An application of the discontinuous Galerkin method for solving kinetic equations A. Majorana Dipartimento di Matematica e Informatica, Università di Catania, Italy Novel Applications of Kinetic Theory and Computations (October 17-21, 2011) ICERM Semester Program on "Kinetic Theory and Computation" Providence, RI A. Majorana (Univ. Catania) DG method & kinetic equations 1 / 54
Outline A kinetic equation 1 The DG method 2 Semiconductor Boltzmann-Poisson equations 3 Numerical examples 4 The radiative transport equation 5 The nonlinear Boltzmann equation 6 References 7 A. Majorana (Univ. Catania) DG method & kinetic equations 2 / 54
A kinetic equation Introduction In recent years, deterministic solvers to the Boltzmann or similar kinetic equations were considered in the literature. These methods provide accurate results which, in general, agree well with those obtained from DSMC simulations, sometimes at a comparable or even less computational time. Here, I wish to show some simulation results (also, in collaboration with Irene Gamba, Chi-Wang Shu and Yingda Chen) to demonstrate the performance of solvers based on the Discontinuous Galerkin (DG) method. A. Majorana (Univ. Catania) DG method & kinetic equations 3 / 54
A kinetic equation Some references Keywords: kinetic equation, discontinuous Galerkin W. H. Reed, Report Los Alamos LA–4769 (1971) W. H. Reed, Report Los Alamos LA-UR-73-479 (1973) F . Rogier and J. Schneider, Transport Theory Statist. (1994) A. Alekseenko, N. Gimelshein and S. Gimelshein, preprint (2000) C. C. Pain, C. R. E. de Oliveira and A. J. H. Goddard, Transport Theory Statist. (2000) V. F . de Almeida, Technical Report (2003) M. K. Gobbert, S. G. Webster and T. S. Cale, Journal of Scientific Computing (2007) Z. Lian, MIT thesis (2007) C. Shanqin, E. Weinan, L. Yunxian and C.-W. Shu, Journal of Computational Physics (2007) L. L. Baker and N. G. Hadjiconstantinou, Int. J. Numer. Meth. Fluids (2008) A. Majorana (Univ. Catania) DG method & kinetic equations 4 / 54
A kinetic equation Some references A. A. Alexeenko, C. Galitzine and A. M. Alekseenko, 40th Thermophysics Conference (2008) Y. Cheng, I. M. Gamba, A. M. and C.-W. Shu, Computer Methods in Applied Mechanics and Engineering (2009) B. Ayuso, J. A.Carrillo, and C.-W. Shu, preprint (2009, 2010) R. E. Heath, I. M. Gamba, P . J. Morrison, and C. Michler, preprint arXiv (2010) N. G. Hadjiconstantinou, G. A. Radtke and L. L. Baker, Journal of Heat Transfer (2010) Y. Cheng, I. M. Gamba and J. Proft, Mathematics of Computation (2010) W. Hoitinga and E. H. van Brummelen, Journal of Computational Physics (2011) A. M., Kinetic and Related models (2011) ....... and other papers A. Majorana (Univ. Catania) DG method & kinetic equations 5 / 54
A kinetic equation Some references A kinetic equation ∂ f ∂ t + η ( v ) · ∇ x f + A ( t , x ) · ∇ v f = C ( f ) + S ( t , x , v ) , (1) where, f = f ( t , x , v ) is the distribution function - the unknown -, η is a given vectorial function of the velocity v , A is given or related to another equation, for instance, the Poisson equation, C ( f ) is the linear or nonlinear collision operator, and S ( t , x , v ) is the source term. A. Majorana (Univ. Catania) DG method & kinetic equations 6 / 54
The DG method We denote by Ω v ∈ R d ( d = 1 , 2 , 3 ) the domain of the velocity. If the set Ω v is not bounded, it is necessary to choose a new reasonable large but bounded subset of Ω v and modify the collision operator C in order to avoid that particles, having velocities belonging to this new set, after the collision, assume velocities outside this set. We assume that Ω v is bounded, and the distribution function f vanishes on the boundary of Ω v . The physical domain depends on the problem, strongly. Moreover, in order to solve the kinetic equation, we need to assume initial and boundary conditions. A. Majorana (Univ. Catania) DG method & kinetic equations 7 / 54
The DG method The DG method allows to find approximate solutions of the kinetic equation by solving a (usually large) set of ordinary differential equations in time. Then, we easily derive an approximation of the distribution function f and we can evaluate its moments. There are some reasons to introduce an intermediate step. the variables x and v have a different physical meaning, and, usually, we are interested in the main moments of f instead of f itself; usually, the boundary conditions on ∂ Ω v do not change. A. Majorana (Univ. Catania) DG method & kinetic equations 8 / 54
