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Techniques - Institut Camille Jordan
Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Operator Splitting Techniques for Multi-Scale Reacting Waves And Application to Low Mach Number Flames with Complex Chemistry Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results S. Descombes 2 T. Dumont 1 , J. Beaulaurier 3 F. Laurent-Nègre 3 V. Louvet 1 M. Massot 3 1 Camille 2 J. Jordan Institute - Claude Bernard Lyon 1 University - France Dieudonné Laboratory, Nice-Sophia Antipolis University - France 3 EM2C Laboratory - Ecole Centrale Paris - France Conclusion 12th International Conference on Numerical Combustion Outline Splitting Techniques for Low Mach Number Flames 1 Context and Motivation Application Background Numerical and computational difficulties Numerical Strategies 2 Operator splitting and stiffness Aim of splitting methods Standard numerical analysis of operator splitting Stiffness comes into play 3 Laminar Premixed Counter-Flow flames simulations Premixed counter-flow flames model Application of splitting method : chemical correction Numerical Results 4 Conclusion Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Outline Splitting Techniques for Low Mach Number Flames 1 Context and Motivation Application Background Numerical and computational difficulties Numerical Strategies 2 Operator splitting and stiffness Aim of splitting methods Standard numerical analysis of operator splitting Stiffness comes into play 3 Laminar Premixed Counter-Flow flames simulations Premixed counter-flow flames model Application of splitting method : chemical correction Numerical Results 4 Conclusion Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Application Background : Numerical simulation of unsteady reactive phenomena Splitting Techniques for Low Mach Number Flames Flames (Instabilities, dynamics, pollutants) Context Chemical “waves” (spiral waves, scroll waves, ...), example in medicine Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Dynamics involving many species and reactions : complex chemistry ,→ ,→ Solution with high gradients Time and space multi-scales problems Application Background : Numerical simulation of unsteady reactive phenomena Splitting Techniques for Low Mach Number Flames Flames (Instabilities, dynamics, pollutants) Context Instationnarities (S. Ducruix, S. Candel, N. Tran, EM2C) Background Chemical “waves” (spiral waves, scroll waves, ...), example in medicine Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Dynamics involving many species and reactions : complex chemistry ,→ ,→ Solution with high gradients Time and space multi-scales problems Application Background : Numerical simulation of unsteady reactive phenomena Splitting Techniques for Low Mach Number Flames Flames (Instabilities, dynamics, pollutants) Chemical “waves” (spiral waves, scroll waves, ...), example in medicine Context Background Numerical Difficulties Strategies Potassium ions during stroke (T. Dumont, ICJ) Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Dynamics involving many species and reactions : complex chemistry ,→ ,→ Solution with high gradients Time and space multi-scales problems Application Background : Numerical simulation of unsteady reactive phenomena Splitting Techniques for Low Mach Number Flames Flames (Instabilities, dynamics, pollutants) Context Chemical “waves” (spiral waves, scroll waves, ...), example in medicine Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Dynamics involving many species and reactions : complex chemistry ,→ ,→ Solution with high gradients Time and space multi-scales problems Numerical and computational difficulties Splitting Techniques for Low Mach Number Flames Context Difficulties to be treated : Background Numerical Difficulties Strategies Splitting Spatial Stiffness Resolving a large scale spectrum Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Need for dedicated solver able to solve a large scale spectrum and large spatial gradients on multi-dimensional configuration Numerical and computational difficulties Splitting Techniques for Low Mach Number Flames Difficulties to be treated : Spatial Stiffness Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Resolving a large scale spectrum ,→ Need for dedicated solver able to solve a large scale Numerical and computational difficulties Splitting Techniques for Low Mach Number Flames Difficulties to be treated : Context Background Numerical Difficulties Strategies Splitting Aim of splitting Spatial Stiffness