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Dense Output for Strong Stability Preserving Runge–Kutta Methods

Abstract

We investigate dense output formulae (also known as continuous extensions) for strong stability preserving (SSP) Runge–Kutta methods. We require that the dense output formula also possess the SSP property, ideally under the same step-size restriction as the method itself. A general recipe for first-order SSP dense output formulae for SSP methods is given, and second-order dense output formulae for several optimal SSP methods are developed. It is shown that SSP dense output formulae of order three and higher do not exist, and that in any method possessing a second-order SSP dense output, the coefficient matrix A has a zero row.

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Acknowledgements

We thank an anonymous referee for several suggestions that improved the presentation of this work.

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Correspondence to David I. Ketcheson.

Additional information

Adil Kusmanov: This work was supported by the King Abdullah University of Science and Technology (KAUST), 4700 Thuwal, 23955-6900, Saudi Arabia. The second author was also supported by the Department of Numerical Analysis, Eötvös Loránd University, and the Department of Differential Equations, Budapest University of Technology and Economics, Hungary. The last two authors were supported by the KAUST Visiting Student Research Program.

Appendix: Some Complete Methods in Shu–Osher Form

Appendix: Some Complete Methods in Shu–Osher Form

We have worked with methods in Butcher form because of the simplicity of the order conditions in that form. In this “Appendix” we write out some of the methods with dense output in Shu–Osher form, since this is usually the most convenient form for implementation. For the sake of brevity we write the methods in autonomous form and let \({\mathcal {C}}={\mathcal {C}}(A,b,\overline{{{\varvec{b}}}})\).

The dense output in Shu–Osher form is

$$\begin{aligned} u_{n+\theta }&= \mu u_n + \sum _{j=1}^s {\overline{\beta }}_j(\theta ) \left( y_j + \frac{h}{{\mathcal {C}}}f(y_j)\right) \end{aligned}$$

where

$$\begin{aligned} {\overline{\beta }}^\top&:= \overline{{{\varvec{b}}}}^\top \left( I+{\mathcal {C}}A\right) ^{-1} \\ \mu&:= 1 - {\mathcal {C}}\overline{{{\varvec{b}}}}^\top \left( I + {\mathcal {C}}A\right) ^{-1}. \end{aligned}$$

We denote each method by SSP\((s,p,{\overline{p}})\), where s is the number of stages, p is the order of the method (Ab), and \({\overline{p}}\) is the order of the dense output.

SSP(2,2,2)

This method has \({\mathcal {C}}(A,b,\overline{{{\varvec{b}}}}) = 1\).

$$\begin{aligned} y_1&= u_n \\ y_2&= y_1 + h f(y_1) \\ u_{n+1}&= \frac{1}{2} u_n + \frac{1}{2} \left( y_2 + hf(y_2)\right) \\ u_{n+\theta }&= \left( 1 - \theta + \frac{1}{2}\theta ^2\right) u_n + (\theta -\theta ^2)\left( y_1+hf(y_1)\right) + \frac{1}{2}\theta ^2\left( y_2 + hf(y_2)\right) . \end{aligned}$$

SSP(3,2,2)

This method has \({\mathcal {C}}(A,b,\overline{{{\varvec{b}}}}) = 2\).

$$\begin{aligned} y_1&= u_n \\ y_2&= y_1 + \frac{h}{2} f(y_1) \\ y_3&= y_2 + \frac{h}{2} f(y_2) \\ u_{n+1}&= \frac{1}{3} u_n + \frac{2}{3} \left( y_3 + \frac{h}{2} f(y_3)\right) \\ u_{n+\theta }&= \left( 1-2\theta + \frac{4}{3}\theta ^2\right) u_n + 2(\theta -\theta ^2)\left( y_1+\frac{h}{2}f(y_1)\right) + \frac{2}{3}\theta ^2\left( y_3 + \frac{h}{2}f(y_3)\right) . \end{aligned}$$

SSP(3,3,2)

This method has \({\mathcal {C}}(A,b,\overline{{{\varvec{b}}}}) = 1\).

$$\begin{aligned} y_1&= u_n \\ y_2&= y_1 + h f(y_1) \\ y_3&= \frac{3}{4} u_n + \frac{1}{4} \left( y_2 + h f(y_2)\right) \\ u_{n+1}&= \frac{1}{3} u_n + \frac{2}{3} \left( y_3 + h f(y_3)\right) \\ u_{n+\theta }&= \left( 1 - \theta + \frac{1}{3}\theta ^2\right) u_n + (\theta -\theta ^2)\left( y_1+hf(y_1)\right) + \frac{2}{3}\theta ^2\left( y_3 + hf(y_3)\right) . \end{aligned}$$

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Ketcheson, D.I., Lóczi, L., Jangabylova, A. et al. Dense Output for Strong Stability Preserving Runge–Kutta Methods. J Sci Comput 71, 944–958 (2017). https://doi.org/10.1007/s10915-016-0331-5

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Keywords

  • Runge-Kutta methods
  • SSP methods
  • Dense output
  • Continuous extension