✬ ✩ 1 High Order Asymptotic Preserving Schemes for Some Discrete-Velocity Kinetic Equations Fengyan Li Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, NY, USA Joint work with : J. Jang (UC Riverside), J.-M. Qiu (Houston), T. Xiong (Houston) ✫ ✪ lif@rpi.edu
✬ ✩ 2 Outline • Discrete-velocity kinetic equations • High order asymptotic preserving (AP) methods • Theoretical results • Numerical examples ✫ ✪ lif@rpi.edu
✬ ✩ 3 Discrete-velocity Kinetic Equations Consider the discrete-velocity model ∂ t f + v · ∇ x f = C ( f ) (1) f = f ( x, v, t ): distribution function of particles dependent of time t > 0, position x , and velocity v ∈ {− 1 , 1 } . C ( f ): collision operator � ( C ( f ) dv = 0). ✫ ✪ lif@rpi.edu
✬ ✩ 4 We are particularly interested in the discrete-velocity model in a diffusive scaling (under the scaling of t ′ := ε 2 t and x ′ := εx ): ε∂ t f + v · ∇ x f = 1 ε C ( f ) (2) The parameter ε > 0 can be regarded as the mean free path of the particles, and it measures the distance of the system to the equilibrium state. The smaller ε is, the closer the system is to the equilibrium. ✫ ✪ lif@rpi.edu
✬ ✩ 5 Examples: Jin-Pareschi-Toscani (1998) dµ : discrete Lebesgue measure on {− 1 , 1 } � fdµ = 1 �·� : � f � = 2 ( f ( x, v = − 1 , t ) + f ( x, v = 1 , t )) C ( f ) when ε → 0, with ρ = � f � (a.1) � f � − f ∂ t ρ = ∂ xx ρ (a.2) � f � − f + Aεv � f � ∂ t ρ + A∂ x ρ = ∂ xx ρ C C � f � m ( � f � − f ) 1 − m ∂ xx ( ρ 1 − m ) , m � = 1 (a.3) ∂ t ρ = � � f � 2 − ( � f � − f ) 2 � ∂ t ρ + C∂ x ρ 2 = ∂ xx ρ (a.4) � f � − f + Cε v Note : • (a.1): telegraph equation 1 1 − m ∂ xx ( ρ 1 − m ) with m < 0: porous medium equation • ∂ t ρ = ✫ ✪ lif@rpi.edu
✬ ✩ 6 In some applications, ε may differ in several orders of magnitude from the rarefied regime ( ε = O (1)) to the hydrodynamic (diffusive) regime ( ε << 1). It is desirable to design a class of numerical methods which work uniformly with respect to the parameter ε . ✫ ✪ lif@rpi.edu
✬ ✩ 7 Objective : to design high order asymptotic preserving methods for the discrete-velocity kinetic equation in the diffusive scaling Asymptotically preserving (AP) methods: Jin, Levermore, Naldi, Pareschi, Degond, Toscani, Klar, Filbet, Carrillo, Lemou, Mieussens, Hauck, Liu, · · · - uniformly stable with respect to ε ranging from O (1) to 0; - When ε → 0, the methods are consistent for the limiting equation on fixed mesh. ✫ ✪ lif@rpi.edu
