High Order Semi-Lagrangian Schemes And Operator Splitting For The Boltzmann Equation Yaman Güçlü 1 Andrew J. Christlieb 1 William N.G. Hitchon 2 1 Department of Mathematics, Michigan State University, East Lansing (MI) 2 Department of Electrical and Computer Engineering, University of Wisconsin, Madison (WI) 7 June 2013 Issues in Solving the Boltzmann Equation for Aerospace Applications ICERM topical workshop, Providence (RI), 3-7 June 2013 Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 1
Contents Model Equations 1 Numerical Challenges 2 Convected Scheme 3 Numerical Results 4 Conclusions 5 Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 2
Maxwell-Boltzmann system Maxwell’s equations: ∇ × E = − ∂ B ∇ · E = ρ ∂ t ε 0 ∂ E ∇ · B = 0 ∇ × B = µ 0 J + µ 0 ε 0 ∂ t Sources: charge and current density: � � ρ ( r , t ) = q α n α ( r , t ) , J ( r , t ) = q α n α ( r , t ) u α ( r , t ) . α α Number density and mean velocity of each species: 1 � � n α ( r , t ) = R 3 f α ( r , v , t ) d v , u α ( r , t ) = R 3 v f α ( r , v , t ) d v . n α ( r , t ) Boltzmann’s equation for each species: ∂ f α ∂ t + v · ∇ f α + q α � � � ( E + v × B ) · ∇ v f α = Q α f α , f β ( r , v , t ) m α β Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 3
Boltzmann’s equation Eulerian formulation: ( t , x , v ) independent variables ∂ f α ∂ t + v · ∇ f α + F α · ∇ v f α = ∂ f α � � � m α ∂ t � coll Lagrangian formulation: follow trajectory ( x ( t ) , v ( t )) in phase space 1 d x d v dt = v ( t ) , dt = F α ( t , x ( t ) , v ( t )) m α � Df α = ∂ f α • Substituting into Boltzmann’s equation: � � ∂ t Dt � coll • Time rate of change of f α ( t , x ( t ) , v ( t )) along phase-space trajectory only determined by collision operator • Without collisions, f α constant along phase-space trajectory: fluid motion in phase-space is incompressible Semi-Lagrangian method: • f α ( t , x , v ) lies on Eulerian mesh • Evolution within time step uses Lagrangian formulation ( method of characteristics ) Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 4
Modeling challenges W EAKLY COLLISIONAL PLASMA : • Electrons can be far from equilibrium and involved in strongly non-linear processes (e.g. ionization near threshold) • Multiple species: electrons, multiple ions, neutrals; • Multiple time and spatial scales; • Complex geometries, different boundary conditions (perfect/real conductors, dielectrics, absorbing), often time varying and coupled to domain (plasma feedbacks into circuit); • Complex collisional processes: elastic, inelastic (excitation, ionization, recombination, attachment, dissociation etc.); • External magnetic fields: electrons may be strongly magnetized, possibly ions too; • Other important processes: radiation transport, gas-phase chemical reactions, plasma-surface interaction, aggregates (dusty plasmas). C HALLENGES FOR LOW - ORDER E ULERIAN CODES : • For electrons, need high resolution over large velocity mesh • Impressive memory requirement in multiple dimensions • Explicit time-stepping imposes non-physical time-step restriction (CFL limit) • Method of lines (MOL): multistep and multi-stage methods require additional storage Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 5
Convected Scheme The Convected Scheme [ a ] is a forward semi-Lagrangian method for Boltzmann’s equation. Employs operator splitting: 1. Collision operator is local in configuration space, solves ∂ f α = ∂ f α � � � ∂ t ∂ t � coll 2. Ballistic operator advects f α ( t , x , v ) along characteristic trajectories in phase space according to Df α = 0, integrated over a moving cell (MC). Dt f α ( t , x , v ) assumed uniform over MC, allowing for ’area remapping rule’ a W.N.G. H ITCHON , D. K OCH , AND J. A DAMS . An efficient scheme for convection-dominated transport . Journal of Computational Physics , 83(1): 79-95, 1989. Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 6
