Implicit-Explicit Runge-Kutta schemes for the Boltzmann-Poisson equation for semiconductors Vittorio Rispoli In collaboration with: G. Dimarco (Univ. of Toulouse, France) L. Pareschi (Univ. of Ferrara, Italy) HYP2012 University of Padova ITALY Padova, 28.06.2012
MOSFET modelling Figure: Metal-Oxide-Semiconductor Field-Effect Transistor (MOSFET) device modelling from http://nanohub.org/topics/MOSFETLabPage V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 2 / 34
Outline 1 General framework 2 Application to kinetic equations for semiconductor Diffusive limit Parity equations 3 Discretization Time discretization: IMEX schemes Phase-space discretization 4 Results 5 Penalization technique for the collision term V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 3 / 34
Prototype example A prototype hyperbolic system of conservation laws with diffusive relaxation that we will use to illustrate the subsequent theory is the following: u t + v x = 0 , (1) v t + 1 ε 2 p ( u ) x = − 1 ε 2 ( v − q ( u )) , where p ( u ) ′ > 0 . Characteristic speeds: � p ( u ) ′ ± ε V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 4 / 34
Prototype example In the small relaxation limit ε → 0 , the behavior of the solution is, at least formally, governed by the convection-diffusion equation u t + q ( u ) x = p ( u ) xx , (2) v = q ( u ) − p ( u ) x . Ensure that the schemes possess the correct zero-relaxation limit : the numerical scheme applied to system (1) should be a consistent and stable scheme for the limit system (2) as the parameter ε → 0 . asymptotic preservation (AP) V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 5 / 34
Problems 1 Characteristic speeds of the hyperbolic part is of order 1 /ε , standard IMEX R-K schemes developed for hyperbolic systems with stiff relaxation become useless in such parabolic scaling, because the CFL condition would require ∆ t = O ( ε ∆ x ) . 2 Most previous works on asymptotic preserving schemes, in the limit of infinite stiffness become consistent explicit schemes for the diffusive limit equation: such explicit schemes clearly suffer from the usual stability restriction ∆ t = O (∆ x 2 ) . V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 6 / 34
IMEX Runge-Kutta schemes for diffusive relaxation The starting point is to reformulate problem (1) as the equivalent system 1 u t = − ( v + µ p ( u ) x ) x + µ p ( u ) xx , � �� � � �� � Explicit Implicit ε 2 v t = − p ( u ) x − v + q ( u )) . � �� � Implicit Here µ = µ ( ε ) ∈ [0 , 1] is a free parameter such that µ (0) = 1 . 1 Boscarino, Pareschi, Russo (2011) V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 7 / 34
IMEX Runge-Kutta schemes for diffusive relaxation This system can be written in the form = f 1 ( u , v ) + f 2 ( u ) u t � �� � ���� Explicit Implicit ε 2 v t = g ( u , v ) , � �� � Implicit to which we can apply and study the properties of an IMEX-RK scheme. V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 8 / 34
IMEX Runge-Kutta schemes for diffusive relaxation Applying an Implicit-Explicit (IMEX) Runge-Kutta scheme 2 we obtain: k − 1 � � k � U j , V j � � U j � u n + ∆ t U k = + ∆ t , � a kj f 1 a kj f 2 j =1 j =1 k � � U j , V j � ε 2 v n + ∆ t ε 2 V k = a kj g j =1 for the internal stages for k = 1 , . . . , ν and ν ν � � � U k , R k � � U k � u n + ∆ t u n +1 = + ∆ t w k f 1 � w k f 2 k =1 k =1 ν � � U k , V k � ε 2 v n + ∆ t ε 2 v n +1 = w k g k =1 for the numerical solution . 2 Pareschi, Russo (2010); Asher, Ruuth, Spiteri (1997) V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 9 / 34
IMEX Runge-Kutta schemes for diffusive relaxation b , b ∈ R ν characterize The ν × ν matrices � a ik ) and A = ( a ik ) and vectors � A = ( � the scheme and can be represented by a double tableau in the usual Butcher notation: � � c A c A w t w t � Matrix A is lower triangular, i.e. the implicit scheme is a Diagonally Implicit Runge-Kutta ( DIRK ) scheme. This choice guarantees that implicit terms are, indeed, always explicitly evaluated. V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 10 / 34