Resolving a large scale spectrum Pagoda Flame (D. Durox, S. Ducruix, S. Candel) Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Need for dedicated solver able to solve a large scale spectrum and large spatial gradients on multi-dimensional configuration Numerical and computational difficulties Splitting Techniques for Low Mach Number Flames Context Difficulties to be treated : Background Numerical Difficulties Strategies Splitting Spatial Stiffness Resolving a large scale spectrum Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Need for dedicated solver able to solve a large scale spectrum and large spatial gradients on multi-dimensional configuration Strategies for Stiffness Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Stiffness due to Large spectrum of temporal scales in chemical sources Large spatial gradients of the solutions Resolving Stiffness Explicit methods in time (high order in space) Fully implicit methods with adaptative time stepping Semi-implicit methods (source explicit in time) Method of lines coupled to a stiff ODE solver The computational cost and memory requirement have suggested the study of alternative methods : decoupling Diffusion-Convection and Reaction =⇒ operator splitting “Divide and conquer” Strategies for Stiffness Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Stiffness due to Large spectrum of temporal scales in chemical sources Large spatial gradients of the solutions Resolving Stiffness Explicit methods in time (high order in space) Fully implicit methods with adaptative time stepping Semi-implicit methods (source explicit in time) Method of lines coupled to a stiff ODE solver The computational cost and memory requirement have suggested the study of alternative methods : decoupling Diffusion-Convection and Reaction =⇒ operator splitting “Divide and conquer” Strategies for Stiffness Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Stiffness due to Large spectrum of temporal scales in chemical sources Large spatial gradients of the solutions Resolving Stiffness Explicit methods in time (high order in space) Fully implicit methods with adaptative time stepping Semi-implicit methods (source explicit in time) Method of lines coupled to a stiff ODE solver The computational cost and memory requirement have suggested the study of alternative methods : decoupling Diffusion-Convection and Reaction =⇒ operator splitting “Divide and conquer” Strategies for Stiffness Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Stiffness due to Large spectrum of temporal scales in chemical sources Large spatial gradients of the solutions Resolving Stiffness Explicit methods in time (high order in space) Fully implicit methods with adaptative time stepping Semi-implicit methods (source explicit in time) Method of lines coupled to a stiff ODE solver The computational cost and memory requirement have suggested the study of alternative methods : decoupling Diffusion-Convection and Reaction =⇒ operator splitting “Divide and conquer” Outline Splitting Techniques for Low Mach Number Flames 1 Context and Motivation Application Background Numerical and computational difficulties Numerical Strategies 2 Operator splitting and stiffness Aim of splitting methods Standard numerical analysis of operator splitting Stiffness comes into play 3 Laminar Premixed Counter-Flow flames simulations Premixed counter-flow flames model Application of splitting method : chemical correction Numerical Results 4 Conclusion Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Aim of splitting methods Splitting Techniques for Low Mach Number Flames Operator splitting : separate convection-diffusion and reaction High order methods exist Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Allow the use of dedicated solver for each step No need to identify fast scale Yield lower storage and optimization capability ,→ This methods are already be used for flame computation. Najm, Wyckoff, Knio, Journal of Computational Physics, 1999 Day, Bell, Combust. Theory Modelling, 2000 ,→ This work is dedicated to numerical analysis of such methods on stiff systems. Aim of splitting methods Splitting Techniques for Low Mach Number Flames Operator splitting : separate convection-diffusion and reaction High order methods exist Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Allow the use of dedicated solver for each step No need to identify fast scale Yield lower storage and optimization capability ,→ This methods are already be used for flame computation. Najm, Wyckoff, Knio, Journal