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Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes

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An Erratum to this article was published on 23 March 2016

Abstract

High order strong stability preserving (SSP) time discretizations are advantageous for use with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. The search for high order strong stability time-stepping methods with large allowable strong stability time-step has been an active area of research over the last two decades. Recently, multiderivative time-stepping methods have been implemented with hyperbolic PDEs. In this work we describe sufficient conditions for a two-derivative multistage method to be SSP, and find some optimal SSP multistage two-derivative methods. While explicit SSP Runge–Kutta methods exist only up to fourth order, we show that this order barrier is broken for explicit multi-stage two-derivative methods by designing a three stage fifth order SSP method. These methods are tested on simple scalar PDEs to verify the order of convergence, and demonstrate the need for the SSP condition and the sharpness of the SSP time-step in many cases.

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Acknowledgments

This work was supported by: AFOSR Grants FA9550-12-1-0224, FA9550-12-1-0343, FA9550-12-1-0455, FA9550-15-1-0282, and FA9550-15-1-0235; NSF Grant DMS-1418804; New Mexico Consortium Grant NMC0155-01; and NASA Grant NMX15AP39G.

Author information

Authors and Affiliations

  1. Department of Computational Mathematics Science and Engineering, Department of Electrical Engineering and Department of Mathematics, Michigan State University, East Lansing, MI, USA

    Andrew J. Christlieb

  2. Department of Mathematics, University of Massachusetts, North Dartmouth, MA, USA

    Sigal Gottlieb & Zachary Grant

  3. Department of Mathematics, U.S. Naval Academy, Annapolis, MD, USA

    David C. Seal

Authors
  1. Andrew J. Christlieb
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  2. Sigal Gottlieb
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  3. Zachary Grant
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  4. David C. Seal
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Corresponding author

Correspondence to Zachary Grant.

Appendices

Appendix 1: Coefficients of Three Stage Fourth Order Methods

  1. 1.

    For \(K=\frac{1}{2}\) we obtain an SSP coefficient \(\mathcal{{C}}=r =1.1464\). The Butcher array coefficients are given by

    $$\begin{aligned} A= & {} \left[ \begin{array}{lll} 0 &{} \quad 0 \quad &{} \quad 0\\ 0.436148675945340 &{} \quad 0 &{} \quad 0 \\ 0.546571371212865 &{} \quad 0.156647174804152 &{} \quad 0 \end{array} \right] , \; \; \; \; b = \left[ \begin{array}{l} 0.528992280543542 \\ 0.105732787708912 \\ 0.365274931747546 \end{array} \right] \\ \hat{A}= & {} \left[ \begin{array}{lll} 0 &{} \quad 0 &{} \quad 0 \\ 0.095112833764436 &{} \quad 0 &{} \quad 0 \\ 0.071032477596813 &{} \quad 0.107904226252921 &{} \quad 0 \end{array} \right] , \; \; \; \; \hat{b} = \left[ \begin{array}{lll} 0.074866026156687 \\ 0.073410341982927 \\ 0.048740310097159 \end{array} \right] . \end{aligned}$$

    The Shu–Osher arrays are \(R \mathbf{e}= (1,0,0,0)^T\),

    $$\begin{aligned} P= & {} \left[ \begin{array}{l l l l} 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.5000&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.253176729307242 &{} \quad 0.179580018745470 &{} \quad 0&{} \quad 0\\ 0.181986129712082 &{} \quad 0.000000001889650 &{} \quad 0.418750476492704 &{} \quad 0 \\ \end{array} \right] \\ Q= & {} \left[ \begin{array}{l l l l} 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.5&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0 &{} \quad 0.567243251947287 &{} \quad 0 &{} \quad 0\\ 0.140002406985210 &{} \quad 0.003037359947341&{} \quad 0.256223624973014 &{} \quad 0\ \end{array} \right] \end{aligned}$$
  2. 2.

