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On the Linear Stability of the Fifth-Order WENO Discretization

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Abstract

We study the linear stability of the fifth-order Weighted Essentially Non-Oscillatory spatial discretization (WENO5) combined with explicit time stepping applied to the one-dimensional advection equation. We show that it is not necessary for the stability domain of the time integrator to include a part of the imaginary axis. In particular, we show that the combination of WENO5 with either the forward Euler method or a two-stage, second-order Runge–Kutta method is linearly stable provided very small time step-sizes are taken. We also consider fifth-order multistep time discretizations whose stability domains do not include the imaginary axis. These are found to be linearly stable with moderate time steps when combined with WENO5. In particular, the fifth-order extrapolated BDF scheme gave superior results in practice to high-order Runge–Kutta methods whose stability domain includes the imaginary axis. Numerical tests are presented which confirm the analysis.

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Authors and Affiliations

  1. Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada

    Mohammad Motamed & Steven J. Ruuth

  2. Mathematical Institute, University of Oxford, Oxford, OX1 3LB, UK

    Colin B. Macdonald

Authors
  1. Mohammad Motamed
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  2. Colin B. Macdonald
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  3. Steven J. Ruuth
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Correspondence to Mohammad Motamed.

Additional information

The work of M. Motamed was partially supported by NSERC Canada.

The work of C.B. Macdonald was supported by NSERC Canada, NSF grant number CCF-0321917, and by Award No KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST).

The work of S.J. Ruuth was partially supported by NSERC Canada.

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Motamed, M., Macdonald, C.B. & Ruuth, S.J. On the Linear Stability of the Fifth-Order WENO Discretization. J Sci Comput 47, 127–149 (2011). https://doi.org/10.1007/s10915-010-9423-9

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  • DOI: https://doi.org/10.1007/s10915-010-9423-9

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