of Computational Physics, 1999 Day, Bell, Combust. Theory Modelling, 2000 ,→ This work is dedicated to numerical analysis of such methods on stiff systems. Aim of splitting methods Splitting Techniques for Low Mach Number Flames Operator splitting : separate convection-diffusion and reaction High order methods exist Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Allow the use of dedicated solver for each step No need to identify fast scale Yield lower storage and optimization capability ,→ This methods are already be used for flame computation. Najm, Wyckoff, Knio, Journal of Computational Physics, 1999 Day, Bell, Combust. Theory Modelling, 2000 ,→ This work is dedicated to numerical analysis of such methods on stiff systems. Basis of operator splitting Splitting Techniques for Low Mach Number Flames Context Background Reaction-diffusion-convection system to be solved (t : time interval) ∂t U − ∆U + ∇U = Ω(U) t U(t) = T U0 U(0) = U0 Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Two elementary “blocks”. Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ∂t V − ∆V + ∇V = 0 V (0) = V0 ∂t W = Ω(W ) t W (t) = Y W0 W (0) = W0 t V (t) = X V0 Basis of operator splitting Splitting Techniques for Low Mach Number Flames Context Background Reaction-diffusion-convection system to be solved (t : time interval) ∂t U − ∆U + ∇U = Ω(U) t U(t) = T U0 U(0) = U0 Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Two elementary “blocks”. Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ∂t V − ∆V + ∇V = 0 V (0) = V0 ∂t W = Ω(W ) t W (t) = Y W0 W (0) = W0 t V (t) = X V0 Basis of operator splitting II Splitting Techniques for Low Mach Number Flames First order methods, Lie formalism : Context Background Lie Formulae. Numerical Difficulties Strategies Splitting Aim of splitting Lt1 U0 = X t Y t U0 Lt1 − T t = O(t 2 ), Lt2 U0 = Y t X t U0 Lt2 − T t = O(t 2 ), Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Basis of operator splitting III Splitting Techniques for Low Mach Number Flames Context Background Second order methods, Strang formalism : Strang Formulae. Numerical Difficulties Strategies Splitting Aim of splitting S1t U0 = Y t/2 X t Y t/2 U0 S1t − T t = O(t 3 ), S2t U0 = X t/2 Y t X t/2 U0 S2t − T t = O(t 3 ), Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion But stiffness involves order reduction ! Basis of operator splitting III Splitting Techniques for Low Mach Number Flames Context Background Second order methods, Strang formalism : Strang Formulae. Numerical Difficulties Strategies Splitting Aim of splitting S1t U0 = Y t/2 X t Y t/2 U0 S1t − T t = O(t 3 ), S2t U0 = X t/2 Y t X t/2 U0 S2t − T t = O(t 3 ), Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion But stiffness involves order reduction ! Stiffness comes into play Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion order reduction due to fast scale (see reference Descombes, Massot, Numerische Mathematik, 2004) order reduction due to high spatial gradient (see reference Descombes, Dumont, Louvet, Massot, International Journal of Computer Mathematics, 2007) Stiffness comes into play Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Error behaviour. In case of fast scale, the local error for Lie and Strang formalism behave like : √ ! CL0 t 2 CL1 t t kEL (t)U0 k2 ≤ + √ kU0 k2 2 3 2e √ √ (CS0 + 2CS1 ) t 3 CS2 t 2 t CS3 α t t √ kES (t)U0 k2 ≤ + + 12 4 15 2e In case of high spatial gradient, for Lie√ Formalism, there exists an explicit constant θ > 0 depending on k V U0 k2 such that for t ≤ θ, 2 kE √L (t)U0 k2 behaves like t and for t ≥ θ, kEL (t)U0 k2 behaves like t t. Outline Splitting Techniques for Low Mach Number Flames 1 Context and Motivation Application Background Numerical and computational difficulties Numerical Strategies 2 Operator splitting and stiffness Aim of splitting methods Standard numerical analysis of operator splitting Stiffness comes into play 3 Laminar Premixed Counter-Flow flames simulations Premixed counter-flow flames model Application of splitting method : chemical correction Numerical Results 4 Conclusion Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Experimental setup and discretization Splitting Techniques for Low Mach Number Flames fuel + oxidant z Context Background Numerical Difficulties stagnation plane Strategies Splitting Aim of splitting r Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations flame front d Modelisation Splitting : chemical correction Numerical results Conclusion fuel + oxidant Assumptions Splitting Techniques for Low Mach Number