    For \(K=\sqrt{\frac{1}{2}}\) we obtain an SSP coefficient \(\mathcal{{C}}= r= 1.3927\) Butcher formulation

    $$\begin{aligned} A= & {} \left[ \begin{array}{lll} 0 &{} \quad 0 &{} \quad 0\\ 0.443752012194422 &{} \quad 0 &{} \quad 0 \\ 0.543193299768317 &{} \quad 0.149202742858795 &{} \quad 0 \end{array} \right] , \; \; \; \; b = \left[ \begin{array}{l} 0.515040964378407 \\ 0.178821699719783 \\ 0.306137335901811 \end{array} \right] \\ \hat{A}= & {} \left[ \begin{array}{lll} 0 &{} \quad 0 &{} \quad 0\\ 0.098457924163299 &{} \quad 0 &{} \quad 0 \\ 0.062758211639901 &{} \quad 0.110738910914425 &{} \quad 0 \end{array} \right] , \; \; \; \; \hat{b} = \left[ \begin{array}{l} 0.072864982225864 \\ 0.073840478463180 \\ 0.061973770357455 \end{array} \right] . \end{aligned}$$

    The Shu–Osher arrays are \(R \mathbf{e}= (1,0,0,0)^T\),

    $$\begin{aligned} P= & {} \left[ \begin{array}{l l l l} 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.618033988749895&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.362588515112176 &{} \quad 0.207801573327953 &{} \quad 0&{} \quad 0\\ 0.144580879241747&{} \quad 0.110491604448675 &{} \quad 0.426371652664792 &{} \quad 0\\ \end{array} \right] \\ Q= & {} \left[ \begin{array}{l l l l} 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.381966011250105&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0 &{} \quad 0.429609911559871 &{} \quad 0 &{} \quad 0\\ 0.078129569197367 &{} \quad 0&{} \quad 0.240426294447419 &{} \quad 0\ \end{array} \right] . \end{aligned}$$
  3. 3.

    For \(K=1\) we obtain an SSP coefficient \(\mathcal{{C}}=r =1.6185\). The Butcher array coefficients are given by

    $$\begin{aligned} A= & {} \left[ \begin{array}{lll} 0 &{} \quad 0 &{} \quad 0 \\ 0.452297224196082 &{} \quad 0 &{} \quad 0 \\ 0.528050722182308&{} \quad 0.159236998008155 &{} \quad 0 \end{array} \right] , \; \; \; \; b = \left[ \begin{array}{l} 0.502519798444212\\ 0.210741084344740 \\ 0.286739117211047 \end{array} \right] \\ \hat{A}= & {} \left[ \begin{array}{lll} 0 &{} \quad 0 &{} \quad 0 \\ 0.102286389507741 &{} \quad 0 &{} \quad 0 \\ 0.055482128781494 &{} \quad 0.108677624192402 &{} \quad 0 \end{array} \right] , \; \; \; \; \hat{b} = \left[ \begin{array}{lll} 0.071256397204544 \\ 0.069475972085130 \\ 0.066877749079721 \end{array} \right] . \end{aligned}$$

    The Shu–Osher arrays are \(R \mathbf{e}= (1,0,0,0)^T\),

    $$\begin{aligned} P= & {} \left[ \begin{array}{llll} 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.732050807568877 &{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.457580533944888 &{} \quad 0.257727809835459 &{} \quad 0&{} \quad 0\\ 0.137886171629970 &{} \quad 0.176326540063367 &{} \quad 0.464092174540814 &{} \quad 0\\ \end{array} \right] , \\ Q= & {} \left[ \begin{array}{l l l l} 0&{} \quad 0&{} \quad 0&{} \quad 0 \\ 0.267949192431123 &{} \quad 0&{} \quad 0&{} \quad 0 \\ 0 &{}\quad 0.284691656219654 &{} \quad 0 &{} \quad 0\\ 0.046502314960818 &{} \quad 0&{} \quad 0.175192798805030 &{} \quad 0\ \end{array} \right] . \end{aligned}$$

Appendix 2: Two Stage Third Order Method

This code gives the SSP coefficient and the Butcher and Shu Osher arrays for the optimal explicit SSP two stage third order method given the value K.

figure b

Appendix 3: Three Stage Fifth Order Method

This code gives the SSP coefficient and the Butcher and Shu Osher arrays for the optimal explicit SSP three stage fifth order method given the value K.

figure c

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Christlieb, A.J., Gottlieb, S., Grant, Z. et al. Explicit Strong Stability Preserving Multistage Two-Derivative Time-Stepping Schemes. J Sci Comput 68, 914–942 (2016). https://doi.org/10.1007/s10915-016-0164-2

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