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A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources
Francis Filbet, S. Jin
TL;DR
Stiff nonlinear collision terms make kinetic equations difficult to solve efficiently while preserving the fluid-dynamic limit. The paper penalizes the Boltzmann collision operator with a BGK relaxation term, yielding a uniformly stable asymptotic-preserving scheme whose implicit stiff part is explicitly invertible and which captures Euler behavior without resolving the Knudsen scale.
Problem
Small Knudsen numbers create stiff nonlinear collision terms, making explicit schemes unstable and implicit schemes costly because they require nonlinear solves.
Method
The scheme penalizes the Boltzmann collision operator with a BGK operator, treating the less-stiff remainder explicitly and the stiff BGK term implicitly but explicitly invertibly.
Results
The method is uniformly stable in the Knudsen number, preserves the Euler limit without resolving the small scale, and is Navier–Stokes-consistent when viscous terms are resolved.
Takeaways & Limitations
BGK penalization provides a general route to asymptotic-preserving schemes without nonlinear iterative solvers or the Wild Sum procedure.
Takeaways & Limitations
In a mixed-regime test, the authors could not compare the numerical solution with one obtained from a macroscopic model.
Abstract
from arXiv · showhide
In this paper, we propose a general framework to design asymptotic preserving schemes for the Boltzmann kinetic kinetic and related equations. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We propose to penalize the nonlinear collision term by a BGK-type relaxation term, which can be solved explicitly even if discretized implicitly in time. Moreover, the BGK-type relaxation operator helps to drive the density distribution toward the local Maxwellian, thus natually imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver or the use of Wild Sum. It is uniformly stable in terms of the (possibly small) Knudsen number, and can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. It is also consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. The method is applicable to many other related problems, such as hyperbolic systems with stiff relaxation, and high order parabilic equations.
1. Introduction
The paper frames asymptotic-preserving schemes as a way to solve Boltzmann equations across kinetic and fluid regimes despite stiff nonlinear collisions. It proposes BGK penalization to preserve fluid limits while avoiding nonlinear implicit solves.
- Kinetic and fluid limits: The Boltzmann equation models dilute-gas particle distributions, with the Knudsen number measuring the mean free path relative to a typical length scale.Its collision operator is quadratic and conserves mass, momentum, and energy.
- Kinetic and fluid limits: As ε → 0, the distribution converges to a local Maxwellian and the conserved moments satisfy compressible Euler equations.For small nonzero ε, Chapman–Enskog expansion gives Navier–Stokes equations as a second-order approximation.
- Asymptotic-preserving design: An AP scheme must remain suitable for the kinetic equation at fixed mesh and time step while becoming a scheme for the Euler limit as the Knudsen number vanishes.It should also implement implicit collision terms explicitly or more efficiently than Newton-type nonlinear solvers.
- Asymptotic-preserving design: Implicit treatment is needed for uniform stability, but inverting the nonlocal nonlinear collision operator is numerically delicate.Earlier penalization and Wild Sum approaches captured the Euler limit but were reported not to capture the compressible Navier–Stokes asymptotic for small ε.
- Proposed framework: The proposed method penalizes the Boltzmann collision operator with BGK relaxation, treating the less-stiff residual explicitly and the stiff BGK term implicitly but explicitly invertibly.λ is the largest spectrum of the linearized collision operator around the local Maxwellian, and the resulting scheme is uniformly stable in ε.
- Proposed framework: The BGK penalty drives f toward the local Maxwellian, preserving the Euler limit and yielding Navier–Stokes consistency for sufficiently small time steps when ε is small.The framework also applies to nonlinear hyperbolic systems with stiff relaxation and higher-order parabolic equations.
2. An Asymptotic Preserving (AP) stiff ODE solver
The paper develops AP IMEX solvers for stiff nonlinear ODEs that avoid nonlinear solves while remaining stable as ε becomes small. A BGK-like linear penalty preserves the equilibrium limit, and second-order schemes retain accuracy on under-resolved fast–slow systems.
- Scope: The framework is presented as applicable beyond the prototype ODE setting, including kinetic equations and other stiff-source partial differential equations.The ODE solver is introduced independently of spatial discretization because its design relies on the stiff-source structure.
- Motivation: The solver targets stiff nonlinear source terms by combining large implicit time steps with low-cost linear updates, avoiding fully nonlinear solves.The motivation is to retain implicit stability for small ε while keeping each time step computationally inexpensive.
- Method: The source is split using a well-balanced linear operator P with P(M) = 0, while the remaining part is treated explicitly.The resulting first-order IMEX update is linear in the right-hand side and therefore easy to implement.
- Stability: For Q(f) = −λf, the first-order scheme is A-stable and L-stable when ν > 1/2.ν = 0 gives explicit Euler, ν = 1 gives implicit Euler, and ν > 1/2 is the stability condition stated for the theorem.
- Asymptotic preservation: The BGK-type penalty drives f toward the equilibrium M, yielding the AP property for prepared data in both first- and second-order schemes.If f^n = M + O(ε), the intermediate and updated states remain M + O(ε).
