ( ω, k ) = a 2 + 4 ν (( i ( ω + b · k ) + ν | k | 2 + c ) Convergence in 2 iterations (I if I subdomains). Two options : Use the optimal transmission condition (easier in 1D) 9 / 29
Introduction The SWR algorithm for advection diffusion equation The two-dimensional wave equation Conclusion und perspectives The optimal SWR algorithm Ω 1 = ( −∞ , L ) × R n , Ω 2 = (0 , ∞ ) × R n . B j ≡ ∂ x + S j ( ∂ t , ∂ y ) a > 0, Fourier transform t ↔ ω, y ↔ κ S 1 ( i ω, i κ ) = δ 1 / 2 − a , S 2 ( i ω, i κ ) = δ 1 / 2 + a . 2 ν 2 ν δ ( ω, k ) = a 2 + 4 ν (( i ( ω + b · k ) + ν | k | 2 + c ) Convergence in 2 iterations (I if I subdomains). Two options : Use the optimal transmission condition (easier in 1D) Approximate the optimal − > optimized transmission conditions 9 / 29
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