On the time integration of the Boltzmann equation and related problems Giacomo Dimarco Department of Mathematics and Computer Science Universit` a di Ferrara Italy http://perso.math.univ-toulouse.fr/dimarco/ giacomo.dimarco@unife.it Joint research with: Nicolas Crouseilles (INRIA, Rennes, France) Lorenzo Pareschi (University of Ferrara, Italy) Vittorio Rispoli (University of Toulouse, France) Marie-Helene Vignal (University of Toulouse, France) Luc Mieussens (Universit´ e de Bordeaux 1, France) Sharing Higher-order advanced research Know-How on finite volume SHARK-FV 2014 logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 1 / 51
Outline Introduction 1 The Boltzmann equation 2 The hydrodynamic limit 3 The diffusive limit 4 Asymptotic Preserving methods and domain decomposition 5 Multiscale problems 6 logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 2 / 51
Introduction Outline for section 1 Introduction 1 The Boltzmann equation 2 The hydrodynamic limit 3 The diffusive limit 4 Asymptotic Preserving methods and domain decomposition 5 Multiscale problems 6 logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 3 / 51
Introduction Motivations Many problems of interests in applications involve non equilibrium gas flows as hypersonic objects simulations or micro-electro-mechanical devices. These kind of problems are characterized by breakdowns of fluid models, either Euler or Navier-Stokes. When the breakdown is localized both in space and time we must deal with connections of continuum and non equilibrium regions. To face such problems, the most natural approach is to try to combine numerical schemes for continuum models with microscopic kinetic models which guarantee a more accurate description of the physics when far from the thermodynamical equilibrium. Alternatively, we can try to construct numerical methods which address explicitly the multiscale nature of the solutions. Asymptotic Preserving methods represent one class among the possible methodologies. logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 4 / 51
Introduction The AP diagram ε → 0 ✲ P ε P 0 ✻ ✻ ∆ t → 0 ∆ t → 0 ✲ P ε P 0 ∆ t ∆ t ε → 0 In the diagram P ε is the original singular perturbation problem and P ε ∆ t its numerical approximation characterized by a discretization parameter ∆ t . The asymptotic-preserving (AP) property corresponds to the request that P ε ∆ t is a consistent discretization of P 0 as ε → 0 independently of ∆ t . logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 5 / 51
Introduction Near continuum flow Euler or Navier-Stokes region Boltzmann region ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ������������������ ε > 0.01 ε << 0.01 logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 6 / 51
The Boltzmann equation Outline for section 2 Introduction 1 The Boltzmann equation 2 The hydrodynamic limit 3 The diffusive limit 4 Asymptotic Preserving methods and domain decomposition 5 Multiscale problems 6 logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 7 / 51
The Boltzmann equation The kinetic model In the Boltzmann description of RGD 1 , the density f = f ( x, v, t ) of particles follows the equation ∂f ∂t + v · ∇ x f = 1 x ∈ Ω ⊂ R 3 , v ∈ R 3 , εQ ( f, f ) , The parameter ε > 0 is called Knudsen number and it is proportional to the mean free path between collisions. The bilinear collisional operator Q ( f, f ) is given by � � Q ( f, f )( v ) = S 2 B ( | v − v ∗ | , ω )( f ( v ′ ) f ( v ′ ∗ ) − f ( v ) f ( v ∗ )) dv ∗ dω, R 3 where ω is a vector of the unitary sphere S 2 ⊂ R 3 and for simplicity the dependence of f on x and t has been omitted. The collisional velocities ( v ′ , v ′ ∗ ) are given by the relations v ′ = 1 ∗ = 1 v ′ 2( v + v ∗ + | q | ω ) , 2( v + v ∗ + | q | ω ) , where q = v − v ∗ is the relative velocity. logo 1 C.Cercignani ’88 Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 8 / 51
The Boltzmann equation Collision details The kernel B characterizes the details of the binary interactions. The classical Variable Hard Spheres (VHS) model used for RGD simulations is B ( | q | , ω ) = K | q | α , 0 ≤ α < 1 , where K is a positive constant. The case α = 0 corresponds to a Maxwellian gas , while α = 1 is called a Hard Sphere Gas . The collisional operator is such that the H-Theorem holds � R 3 Q ( f, f ) log( f ) dv ≤ 0 . This condition implies that each function f in equilibrium (i.e. Q ( f, f ) = 0 ) has locally the form of a Maxwellian distribution � � −| u − v | 2 ρ M ( ρ, u, T )( v ) = (2 πT ) 3 / 2 exp , 2 T where ρ, u, T are the density , the mean velocity and the gas temperature � � � T = 1 R 3 ( v − u ) 2 fdv. ρ = R 3 fdv, ρu = R 3 fvdv, 3 ρ logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 9 / 51
The Boltzmann equation Hydrodynamic equations If we consider the Boltzmann equation and multiply it for the elementary collisional invariants 1 , v, | v | 2 and integrate in v we obtain a system of conservation laws corresponding to conservation of mass, momentum and energy. Clearly the differential system is not closed since it involves higher order moments of the function f . Formally as ε → 0 the function f is locally replaced by a Maxwellian. In this case it is possible to compute f from its low order moments thus obtaining to leading order the closed system of compressible Euler equations 3 � ∂ρ ∂ + ( ρu i ) = 0 , ∂t ∂x i i =1 3 � ∂ ∂ ∂ ∂t ( ρu j ) + ( ρu i u j ) + p = 0 , j = 1 , 2 , 3 ∂x i ∂x j i =1 � 3 ∂E ∂ + ( Eu i + pu i ) = 0 , ∂t ∂x i i =1 logo where p = ρT . Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 10 / 51
The Boltzmann equation Numerical Challenges Physical conservation properties, positivity and entropy inequality are very important since they characterize the steady states. Methods that do not maintain such properties need special attention in practical applications. The interaction/collision operator may contain an highly dimensional integral in velocity space. In such cases fast solvers are essential to avoid excessive computational cost. The significant velocity range may vary strongly with space position (steady states may not be compactly supported in velocity space). Thus methods that use a finite velocity range require care and may be inadequate in some circumstances. Stiffness of the problem for small free paths and/or large velocities. Stiff solvers for small free path problems may be hard to use when we have to invert a large non linear system. logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 11 / 51
The Boltzmann equation Main goal The goal is to construct simple and efficient time discretizations for the solution of kinetic equations in regions with a large variation of the mean free path. Requirements For large Knudsen numbers, the methods behave as standard explicit methods. For intermediate Knudsen numbers, the methods are capable to speed up the computation, allowing larger time steps, without degradation of accuracy. In the limit of very small Knudsen numbers, the collision step replaces the distribution function by the local Maxwellian. This property is usually referred to as asymptotic preserving (AP) since it implies consistency with the underlying system of Euler equations of gas dynamics. An high order accuracy should be maintained both in space and time by the numerical scheme for all range of Knudsen numbers. We refer in this case to as asymptotic accurate (AA) schemes. logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 12 / 51
The hydrodynamic limit Outline for section 3 Introduction 1 The Boltzmann equation 2 The hydrodynamic limit 3 The diffusive limit 4 Asymptotic Preserving methods and domain decomposition 5 Multiscale problems 6 logo Giacomo Dimarco (Mathematics Department) On the time integration of the Boltzmann equation Ofir, 30 April 2014 13 / 51
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