[Paper Review] Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation

David Coulette, Emmanuel Franck+3 more|arXiv (Cornell University)|Dec 30, 2016

Lattice Boltzmann Simulation Studies5 references3 citations

TL;DR

This paper introduces a high-order, asymptotic-preserving Palindromic Discontinuous Galerkin Lattice Boltzmann Method for kinetic equations with stiff relaxation. It combines implicit upwind DG spatial discretization with palindromic time integration to achieve high-order accuracy (up to order 6), stability at high CFL numbers, and robustness for small relaxation times, demonstrating optimal convergence rates and stability in challenging test cases including Riemann problems with $ au > 0$. The method maintains asymptotic preservation and allows for efficient, low-storage computation via triangular linear systems.

ABSTRACT

We present a high order scheme for approximating kinetic equations with stiff relaxation. The objective is to provide efficient methods for solving the underlying system of conservation laws. The construction is based on several ingredients: (i) a high order implicit upwind Discontinuous Galerkin approximation of the kinetic equations with easy-to-solve triangular linear systems; (ii) a second order asymptotic-preserving time integration based on symmetry arguments; (iii) a palindromic composition of the second order method for achieving higher orders in time. The method is then tested at orders 2, 4 and 6. It is asymptotic-preserving with respect to the stiff relaxation and accepts high CFL numbers.

Motivation & Objective

• To develop a high-order numerical scheme for kinetic equations with stiff relaxation that maintains stability and accuracy even at high CFL numbers.
• To overcome the limitations of explicit schemes and standard LBM by introducing an implicit, upwind Discontinuous Galerkin (DG) spatial discretization with triangular linear systems.
• To achieve high-order temporal accuracy through palindromic composition of a symmetric second-order time integrator, ensuring asymptotic preservation as $ au \to 0$.
• To test the method’s robustness and convergence for stiff relaxation regimes, including viscous and contact discontinuity problems.
• To enable efficient, low-storage computation suitable for parallel and hybrid architectures, with potential extension to higher dimensions.

Proposed method

• The method employs an implicit upwind DG spatial discretization, where the resulting linear system is triangular and solvable explicitly, enabling high CFL numbers and stability.
• A second-order, symmetric time integrator is constructed based on symmetry arguments to preserve asymptotic behavior in the limit $ au \to 0$.
• Higher-order accuracy in time is achieved via palindromic composition of the second-order scheme, tested at orders 4 and 6 using Suzuki and Kahan-Li schemes.
• The kinetic model is based on Graille’s vectorial LBM, with $2m$ discrete velocities $v_i = \pm\lambda$, and equilibrium distributions defined to recover conservative variables $w$ and fluxes $q(w)$.
• The scheme is asymptotic-preserving: as $\tau \to 0$, the solution converges to the original hyperbolic conservation law system.
• Numerical tests are performed on 1D Riemann problems and viscous test cases, with convergence studies at orders 2, 4, and 6, and stability analysis for $\tau > 0$.

Experimental results

Research questions

• RQ1 Can an implicit DG method with upwind fluxes achieve high-order accuracy and stability at high CFL numbers in kinetic equations with stiff relaxation?
• RQ2 Does a palindromic composition of a symmetric second-order time integrator preserve asymptotic preservation for small relaxation times $\tau$?
• RQ3 How do schemes with complex time steps compare in stability and accuracy to real-step schemes when $\tau \approx \Delta t$ or $\tau < \Delta t$?
• RQ4 What is the convergence behavior of the method at orders 4 and 6 under high CFL conditions and stiff relaxation?
• RQ5 Can the method maintain accuracy and stability for viscous test cases with $\tau > 0$, particularly near critical values where standard schemes fail?

Key findings

• The method achieves optimal convergence rates of order 4 and 6 in space for the low Mach number test case at a CFL number of 100, with the fourth-order Suzuki and sixth-order Kahan-Li schemes showing expected convergence behavior.
• At high CFL numbers ($\beta = 100$), schemes with complex time steps become unstable, while the palindromic schemes remain stable and accurate.
• For $\tau = 0.000519$, the sixth-order Kahan-Li scheme fails due to a singularity condition ($2\tau + \gamma_2\Delta t/2 = 0$), but the complex-step scheme remains stable and produces results indistinguishable from the reference solution on both $N_x=100$ and $N_x=1000$ meshes.
• The method remains asymptotic-preserving: as $\tau \to 0$, the solution converges to the original hyperbolic conservation law, and the relaxation term acts like a viscosity for $\tau > 0$, consistent with Chapman-Enskog theory.
• The implicit DG solver produces triangular linear systems that are efficiently solved via forward substitution, enabling low-storage implementation and high efficiency.
• The method is robust for $\tau > 0$, with the complex-step scheme able to cross the $2\tau + \Delta t$ stability barrier at moderate CFL numbers, suggesting potential for broader applicability in stiff regimes.