- Numerical test: The second-order IMEX scheme remains stable and accurate for a large time step Δt = 0.3 on a system with fast eigenvalues near −1000 ± i and a slow oscillatory eigenvalue i.At the same time step, explicit second-order Runge–Kutta blows up, whereas the first-order AP method damps the slow oscillation.
3. Application to the Boltzmann equation
The Boltzmann equation is rewritten with a well-balanced BGK relaxation approximation of the collision operator, enabling an explicitly solvable implicit treatment. The resulting scheme preserves the Euler limit and is consistent with compressible Navier–Stokes behavior when viscous scales are resolved.
- Scheme construction: The collision operator is approximated using a well-balanced relaxation operator P that preserves Maxwellian steady states and can be chosen as a BGK operator.P is constructed from a local expansion of Q around the Maxwellian, with β chosen as an upper bound for the linearized operator.
- Scheme construction: The convection term is treated explicitly, while the stiff source contribution is handled by the implicit-explicit relaxation solver.The implicit BGK operator can nevertheless be inverted explicitly, avoiding nonlinear inversion of Q.
- Scheme construction: Although the update appears nonlinearly implicit, conserved moments determine U^(n+1) explicitly, after which the Maxwellian and distribution update are computed.This establishes an explicit solution procedure for the apparently nonlinear update.
- Asymptotic properties: For ε → 0 and f^n = M^n + O(ε), the method remains asymptotic preserving and its moments consistently discretize the compressible Euler system.The proof checks that Maxwellian data remain Maxwellian in the limit and that the moment update becomes an Euler discretization.
- Asymptotic properties: For ε ≪ 1 and sufficiently small time steps, the scheme becomes a first-order temporal approximation of compressible Navier–Stokes equations.Capturing the O(ε) viscosity and heat conductivity requires mesh size and cΔt to be o(ε), whereas the Euler limit allows scales much larger than ε.
4. Numerical tests
Numerical tests show that the proposed schemes remain accurate and stable across kinetic, Navier–Stokes, Euler, and mixing regimes, including unresolved small Knudsen numbers.
- Smooth solutions: The second-order method achieves uniformly second-order convergence, with an estimated slope of 1.9, while remaining stable without an ε-dependent time-step restriction.The velocity discretization is spectrally accurate and contributes little to the error.
- Sod tube problem: For ε = 10−2, the second-order AP method agrees well with an explicit second-order Runge–Kutta scheme using nx = 100 and nv = 32.In this kinetic regime, the AP method attains comparable accuracy without additional computational effort.
- Sod tube problem: For ε = 10−3, macroscopic quantities and heat flux agree very well with explicit Runge–Kutta results despite a time step at least ten times larger.The explicit comparison uses a time step of order O(ε), whereas the AP method avoids that restriction.
- Mixing regimes: When ε varies from 10−4 to 1, the fully explicit computation would take more than one hundred times longer, while the AP scheme accurately propagates discontinuities.The results provide a compromise between accuracy and stability across the different regimes.
5. Other applications: numerical stability
The framework is also applied to a nonlinear degenerate parabolic equation, where an IMEX discretization removes the severe explicit time-step constraint while preserving stability and nonnegativity.
- Asymptotic behavior: The transformed parabolic solution converges to the Barenblatt–Pattle distribution, whose constant depends on the initial mass but not on details of the initial data.The asymptotic analysis is framed through entropy or relative entropy decay toward equilibrium.
- Problem and motivation: The parabolic test addresses explicit schemes constrained by Δt ≃ O(Δv^2) and implicit schemes requiring a nonlinear solve at every time step.This motivates applying the paper’s splitting technique to a high-order differential operator.
- Numerical method: The collision operator is decomposed using a linearized equilibrium-directed penalty, followed by an IMEX scheme requiring only a linear-system solve per time step.The parameter ε is tied to velocity discretization rather than a physical time scale in this example.
- Numerical results: With Δt = 0.02, much larger than the classical CFL scale, the scheme remains stable and preserves nonnegativity at every time step.The solution converges toward an approximation of the steady state, although the scheme is not exactly well-balanced.
6. Conclusion
The paper proposes BGK penalization for stiff nonlinear collision terms, yielding asymptotic-preserving schemes that are easier to solve and extend beyond the Boltzmann equation.
- Main contribution: The framework replaces the difficult implicit nonlinear Boltzmann collision operator with an implicitly treated BGK operator that is easier to handle numerically.The approach targets problems with multiple time and spatial scales and nonlinear stiff source terms.
- Main findings: The method is asymptotic preserving in the Euler regime and is supported by numerical examples with mixing scales and non-local-Maxwellian initial data.The conclusion also reports uniform convergence in the Knudsen number.
- Scope: The approach extends to PDEs or ODEs with stiff source terms possessing a stable and unique local equilibrium.The Fokker–Planck equation serves as an example of this broader applicability.