✬ ✩ 8 Numerical challenges and considerations ε∂ t f + v · ∇ x f = 1 ε C ( f ) • Stiffness in both the convective and the collision terms • The characteristic speed of the homogeneous hyperbolic part is 1 ε . Standard schemes for hyperbolic problems with stiff relaxation ( implicit treatment for the collision term + explicit treatment for the convection term ) may impose a restrictive stability condition: ∆ t ≈ ε ∆ x (3) • Not all stable schemes can capture the diffusive limit when ε → 0 on under-resolved meshes (∆ t, ∆ x >> ε ) • Easy to solve ✫ ✪ lif@rpi.edu
✬ ✩ 9 High Order Asymptotic Preserving Methods Three components - Micro-macro decomposition of the equation - Discontinuous Galerkin (DG) spatial discretization: numerical flux - Globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta temporal discretization: implicit-explicit strategy Major references: • Jin-Pareschi-Toscani (1998) • Lemou-Mieussens (2010), Liu-Mieussens (2010) • Boscarino-Pareschi-Russo (2013) ✫ ✪ lif@rpi.edu
✬ ✩ 10 a family of high order methods are proposed for Main results: discrete-velocity kinetic equations in the diffusive scaling. - For the telegraph equation, when a first order temporal discretization is applied, uniform stability is established with respect to ε . Error estimates are also obtained for any ε . - Formal asymptotic analysis shows that the proposed schemes in the limit of ε → 0 provide explicit and consistent high order methods for the limiting equations. ✫ ✪ lif@rpi.edu
✬ ✩ 11 1. Micro-macro decomposition Consider the Hilbert space L 2 ( dµ ) with the inner product �· , ·� : � � f 1 , f 2 � = f 1 f 2 dµ, and an orthogonal projection Π: Π f = � f � . Let f = Π f + ( I − Π) f =: ρ + εg , then the micro-macro decomposition of (2) is ∂ t ρ + ∂ x � vg � = 0 (4a) ∂ t g + 1 ε 2 v∂ x ρ + 1 ε { I − Π } ( v∂ x g ) = 1 ε 3 C ( ρ + εg ) (4b) Note : Solving for f from (2) is equivalent to solving for ρ and g from (4). ✫ ✪ lif@rpi.edu
✬ ✩ 12 2. Discontinuous Galerkin (DG) spatial discretization Mesh: I i = [ x i − 1 2 ] , h i = | I i | 2 , x i + 1 U h = U k h = { u : u ∈ P k ( I i ) , ∀ i } , with P k ( I i ) consisting Discrete space: of polynomials of degree at most k u − i + 1 2 u + I i i − 1 2 x i − 1 x i + 1 2 2 Notations: u ± = u ± ( x ) = ∆ x → 0 ± u ( x + ∆ x ) lim { u } = 1 2 ( u − + u + ) average : [ u ] = u + − u − jump : ✫ ✪ lif@rpi.edu
✬ ✩ 13 Semi-discrete DG method : Look for ρ h ( · , t ) , g h ( · , v, t ) ∈ U h , such that ∀ φ, ψ ∈ U h , and i , � � � vg h � ∂ x φdx + � − � 2 φ − 2 φ + ∂ t ρ h φdx − � vg h � i + 1 � vg h � i − 1 = 0 , (5a) i + 1 i − 1 I i I i 2 2 � � ∂ t g h ψdx + 1 ( I − Π) D h ( g h ; v ) ψdx ε I i I i �� � − 1 2 ψ − 2 ψ + vρ h ∂ x ψdx − v � ρ h,i + 1 + v � ρ h,i − 1 i + 1 i − 1 ε 2 I i 2 2 � = 1 C ( ρ h + εg h ) ψdx. (5b) ε 3 I i Numerical flux: Alternating: � ρ = ρ + , or � � vg � = � vg � + , ˆ � vg � = � vg � − , ˆ ρ = ρ − (6a) Central: � � vg � = {� vg �} , ˆ ρ = { ρ } (6b) ✫ ✪ lif@rpi.edu
