Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Time-space adaptive numerical methods for multi-scale reaction waves simulation M. Duarte 1 M. Massot 1 S. Descombes 2 T. Dumont 3 V. Louvet 3 C. Tenaud 4 1 EM2C - Ecole Centrale Paris - France 2 Laboratoire J. A. Dieudonné - Nice - France 3 ICJ - Université Claude Bernard Lyon 1 - France 4 LIMSI - CNRS - France In collaboration with S. Candel 1 and F. Laurent 1 . SMAI 2011 - May 25th 2011 Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 1 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Outline Context and Motivation 1 Time/Space Adaptive Numerical Scheme 2 Numerical Illustration 3 Conclusions and Perspectives 4 Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 2 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Context and Motivation Predictive Simulations for industrial needs Strong evolution of Computer Power and Modeling Tools source: CERFACS. Main Numerical Goals: Resolution of the dynamics of reaction fronts. Reliable accuracy control based on mathematical aspects. New Numerical Strategies for Time/Space Multi-scale Fronts source: R. Vicquelin, EM2C Lab. Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 3 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Context and Motivation Predictive Simulations for industrial needs Strong evolution of Computer Power and Modeling Tools source: CERFACS. Main Numerical Goals: Resolution of the dynamics of reaction fronts. Reliable accuracy control based on mathematical aspects. New Numerical Strategies for Time/Space Multi-scale Fronts source: R. Vicquelin, EM2C Lab. Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 3 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Application Background Flames (dynamics, Plasma (repetitive pollutants, complex chemistry) discharges, streamers) source: Yale Univ. source: A. Bourdon EM2C Biochemical Engineering Chemical waves (migraines, Rolando’s (spiral waves, scroll waves) region, strokes) Time and Space Multi-scale Phenomena Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 4 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Numerical Strategies Coupled resolution of large Alternative methods: decoupling time scale spectrum time scale spectrum Time explicit methods Partitioning methods (high order in space) G. Warnecke et al , ... Implicit methods Operator Splitting techniques (adaptive time stepping) J.B. Bell et al , M.S. Day et al , H.N. Najm, O.M. Knio, ... SANDIA, ISTA/JAXA, CERFACS, ... AMR techniques for flames, detonations, reactive flows R. Deiterding, T. Ogawa et al , D.W. Schwendeman et al , J.W. Banks et al , S. Paolucci et al , K. Schneider et al , ... Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 5 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Numerical Strategies Coupled resolution of large Alternative methods: decoupling time scale spectrum time scale spectrum Time explicit methods Partitioning methods (high order in space) G. Warnecke et al , ... Implicit methods Operator Splitting techniques (adaptive time stepping) J.B. Bell et al , M.S. Day et al , H.N. Najm, O.M. Knio, ... SANDIA, ISTA/JAXA, CERFACS, ... AMR techniques for flames, detonations, reactive flows R. Deiterding, T. Ogawa et al , D.W. Schwendeman et al , J.W. Banks et al , S. Paolucci et al , K. Schneider et al , ... Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 5 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Numerical Strategies Coupled resolution of large Alternative methods: decoupling time scale spectrum time scale spectrum Time explicit methods Partitioning methods (high order in space) G. Warnecke et al , ... Implicit methods Operator Splitting techniques (adaptive time stepping) J.B. Bell et al , M.S. Day et al , H.N. Najm, O.M. Knio, ... SANDIA, ISTA/JAXA, CERFACS, ... AMR techniques for flames, detonations, reactive flows R. Deiterding, T. Ogawa et al , D.W. Schwendeman et al , J.W. Banks et al , S. Paolucci et al , K. Schneider et al , ... Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 5 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Time/Space Adaptive Numerical Strategy Time integration method Adaptive splitting time step ⇓ technique Strang Operator Splitting ֒ → dynamic accuracy control ֒ → based on Numerical based on NA Analysis for stiff PDEs Descombes et al 11 ֒ → splitting time steps larger ֒ → applied to instationary than fastest scales problems Descombes & Massot 04, Descombes et al 07-11 Duarte et al 11 Space adaptive multiresolution technique ֒ → based on wavelet transform and NA Harten 95, Cohen et al 01 ֒ → applied to stiff PDEs Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 6 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Time/Space Adaptive Numerical Strategy Time integration method Adaptive splitting time step ⇓ technique Strang Operator Splitting ֒ → dynamic accuracy control ֒ → based on Numerical based on NA Analysis for stiff PDEs Descombes et al 11 ֒ → splitting time steps larger ֒ → applied to instationary than fastest scales problems Descombes & Massot 04, Descombes et al 07-11 Duarte et al 11 Space adaptive multiresolution technique ֒ → based on wavelet transform and NA Harten 95, Cohen et al 01 ֒ → applied to stiff PDEs Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 6 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Strang Operator Splitting ∂ t u − ∂ x · F ( u ) − ∂ x · ( D ( u ) ∂ x u ) = Ω( u ) ∂ t u = Ω( u ) ∂ t u = ∂ x · ( D ( u ) ∂ x u ) ∂ t u = ∂ x · F ( u ) R ∆ t R → Radau5 (Hairer & Wanner 91) S ∆ t u 0 = R ∆ t / 2 D ∆ t / 2 C ∆ t D ∆ t / 2 R ∆ t / 2 u 0 D ∆ t D → ROCK4 (Abdulle 02) C ∆ t C → OSMP3 (Daru & Tenaud 04) Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 7 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Strang Operator Splitting ∂ t u − ∂ x · F ( u ) − ∂ x · ( D ( u ) ∂ x u ) = Ω( u ) ∂ t u = Ω( u ) ∂ t u = ∂ x · ( D ( u ) ∂ x u ) ∂ t u = ∂ x · F ( u ) R ∆ t R → Radau5 (Hairer & Wanner 91) S ∆ t u 0 = R ∆ t / 2 D ∆ t / 2 C ∆ t D ∆ t / 2 R ∆ t / 2 u 0 D ∆ t D → ROCK4 (Abdulle 02) C ∆ t C → OSMP3 (Daru & Tenaud 04) Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 7 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Time Adaptive Numerical Strategy Time integration method Adaptive splitting time step ⇓ technique Strang Operator Splitting ֒ → dynamic accuracy control ֒ → based on Numerical based on NA Analysis for stiff PDEs Descombes et al 11 ֒ → splitting time steps larger ֒ → applied to instationary than fastest scales problems Descombes & Massot 04, Descombes et al 07-11 Duarte et al 11 Space adaptive multiresolution technique ֒ → based on wavelet transform and NA Harten 95, Cohen et al 01 ֒ → applied to stiff PDEs Duarte et al 10, Tenaud MR CHORUS Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 8 / 29
Context and Motivation Time/Space Adaptive Numerical Scheme Numerical Illustration Conclusions and Perspectives Adaptive Splitting Time Step We define two time integration solvers: S ∆ t u 0 − T ∆ t u 0 = O (∆ t 3 ) = ⇒ Strang formula S ∆ t u 0 − T ∆ t u 0 = O (∆ t 2 ) � = ⇒ embedded Strang formula and considering � � � S ∆ t u 0 − � S ∆ t u 0 � ≈ O (∆ t 2 ) < η yields � η ∆ t new = ∆ t � � � S ∆ t u 0 − � � S ∆ t u 0 Duarte, Massot, Descombes, Dumont, Louvet, Tenaud Time-Space Adaptive Numerical Methods 9 / 29
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