Flames Complex chemistry Context 2D axisymetrical configuration Similarity assumption : Detailed transport Background Numerical Difficulties Strategies ρ = ρ(z, t) Splitting T = T (z, t) Yk = Yk (z, t) ur = rU(z, t) p̃ = −J(t) Aim of splitting ρuz = V (z, t) Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations ,→ r2 + p̂(z, t) 2 Experimental studies in EM2C laboratory, Ecole Centrale Paris (T. Schuller, P. Duchaine) Modelisation Splitting : chemical correction Numerical results Conclusion See N. Darabiha, “Transient behaviour of laminar counterflow hydrogen-air diffusion flames with complex chemistry”, Combust. Sci. and Tech., 1992 Assumptions Splitting Techniques for Low Mach Number Flames Complex chemistry Context 2D axisymetrical configuration Similarity assumption : Detailed transport Background Numerical Difficulties Strategies ρ = ρ(z, t) Splitting T = T (z, t) Yk = Yk (z, t) ur = rU(z, t) p̃ = −J(t) Aim of splitting ρuz = V (z, t) Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations ,→ r2 + p̂(z, t) 2 Experimental studies in EM2C laboratory, Ecole Centrale Paris (T. Schuller, P. Duchaine) Modelisation Splitting : chemical correction Numerical results Conclusion See N. Darabiha, “Transient behaviour of laminar counterflow hydrogen-air diffusion flames with complex chemistry”, Combust. Sci. and Tech., 1992 Isobaric Flame Equations Splitting Techniques for Low Mach Number Flames Context System of equation. ρcp Background ∂T ∂ ∂T + cp V − ∂t ∂z ∂z „ λ ∂T ∂z « =− Strategies − Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion nS X k =1 Aim of splitting ρ hk mk ωk k =1 Numerical Difficulties Splitting nS X ρYk cp,k Vz,k ∂T , ∂z ´ ∂Yk ∂Yk ∂ ` +V + ρYk Vz,k = mk ωk , ∂t ∂z ∂z ∂J = 0, ∂z „ « ∂U ∂U ∂ ∂U + ρU 2 + V =J+ µ , ρ ∂t ∂z ∂z ∂z ∂ρ ∂V + 2ρU + = 0. ∂t ∂z Application of splitting method : chemical correction I Splitting Techniques for Low Mach Number Flames ,→ Eliminating chemical terms : important variation of the temporal derivative of the density ρ, instant perturbation of the velocity field. ,→ Necessity to take into account the chemical contribution in the diffusion-convection step Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion n S X ∂T ρcp =− hk mk ωk ∂t k =1 ∂Yk ρ = mk ωk ∂t Application of splitting method : chemical correction I Splitting Techniques for Low Mach Number Flames ,→ Eliminating chemical terms : important variation of the temporal derivative of the density ρ, instant perturbation of the velocity field. ,→ Necessity to take into account the chemical contribution in the diffusion-convection step Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion n S X ∂T ρcp =− hk mk ωk ∂t k =1 ∂Yk ρ = mk ωk ∂t Application of splitting method : chemical correction I Splitting Techniques for Low Mach Number Flames ,→ Eliminating chemical terms : important variation of the temporal derivative of the density ρ, instant perturbation of the velocity field. ,→ Necessity to take into account the chemical contribution in the diffusion-convection step Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion n S X ∂T ρcp =− hk mk ωk ∂t k =1 ∂Yk ρ = mk ωk ∂t Application of splitting method : chemical correction II Splitting Techniques for Low Mach Number Flames Using the ideal gas law, chemical contribution on the temporal derivative of the density : Context Background Numerical Difficulties Strategies Splitting Aim of splitting ∂ρ ∂t chemical nS nS X 1 X = · hk mk ωk − m ωk ρcp k =1 k =1 This contribution must be added to the convection-diffusion step Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Density equation of the convection-diffusion system. ∂ρ + ∂t ∂ρ ∂t + 2ρU + chimie ∂V =0 ∂z Application of splitting method : chemical correction II Splitting Techniques for Low Mach Number Flames Using the ideal gas law, chemical contribution on the temporal derivative of the density : Context Background Numerical Difficulties Strategies Splitting Aim of splitting ∂ρ ∂t chemical nS nS X 1 X = · hk mk ωk − m ωk ρcp k =1 k =1 This contribution must be added to the convection-diffusion step Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Density equation of the convection-diffusion system. ∂ρ + ∂t ∂ρ ∂t + 2ρU + chimie ∂V =0 ∂z Numerical Resolution Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Finite difference spatial discretization : • diffusion : centered schemes • convection : upwind schemes Boundary value problem solved by Newton method Time discretization : • convection-diffusion : 2nde order implicit finite difference scheme • reaction : radauIIA, implicit Runge Kutta Numerical Results I Splitting Techniques for Low Mach Number Flames Unstationary simulations : the perturbation of the flame is a sinusoidal acoustic oscillation Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Some axial velocity and reduced radial velocity profiles for a value of the period T of the sinusoidal perturbation from 10% of the inlet velocity at a frequency of 100Hz. Numerical Results II Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Some mass fraction YCH and Temperature profiles for a value of the period T of the sinusoidal perturbation from 10% of the inlet velocity at a frequency of 100Hz. Convergence order Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion ,→ Reduction order as predicted by the theory Outline Splitting Techniques for Low Mach Number Flames 1 Context and Motivation Application Background Numerical and computational difficulties Numerical Strategies 2 Operator splitting and stiffness Aim of splitting methods Standard numerical analysis of operator splitting Stiffness comes into play 3 Laminar Premixed Counter-Flow flames simulations Premixed counter-flow flames model Application of splitting method : chemical correction Numerical Results 4 Conclusion Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Conclusion and perspectives Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Splitting techniques : numerical analysis and computational validation −→ Numerical analysis of temporal and spatial origins of order loss −→ Chemical correction for operator splitting of isobaric flame equations −→ Computational validation of order loss Work in progress Multi-dimensional configurations : use of splitting techniques in DNS simulations on 2D with wall interactions parareal algorithms Conclusion and perspectives Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Splitting techniques : numerical analysis and computational validation −→ Numerical analysis of temporal and spatial origins of order loss −→ Chemical correction for operator splitting of isobaric flame equations −→ Computational validation of order loss Work in progress Multi-dimensional configurations : use of splitting techniques in DNS simulations on 2D with wall interactions parareal algorithms References I Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion S. Descombes, T. Dumont, M. Massot Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation, order reduction and application to spiral waves Proceeding of the Workshop “Patterns and waves : theory and applications”, Saint-Petersbourg (2003) S. Descombes and M. Massot Operator splitting for nonlinear reaction-diffusion systems with an entropic structure : singular perturbation and order reduction Numerische Mathematik (2004) C. Besse, B. Bidégaray, S. Descombes Order estimates in time of splitting methods for the nonlinear Schrödinger equation SIAM J. Numer. Anal. (2002) M. Massot Singular perturbation analysis for the reduction of complex chemistry in gaseous mixtures using the entropic structure DCDS - B (2002) References II Splitting Techniques for Low Mach Number Flames V. Giovangigli and M. Massot Multicomponent reactive flows : reduced chemistry and entropic structure on partial equilibrium manifolds M2AS (2004) Context Background Numerical Difficulties Strategies Splitting S. Descombes and M. Massot On the local error of splitting approximations of reaction-diffusion equations Preprint (2006) Aim of splitting Standard analysis S. Descombes, T. Dumont, M. Massot, V. Louvet Stiffness on the local and global errors of splitting approximation of reaction-diffusion equations with high spatial gradients Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion International Journal of Computer Mathematics (2007) S. Descombes, T. Dumont, J. Beaulaurier, M. Massot, F. Laurent-Nègre, V. Louvet Operator splitting techniques for multi-scale reacting waves and applications to low mach number flames with complex chemistry : theorical and numerical aspectes Forthcoming paper References III Splitting Techniques for Low Mach Number Flames Context Background Numerical Difficulties Strategies Splitting Aim of splitting Standard analysis Stiffness Laminar Premixed Counter-Flow Flames simulations Modelisation Splitting : chemical correction Numerical results Conclusion Grant PEPS (Projet Exploratoire Pluridisciplinaire) 2007-2008 (Frédérique Laurent, Anne Bourdon) Young Investigator Award (S. Descombes, M. Massot) “New Interfaces of Mathematics”, French Ministry of Research 2003-2006