Convected Scheme PROs: CONs: • Preserves positivity (good as f α > 0) • Difficult to handle boundary conditions • No CFL restriction on ∆ t • Numerical diffusion : local remapping ∆ x 2 � error O � • Very simple implementation • Can enforce total energy conservation for stationary electric field Reduced Numerical Diffusion Numerical diffusion mitigated by reducing remapping frequency ⇒ “long-lived moving cells” [ a ]. Recently [ b ], we devised a high-order version of the Convected Scheme, for neutral gas kinetics: Model equation: uniform velocity advection: n t + u 0 n x = 0 Basic idea: compensating remapping error by applying small corrections to final position of moving cells prior to remapping ⇒ antidiffusive velocity field Tool: modified equation analysis, perturbation analysis aA.J. C HRISTLIEB , W.N.G. H ITCHON AND E.R. K EITER . A computational investigation of the effects of varying discharge geometry for an inductively coupled plasma . IEEE T. Plasma Sci. , 28(6): 2214-2231, 2000. bY. G ÜÇLÜ AND W.N.G. H ITCHON . A high order cell-centered semi-Lagrangian scheme for multi-dimensional kinetic sim- ulations of neutral gas flows . Journal of Computational Physics , 231(8): 3289-3316, Apr 2012. Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 7
High-order semi-Lagrangian solution of the Vlasov-Poisson system P ROBLEM : Difficult to construct high-order semi-Lagrangian ballistic operator when mean force is present (no straight trajectories) S OLUTION : • Further split ballistic operator into separate constant advection operators along x and v [ a ] • Apply favorite high-order semi-Lagrangian solver to each operator • Combine operators to high-order in time using Runge-Kutta-Nyström methods [ b , c ] ( symplectic ⇒ energy stable) aC.Z. C HENG AND G. K NORR . The integration of the Vlasov equation in configuration space . J. Comput. Phys., 22: 330-351, 1976 . bJ.A. R OSSMANITH AND D.C. S EAL . A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations . J. Comput. Phys., 227: 9527-9553, 2011 . cN. C ROUSEILLES , E. F AOU AND M. M EHRENBERGER . High order Runge-Kutta-Nyström splitting methods for the Vlasov- Poisson equation . INRIA-00633934, 2011 . Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 8
Arbitrarily High-Order Convected Scheme (1) 1D CONSTANT ADVECTION EQUATION � ∂ ∂ t + u ∂ � n ( x , t ) = 0 ∂ x • Exact solution (method of characteristics): n ( x , t + ∆ t ) ≡ n ( x − u ∆ t , t ) • Courant parameter: α := u ∆ t / ∆ x C ONVECTED S CHEME UPDATE • Discretize time (arbitrary ∆ t ) and space (uniform ∆ x ): n k i ≈ n ( x i , t k ) • Because of uniform ∆ x , solution can be shifted exactly by integer number of cells • Without loss of generality, assume 0 ≤ α ≤ 1 (this is not a CFL limit) • Under these assumptions, CS update is � � n k + 1 = U k i − 1 n k 1 − U k n k i − 1 + i i i • As long as 0 ≤ U k i ≤ 1, CS is mass and positivity preserving • With no high-order corrections, U ( x , t ) ≡ α 1st-order Upwind scheme ⇒ • With high-order corrections, U ( x , t ) = � u + ˜ � u ( x , t ) ∆ t / ∆ x = α + ˜ α ( x , t ) α ( x , t ) is anti-diffusive Courant parameter • ˜ Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 9
Arbitrarily High-Order Convected Scheme (2) L OCAL TRUNCATION ERROR (LTE) • Exact solution, Taylor expand in space (smooth initial conditions): N − 1 ( − α ) p (∆ x ) p ∂ p � ∆ x N � � n ( x , t ) + O n ( x , t + ∆ t ) = n ( x , t ) + , ∂ x p p ! p = 1 • CS solution, Taylor expand in space about ( x , t ) = ( x i , t k ) : N − 1 ( − 1 ) p (∆ x ) p ∂ p � ∆ x N � � U ( x , t ) n ( x , t ) + O n CS ( x , t + ∆ t ) = n ( x , t ) + ∂ x p p ! p = 1 • We want the local truncation error E ( x , t , ∆ t ) := n ( x , t + ∆ t ) − n CS ( x , t + ∆ t ) = O (∆ x N ) , hence we find ˜ α ( x , t ) by imposing the order condition N − 1 N − 1 ( − α ) p (∆ x ) p ∂ p n ( − 1 ) p (∆ x ) p ∂ p ( Un ) = O (∆ x N ) � � ∂ x p − ∂ x p p ! p ! p = 1 p = 1 Y. Güçlü & A.J. Christlieb (MSU), W.N.G. Hitchon (UW) High Order Semi-Lagrangian For Boltzmann’s eq. Providence, 7 Jun 2013 10
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