Boltzmann-Poisson equation for semiconductors Let f ( t , x , v ) be the density distribution function for particles at time t ≥ 0 , space point x ∈ R d and that travel with velocity v ∈ R d , where d = 1 , 2 , 3 is the dimension. Distribution function f solves a kinetic equation with diffusive scaling 3 m E · ∇ v f = 1 ε ∂ t f + v · ∇ x f − q ε Q ( f ) . (3) In this formula: ε is the mean free path, q is the elementary charge and m the effective mass of the electron. E ( t , x ) = −∇ x Φ( t , x ) is the electric field. E is obtained from the electric potential Φ given by the solution of a Poisson equation . 3 Poupaud (1991); Klar (1998); Jin, Pareschi (2000) V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 11 / 34
Collision term The anisotropic collision term Q ( f ) is defined by � � � Q ( f ) = σ ( v , w ) M ( v ) f ( w ) − M ( w ) f ( v ) d w , where σ is the scattering kernel and M is the normalized Maxwellian at the temperature θ of the semiconductor � � −| v | 2 1 M ( v ) = (2 πθ ) d / 2 exp . 2 θ The collision frequency λ is defined by: � λ ( v ) = σ ( v , w ) M ( w ) d w . V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 12 / 34
Diffusive limit Define the total mass ρ = ρ ( t , x ) as � ρ = f ( v ) d v . As ε → 0 , one can show that f is approximated by f ( t , x , v ) ≈ ρ ( t , x ) M ( v ) ( ε → 0) with ρ satisfying the drift-diffusion equation: ∂ t ρ = ∇ x · ( D ∇ x ρ + ηρ E ) . (4) In this equation, D is the diffusion coefficient defined implicitly in terms of the cross section and constant η is the mobility given by the Einstein relation q D = η m θ . V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 13 / 34
Even and Odd parities formalism Define the even parity r and the odd parity j by: r ( t , x , v ) = f ( t , x , v ) + f ( t , x , − v ) , 2 j ( t , x , v ) = f ( t , x , v ) − f ( t , x , − v ) . 2 ε Splitting eq. (3) into two equations, one for v and one for − v and adding and subtracting them we obtain: m E · ∇ v j = 1 ∂ t r + v · ∇ x j − q ε 2 Q ( r ) , (5) � � ∂ t j + 1 = − 1 v · ∇ x r − q m E · ∇ v r ε 2 λ j , ε 2 V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 14 / 34
Even and Odd parities formalism In the fluid-dynamic limit, i.e. as ε → 0 , we obtain: Q ( r ) = 0 , ( ε = 0) λ j = − v · ∇ x r + q m E · ∇ v r , from which r ( t , x , v ) = ρ ( t , x ) M ( v ) and � � � � 1 − v · ∇ x r + q 1 − M v · ∇ x ρ + q j = m E · ∇ v r = m ρ E · ∇ v M . λ ( v ) λ ( v ) Substituting this relations into the first equation in (5) and integrating over v , we obtain the drift-diffusion equation (4). V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 15 / 34
Even and Odd parities formalism Using parities formalism, our problem now reads: 1 ∂ t r + v · ∇ x j − E · ∇ v j = ε 2 Q ( r ) , ∂ t j + 1 − 1 ε 2 ( v · ∇ x r − E · ∇ v r ) = ε 2 λ j , As was done before, we add and subtract in the first equation the term � � µ v v · ∇ x λ · ∇ x r , where µ = µ ( ε ) is a real positive parameter, such that µ (0) = 1 . The idea behind this choice is that we want to compute such term with the implicit solver when the equation is in the fluid-dynamic regime , in order to use the appropriate solver for the diffusion term. V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 16 / 34
IMEX scheme for parity equations The system we are going to solve now reads as follows: � � � � j + µ v = 1 µ v ∂ t r + v · ∇ x λ · ∇ x r − E · ∇ v j ε 2 Q ( r ) + v · ∇ x λ · ∇ x r , � �� � � �� � Explicit Implicit ∂ t j + 1 = − 1 ε 2 ( v · ∇ x r − E · ∇ v r ) ε 2 λ j . � �� � � �� � Implicit Implicit Remark: such scheme can be used for easy to invert collision terms Q ; this is true, e.g., for the relaxed time approximation (RTA), which is the case when σ ≡ 1 , which implies: Q ( r ) = M ρ − r . V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 17 / 34
Hermite approximation of the velocity space Because of the structure of the problem, it is convenient to write unknowns r and j as follows: r = φ M and j = ψ M , with φ = φ ( t , x , v ) and ψ = ψ ( t , x , v ) . Set r = φ M , j = ψ M , with N N � � φ k � ψ k � φ ( v ) = H k ( v ) , ψ ( v ) = H k ( v ) , k =0 k =0 being the Hermite expansion. V. Rispoli (University of Ferrara) IMEX-RK schemes for semiconductors Padova, 28.06.2012 18 / 34
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