The DG method Therefore, we discretize the kinetic equation with respect only to the velocity variable v and we obtain a set of equations, where the new unknowns depend only on the time t and the spatial coordinates x . To this scope, we introduce a partition of the set Ω v , by means of a finite family of open cells C α , such that N � C α ⊆ Ω v ∀ α , C α ∩ C β = ∅ ∀ α � = β , C α = Ω v . α = 1 Remark: No restriction on the partition. A. Majorana (Univ. Catania) DG method & kinetic equations 9 / 54
The DG method v 2 C C β γ C α C ε C δ v 1 An example of a grid in two-dimensional velocity space. A. Majorana (Univ. Catania) DG method & kinetic equations 10 / 54
The DG method DG assumption We choose in each cell C α a finite dimensional vector space V α of function of v defined in C α , and we assume that, in C α , the distribution function f can be approximated by means a linear combination of elements of V α with coefficients depending on space and time. These coefficients are the new unknowns. � 1 , v , v 2 � For instance, if the basis of V α is the set , then f ( t , x , v ) ≈ a α ( t , x ) + b α ( t , x ) · v + c α ( t , x ) v 2 ∀ v ∈ C α and ∀ t , x . Remark: We can change vector space from a cell to another. A. Majorana (Univ. Catania) DG method & kinetic equations 11 / 54
The DG method A linear kinetic equation To make clear the application of DG method, we consider the linear kinetic equation ∂ f ∂ t + η ( v ) · ∇ x f + A ( t , x ) · ∇ v f � K ( v ′ , v ) f ′ − K ( v , v ′ ) f d v ′ + S ( t , x , v ) . � � = (2) Ω v Here, K ( v ′ , v ) is the kernel of the integral operator and, as usual, f ′ = f ( t , x , v ′ ) . A. Majorana (Univ. Catania) DG method & kinetic equations 12 / 54
The DG method A linear kinetic equation � � If ψ α, i ( v ) : i = 1 , .., n α is the basis of the vector space V α , then in the cell C α we have n α � f ( t , x , v ) ≈ f α, i ( t , x ) ψ α, i ( v ) . (3) i = 1 Now, if we multiply both sides of the equation by ψ α, k ( v ) and integrate in C α we obtain the exact equation ∂ � � f ( t , x , v ) ψ α, k ( v ) d v + ∇ x η ( v ) f ( t , x , v ) ψ α, k ( v ) d v ∂ t C α C α � � + A ( t , x ) · [ ∇ v f ( t , x , v )] ψ α, k ( v ) d v = S ( t , x , v ) ψ α, k ( v ) d v C α C α �� � � � K ( v ′ , v ) f ( t , x , v ′ ) − K ( v , v ′ ) f ( t , x , v ) � d v ′ + ψ α, k ( v ) d v . (4) C α Ω v A. Majorana (Univ. Catania) DG method & kinetic equations 13 / 54
The DG method A linear kinetic equation The approximation is introduced using Eq, (3). So, for instance, we have � ∂ f α, i ( t , x ) n α �� ∂ � � f ( t , x , v ) ψ α, k ( v ) d v ≈ ψ α, i ( v ) ψ α, k ( v ) d v ∂ t ∂ t C α C α i = 1 n α � �� � � ∇ x η ( v ) f ( t , x , v ) ψ α, k ( v ) d v ≈ η ( v ) ψ α, i ( v ) ψ α, k ( v ) d v ∇ x f α, i ( t , x ) C α C α i = 1 We note that the coefficients of the partial derivatives in the r.h.s are numerical constant parameters. We show the complete equation in the simplest case: n α = 1 and ψ α, 1 ( v ) = 1. A. Majorana (Univ. Catania) DG method & kinetic equations 14 / 54
The DG method A linear kinetic equation Now, we have �� � �� � ∂ f α M α ∂ t + η ( v ) d v · ∇ x f α + A ( t , x ) · f ( t , x , v ) n d σ C α ∂ C α N �� � �� � � � d v ′ K ( v ′ , v ) � d v ′ K ( v , v ′ ) ≈ d v f β − d v f α C α C β C α Ω v β = 1 � + S ( t , x , v ) d v , (5) C α where M α is the measure of the cell C α . In order to have only the unknowns f α ( t , x ) in Eq. (5), we must find, for fixed t and x , a suitable relationship between the value of the distribution function f on boundary of the cell C α and the new unknowns. A. Majorana (Univ. Catania) DG method & kinetic equations 15 / 54
The DG method A linear kinetic equation Hence, we have a set of partial differential equations, which can be solved applying again the DG method or another technique. A. Majorana (Univ. Catania) DG method & kinetic equations 16 / 54
Semiconductor Boltzmann-Poisson equations The Transport Equation: the physical parameters The Boltzmann equation for an electron gas in a semiconductor ∂ f ∂ t + 1 � ∇ k ε · ∇ x f − q � E · ∇ k f = Q ( f ) . � is the Planck constant divided by 2 π ∇ k is the gradient with respect to the wave vector k ∇ x is the gradient with respect to the space coordinates x q is the positive electric charge A. Majorana (Univ. Catania) DG method & kinetic equations 17 / 54
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