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Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods

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Abstract

Discontinuous Galerkin finite element spatial discretizations are often used in a method-of-lines approach with an ordinary differential equation solver to step the solution forward in time. Explicit strong-stability-preserving time steppers are a popular choice because they provably preserve the nonlinear stability properties of the forward Euler method applied to discontinuous Galerkin semi-discretized equations, subject to a time step constraint. While nonlinear stability is guaranteed by strong-stability-preservation, a separate condition for linear stability of the combined scheme must also be satisfied. In this work, we assess the linear stability properties of discontinuous Galerkin spatial discretizations with a set of strong-stability-preserving multistep Runge–Kutta methods. We find that, in all cases, the constraint for linear stability is more strict than that for strong-stability-preservation. For each order, from the set of multistep Runge–Kutta methods, we select an optimal time stepper that requires the fewest evaluations of the discontinuous Galerkin operator. All methods are tested for convergence in application to both a linear and a nonlinear partial differential equation, and all methods are found to converge in both cases using the maximum stable time step as determined by the stability constraints found in this paper.

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Funding

The authors would like to acknowledge funding from NSF Grants EAR-1520870 and ICER-1855047.

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Authors and Affiliations

  1. New York University, New York, USA

    Benjamin Yeager

  2. The Ohio State University, Columbus, OH, USA

    Ethan Kubatko

  3. Notre Dame University, South Bend, IN, USA

    Dylan Wood

Authors
  1. Benjamin Yeager
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  2. Ethan Kubatko
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Correspondence to Benjamin Yeager.

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A Time Step Restrictions for SSP MSRK Methods with DG Spatial Discretizations

A Time Step Restrictions for SSP MSRK Methods with DG Spatial Discretizations

See Tables 2, 3, 4, 5, 6, 7, 8, 9 and 10.

Table 2 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=2\) with DG spatial discretization of degree \(p=1\)
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Table 3 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=3\) with DG spatial discretization of degree \(p=2\)
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Table 4 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=4\) with DG spatial discretization of degree \(p=3\)
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Table 5 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=5\) with DG spatial discretization of degree \(p=4\)
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Table 6 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=6\) with DG spatial discretization of degree \(p=5\)
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Table 7 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=7\) with DG spatial discretization of degree \(p=6\)
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Table 8 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=8\) with DG spatial discretization of degree \(p=7\)
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Table 9 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=9\) with DG spatial discretization of degree \(p=8\)
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Table 10 Effective stability constraints \(\mu _\mathrm {eff}\) and percent differences \(\varsigma \) between numerical and theoretical stability constraint estimates for SSP RK, LM, and MSRK methods of order \(q=10\) with DG spatial discretization of degree \(p=9\)
Full size table

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Yeager, B., Kubatko, E. & Wood, D. Time Step Restrictions for Strong-Stability-Preserving Multistep Runge–Kutta Discontinuous Galerkin Methods. J Sci Comput 89, 29 (2021). https://doi.org/10.1007/s10915-021-01635-4

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  • DOI: https://doi.org/10.1007/s10915-021-01635-4

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