✬ ✩ 14 D h ( g h ; v ) ∈ U h is an upwind approximation of v∂ x g , defined by � � � � � vg h ∂ x ψdx + � − � 2 ψ − 2 ψ + D h ( g h ; v ) ψdx = − ( vg h ) i + 1 ( vg h ) i − 1 i + 1 i − 1 I i 2 2 i (7) with vg − h , if v > 0 = v { g h } − | v | � vg h := 2 [ g h ] . (8) vg + h , if v < 0 ✫ ✪ lif@rpi.edu
✬ ✩ 15 Method in a compact form : Look for ρ h ( · , t ) , g h ( · , v, t ) ∈ U h , such that ∀ φ, ψ ∈ U h , and i , � ∂ t ρ h φdx + a h ( g h , φ ) = 0 � ∂ t g h ψdx + 1 ε 2 d h ( ρ h , ψ ) = − 1 εb h,v ( g h , ψ ) − v ε 2 s h,v ( ρ h , g h , ψ ) − s ′ h,v ( g h , ψ ) where � � � � a h ( g h , φ ) = − � vg h � ∂ x φdx − � vg h � i − 1 2 [ φ ] i − 1 2 , I i i i � � b h,v ( g h , ψ ) = ( I − Π) D h ( g h ; v ) ψdx = ( D h ( g h ; v ) − �D h ( g h ; v ) � ) ψdx, � � � d h ( ρ h , ψ ) = ρ h ∂ x ψdx + 2 [ ψ ] i − 1 ρ h,i − 1 2 , � I i i i ✫ ✪ lif@rpi.edu
✬ ✩ 16 and ( g h , ψ ) for ( a. 1) ( g h − Avρ h , ψ ) for ( a. 2) s h,v ( ρ h , g h , ψ ) = ( Cρ m h g h , ψ ) for ( a. 3) ( g h − Cvρ 2 h , , ψ ) for ( a. 4) 0 for ( a. 1) 0 for ( a. 2) s ′ h,v ( g h , ψ ) = 0 for ( a. 3) ( Cvg 2 h , ψ ) for ( a. 4) ✫ ✪ lif@rpi.edu
✬ ✩ 17 3. Globally stiffly accurate implicit-explicit (IMEX) temporal discretization DG-IMEX1: � ρ n +1 − ρ n h φdx + a h ( g n h h , φ ) = 0 , (9a) ∆ t � g n +1 − g n ψdx +1 h , ψ ) − v h εb h,v ( g n ε 2 d h ( ρ n +1 h , ψ ) = h ∆ t − 1 ε 2 s h,v ( ρ n +1 , g n +1 , ψ ) − s ′ h,v ( g n h , ψ ) . (9b) h h Note: (1) The terms d h and s h,v are treated implicitly. (2) One first solves ρ n +1 from (9a), and then g n +1 from (9b). h h ✫ ✪ lif@rpi.edu
✬ ✩ 18 Formal asymptotic analysis: As an example, apply the DG-IMEX1 to the telegraph equation with C ( f ) = � f � − f = − εg : - The equation in its micro-macro formulation: ∂ t ρ + ∂ x � vg � = 0 ∂ t g + 1 ε 2 v∂ x ρ + 1 ε { I − Π } ( v∂ x g ) = − 1 ε 2 g - DG-IMEX1 scheme: ∀ φ, ψ ∈ U h , � ρ n +1 − ρ n h φdx + a h ( g n h h , φ ) = 0 ∆ t � g n +1 � − g n ψdx +1 h , ψ ) − v , ψ ) = − 1 h εb h,v ( g n ε 2 d h ( ρ n +1 g n +1 h ψdx h h ε 2 ∆ t ✫ ✪ lif@rpi.edu
✬ ✩ 19 Formal asymptotic analysis: As an example, apply the DG-IMEX1 to the telegraph equation with C ( f ) = � f � − f = − εg : - The equation in its micro-macro formulation: ∂ t ρ + ∂ x � vg � = 0 (12a) ∂ t g + 1 ε 2 v∂ x ρ + 1 ε { I − Π } ( v∂ x g ) = − 1 (12b) ε 2 g - DG-IMEX1 scheme: ∀ φ, ψ ∈ U h , � ρ n +1 − ρ n h φdx + a h ( g n h h , φ ) = 0 (13a) ∆ t � g n +1 � − g n ψdx +1 , ψ ) = − 1 h , ψ ) − v h εb h,v ( g n ε 2 d h ( ρ n +1 g n +1 h ψdx h h ε 2 ∆ t (13b) ✫ ✪ lif@rpi.edu
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