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Implicit Multistage Two-Derivative Discontinuous Galerkin Schemes for Viscous Conservation Laws

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Abstract

In this paper we apply implicit two-derivative multistage time integrators to conservation laws in one and two dimensions. The one dimensional solver discretizes space with the classical discontinuous Galerkin method, and the two dimensional solver uses a hybridized discontinuous Galerkin spatial discretization for efficiency. We propose methods that permit us to construct implicit solvers using each of these spatial discretizations, wherein a chief difficulty is how to handle the higher derivatives in time. The end result is that the multiderivative time integrator allows us to obtain high-order accuracy in time while keeping the number of implicit stages at a minimum. We show numerical results validating and comparing methods.

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Acknowledgments

We would like to thank the anonymous reviewers for their helpful comments and suggestions to improve the quality of this work.

Author information

Authors and Affiliations

  1. Vakgroep wiskunde en statistiek, Hasselt University, Agoralaan Gebouw D, 3590, Diepenbeek, Belgium

    Alexander Jaust & Jochen Schütz

  2. Department of Mathematics, United States Naval Academy, 121 Blake Road, Annapolis, MD, 21402, USA

    David C. Seal

Authors
  1. Alexander Jaust
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  2. Jochen Schütz
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  3. David C. Seal
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Correspondence to Alexander Jaust.

Tabulated results

Tabulated results

In this section we present complete error tables for the data presented summarily throughout the paper (Tables 1, 2, 3, 4, 5, 6, 7 and 8).

Table 1 Numerical results for the heat equation
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Table 2 Numerical results for the convection equation
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Table 3 Numerical results for the convection–diffusion equation with smooth initial condition
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Table 4 Results for the convection–diffusion equation
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Table 5 Numerical results for Burgers equation
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Table 6 Numerical results for the linear coupled advection equation
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Table 7 Numerical results for the linear coupled advection equation
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Table 8 Numerical results for the Euler equations
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Jaust, A., Schütz, J. & Seal, D.C. Implicit Multistage Two-Derivative Discontinuous Galerkin Schemes for Viscous Conservation Laws. J Sci Comput 69, 866–891 (2016). https://doi.org/10.1007/s10915-016-0221-x

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