High-order Asymptotic-Preserving schemes for the Boltzmann equation - PowerPoint PPT Presentation

Introduction IMEX Runge-Kutta Exponential schemes Further developments High-order Asymptotic-Preserving schemes for the Boltzmann equation and related problems Lorenzo Pareschi Department of Mathematics & Computer Science University of Ferrara, Italy http://lorenzopareschi.com Joint research with: S. Boscarino, G.Russo (University of Catania, Italy) G. Dimarco (University of Toulouse, France) Q. Li (University of Maryland, USA) Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 1 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments Motivations x 10 4 11 The computation of fluid-kinetic interfaces and asymptotic 2 10 9 8 1.5 behaviors involves multiple scales where most numerical 7 y 6 5 1 methods lose their efficiency because they are forced to 4 3 2 0.5 operate on a very short time scale. 1 2 4 6 8 10 12 14 x 0 Partitioned time discretizations represent a powerful tool for the numerical treatment of stiff terms in PDEs. When necessary they can be designed in order to achieve suitable asymptotic preserving ( AP ) properties. Similar techniques can be adopted when dealing with kinetic equation of Boltzmann-type . Here, however, the major challenge is represented by the complicated nonlinear structure of the collisional operator which makes prohibitively expensive the use of implicit solvers for the stiff collision term. Additional difficulties are given by the need to preserve some relevant physical properties like conservation of mass, momentum and energy, nonnegativity of the solution, and entropy inequality. Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 2 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments Outline Introduction 1 The Implicit-Explicit (IMEX) paradigm The asymptotic-preserving (AP) property Kinetic equations IMEX Runge-Kutta 2 IMEX-RK for easy invertible collision operators Penalized IMEX-RK for the Boltzmann equation Exponential schemes 3 Exponential schemes for homogeneous equations Extension to non homogeneous problems Further developments 4 Multistep IMEX schemes Final considerations Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 3 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments The Implicit-Explicit (IMEX) paradigm The Implicit-Explicit (IMEX) paradigm Many practical application involves systems of differential equations of the form U ′ = F ( U ) + G ( U ) , � �� � � �� � non stiff term stiff term where F and G , eventually obtained as suitable finite-difference or finite-element approximations of spatial derivatives ( method of lines ), induce considerably different time scales. The use of fully implicit solvers originates a nonlinear system of equations involving also the non stiff term F . Thus it is highly desirable to have a combination of implicit and explicit discretization terms to resolve stiff and non–stiff dynamics accordingly. IMEX methods have been developed to deal with the numerical integration of hyperbolic balance laws , kinetic equations , convection–diffusion equations and singular perturbed problems . Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 5 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments The Implicit-Explicit (IMEX) paradigm A simple example Consider the singularly perturbed problem 1 Singularly perturbed problem � u ′ ( t ) = f ( u, v ) , P ε : εv ′ ( t ) = g ( u, v ) , ε > 0 . As ε → 0 we get the index 1 differential algebraic equation (DAE) u ′ ( t ) = f ( u, v ) , 0 = g ( u, v ) . Assuming that g ( u, v ) = 0 ⇔ v = E ( u ) we obtain P 0 : u ′ ( t ) = f ( u, E ( u )) . Explicit methods : restricted to ∆ t ∼ ε . Implicit methods : require the numerical inversion of g ( u, v ) and as ε → 0 must satisfy the algebraic condition g ( u, v ) = 0 ⇔ v = E ( u ) . 1 E.Hairer, C.Lubich, M.Roche ’89 Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 6 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments The asymptotic-preserving (AP) property The AP diagram ε → 0 ✲ P ε P 0 ✻ ✻ ∆ t → 0 ∆ t → 0 ✲ P 0 P ε ∆ 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 . Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 8 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments The asymptotic-preserving (AP) property Numerical approaches The simplest approach is based on splitting methods where we solved separately the subproblems U ′ = F ( U ) , U ′ = G ( U ) . Easy to analyze and achieve AP property, possible to use existing solvers for the simplified problems and to preserve some relevant physical properties. Main drawback: order reduction in stiff regimes. Different approaches to achieve high-order AP schemes IMEX Runge-Kutta methods Exponential methods IMEX Multistep methods All the different approaches share the difficulty of the inversion of the collision operator due to its implicit evaluation. Here we will not discuss problems related to the discretization of other variables in the systems (like space and velocity), we will just mention in the numerical results the different choices we used (both deterministic and DSMC). Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 9 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments Kinetic equations Kinetic equations in the fluid-dynamic scaling The density f = f ( x, v, t ) ≥ 0 of particles follows 2 Kinetic model ∂f ∂t + v · ∇ x f = 1 x ∈ Ω ⊂ R d x , v ∈ R 3 , εQ ( f, f ) , which is written in this form after the scaling x → x/ε , t → t/ε where ε > 0 is a nondimensional parameter ( Knudsen number ) proportional to the mean free path. The structure of the collision operator Q ( f, f ) depends on the particular model. For example, the classical Boltzmann collision operator reads � � S 2 B ( | v − v ∗ | , ω )( f ( v ′ ) f ( v ′ Q ( f, f )( v ) = ∗ ) − f ( v ) f ( v ∗ )) dv ∗ dω, R 3 where B is a nonnegative kernel characterizing the binary interactions and v ′ = 1 ∗ = 1 v ′ 2( v + v ∗ + | v − v ∗ | ω ) , 2( v + v ∗ + | v − v ∗ | ω ) . 2 C.Cercignani ’88 Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 11 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments Kinetic equations Main properties The collision operator satisfies local conservation properties � Q ( f, f ) φ ( v ) dv = 0 , R dv where φ ( v ) = (1 , v, | v | 2 2 ) are the collision invariants and the entropy inequality � Q ( f, f ) log( f ) dv ≤ 0 . R dv From this we get Q ( f, f ) = 0 ⇔ f = M [ f ] where Maxwellian distribution � � −| u − v | 2 ρ M [ f ]( v ) = (2 πT ) 3 / 2 exp , 2 T with ρ, u, T the density , the mean velocity and the gas temperature � T = 1 3 ρ ( E − ρ | u | 2 ) . ( ρ, u, E ) = fφ ( v ) dv, R dv Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 12 / 50

Introduction IMEX Runge-Kutta Exponential schemes Further developments Kinetic equations Hydrodynamic equations If we multiply the kinetic equation for its collision invariants 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 . As ε → 0 formally Q ( f, f ) = 0 which implies f = M [ f ] and we get the closed system 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 , p = ρT. ∂t ∂x i i =1 Lorenzo Pareschi (University of Ferrara) AP schemes for the Boltzmann equation ICERM, June 3-8, 2013 13 / 50