[Paper Review] High order conservative Semi-Lagrangian scheme for the BGK model of the Boltzmann equation

Sebastiano Boscarino, Seung Yun Cho+2 more|arXiv (Cornell University)|May 9, 2019

Gas Dynamics and Kinetic Theory19 references4 citations

TL;DR

This paper presents a high-order conservative semi-Lagrangian finite-difference scheme for the BGK model of the Boltzmann equation, addressing two key conservation failures: (1) discrete velocity space resolution errors via the use of a discrete Maxwellian, and (2) reconstruction-induced conservation loss via a flux-difference conservative correction. The scheme achieves machine-precision conservation of mass, momentum, and energy with minimal velocity grid points and is asymptotic preserving for the Euler limit as the Knudsen number vanishes.

ABSTRACT

In this paper, we present a conservative semi-Lagrangian finite-difference scheme for the BGK model. Classical semi-Lagrangian finite difference schemes, coupled with an L-stable treatment of the collision term, allow large time steps, for all the range of Knudsen number. Unfortunately, however, such schemes are not conservative. There are two main sources of lack of conservation. First, when using classical continuous Maxwellian, conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points. However, for a small number of grids in velocity space such error is not negligible, because the parameters of the Maxwellian do not coincide with the discrete moments. Secondly, the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme. As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number. To treat the first problem and ensure machine precision conservation of mass, momentum and energy with a relatively small number of velocity grid points, we replace the continuous Maxwellian with the discrete Maxwellian introduced by Mieussens. The second problem is treated by implementing a conservative correction procedure based on the flux difference form. In this way we can construct a conservative semi-Lagrangian scheme which is Asymptotic Preserving (AP) for the underlying Euler limit, as the Knudsen number vanishes. The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

Motivation & Objective

• Address the lack of conservation in classical semi-Lagrangian schemes for the BGK model, particularly in low-velocity-grid regimes.
• Correct the loss of translation invariance caused by non-linear reconstruction, which distorts shock speeds in the small Knudsen number limit.
• Ensure machine-precision conservation of mass, momentum, and energy using a discrete Maxwellian instead of the continuous Maxwellian.
• Develop a scheme that is asymptotic preserving (AP) for the Euler limit as the Knudsen number approaches zero.
• Achieve high-order accuracy while maintaining conservation and stability over large time steps via L-stable treatment of the collision term.

Proposed method

• Replace the continuous Maxwellian with the discrete Maxwellian from [17] to ensure exact conservation of moments even with coarse velocity grids.
• Implement a conservative correction procedure based on the flux difference form, as in [21], to restore conservation properties lost during non-linear reconstruction.
• Use a DIRK (Diagonally Implicit Runge-Kutta) or BDF (Backward Differentiation Formula) time discretization with L-stable treatment of the BGK collision term to allow large time steps.
• Apply high-order interpolation in space and time to compute the characteristic trajectories in the semi-Lagrangian framework.
• Enforce periodic boundary conditions to preserve total mass, momentum, and energy across the domain.
• Derive discrete conservation error estimates using the flux-difference correction and moment consistency bounds, proving convergence to machine precision under appropriate stability conditions.

Experimental results

Research questions

• RQ1 How can a semi-Lagrangian scheme for the BGK model achieve machine-precision conservation of mass, momentum, and energy with a small number of velocity grid points?
• RQ2 What causes the loss of conservation in classical semi-Lagrangian schemes when using continuous Maxwellians and non-linear reconstruction?
• RQ3 Can a conservative correction based on flux differences restore conservation while preserving high-order accuracy and stability?
• RQ4 Does the proposed scheme maintain asymptotic preserving (AP) properties in the small Knudsen number limit, ensuring correct Euler dynamics?
• RQ5 What is the theoretical error bound for moment conservation in the discrete scheme, and how does it scale with time step and tolerance?

Key findings

• The use of the discrete Maxwellian instead of the continuous Maxwellian eliminates moment conservation errors due to coarse velocity grid resolution, enabling machine-precision conservation even with few velocity points.
• The flux-difference conservative correction successfully restores translation invariance and prevents spurious shock speed errors in the small Knudsen number regime.
• The scheme achieves asymptotic preserving (AP) behavior: as the Knudsen number κ → 0, the moments of the solution converge to the compressible Euler equations.
• Conservation error estimates are derived for both DIRK and BDF schemes, showing that the total moment error is bounded by a term proportional to the tolerance of the reconstruction and the time step, with explicit constants depending on the Butcher tableau coefficients.
• For DIRK schemes of order s=1,2,3, the conservation error is bounded by (Σ|bₖ|) * (NₜΔt)/(κ + bₛΔt) * (x_max - x_min) * tol, ensuring convergence to machine precision as tol → 0.
• For BDF schemes of order s=2,3, the error bound includes additional factors γₛ and βₛ, with γ₂ = 3/2 and γ₃ = 146/11, reflecting increased complexity in the time-stepping but still ensuring bounded conservation error.