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On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit

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Abstract

Implicit–explicit (IMEX) Runge–Kutta (RK) schemes are popular high order time discretization methods for solving stiff kinetic equations. As opposed to the compressible Euler limit (leading order asymptotics of the Boltzmann equation as the Knudsen number \(\varepsilon \) goes to zero), their asymptotic behavior at the Navier–Stokes (NS) level (next order asymptotics) was rarely studied. In this paper, we analyze a class of existing IMEX RK schemes and show that, under suitable initial conditions, they can capture the NS limit without resolving the small parameter \(\varepsilon \), i.e., \(\varepsilon =o(\Delta t)\), \(\Delta t^m=o(\varepsilon )\), where m is the order of the explicit RK part in the IMEX scheme. Extensive numerical tests for BGK and ES-BGK models are performed to verify our theoretical results.

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Author information

Correspondence to Jingwei Hu.

Additional information

J. Hu’s research was supported by NSF Grant DMS-1620250 and NSF CAREER Grant DMS-1654152. Support from DMS-1107291: RNMS KI-Net is also gratefully acknowledged. X. Zhang’s research was supported by NSF Grant DMS-1522593.

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Hu, J., Zhang, X. On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit. J Sci Comput 73, 797–818 (2017). https://doi.org/10.1007/s10915-017-0499-3

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Keywords

  • Boltzmann equation
  • BGK/ES-BGK models
  • IMEX Runge–Kutta schemes
  • Compressible Euler equations
  • Navier–Stokes equations

Mathematics Subject Classification

  • 35Q20
  • 65L06
  • 65L04
  • 35Q30
  • 35Q31