Abstract
In this paper, we develop the Runge-Kutta generalized convolution quadrature with variable time stepping for the numerical solution of convolution equations for time and space-time problems and present the corresponding stability and convergence analysis. For this purpose, some new theoretical tools such as tensorial divided differences, summation by parts with Runge-Kutta differences and a calculus for Runge-Kutta discretizations of generalized convolution operators such as an associativity property will be developed in this paper. Numerical examples will illustrate the stable and efficient behavior of the resulting discretization.
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Notes
The generic constant C in the following estimates will depend on
but not explicitly on \(\sigma \). Hence, if 
is independent of \(\sigma \) so is the constant C.Note that the notation \(\mathbf {v}\otimes {\displaystyle \bigotimes \nolimits _{j=1}^{k}} \mathbf {w}^{\left( j\right) }\) sometimes appears also for \(k\le 0\). In this case we set \(\mathbf {v}\otimes {\displaystyle \bigotimes \nolimits _{j=1}^{k}} \mathbf {w}^{\left( j\right) }:=\mathbf {v}\) if \(k\le 0\).
We prefer the notation
instead of \(\left[ \mathbf {C}^{\left( 1\right) },\mathbf {C}^{\left( 2\right) },\ldots ,\mathbf {C}^{\left( n\right) }\right] f\) because of brevity.By \(\mathbf {V}^{\intercal }\) we denote the transposed of the matrix \(\mathbf {V}\) (without complex conjugation) and by \(\mathbf {V}^{-\intercal }=\left( \mathbf {V}^{-1}\right) ^{\intercal }\).
To derive the third equality, we have inserted
$$\begin{aligned} 0= & {} -\sum \nolimits _{j=2}^{n}\left[ \mathbf {D}^{\left( j\right) },\ldots ,\mathbf {D} ^{\left( n\right) }\right] f\circ \left( \left[ \mathbf {D}^{\left( 1\right) },\ldots ,\mathbf {D}^{\left( j-1\right) }\right] g\otimes \mathbf {I}\right) \!+\!\sum \nolimits _{j=1}^{n-1}\left( \mathbf {I}\otimes \left[ \mathbf {D}^{\left( j+1\right) },\ldots ,\mathbf {D}^{\left( n\right) }\right] f\right) \\&\circ \left[ \mathbf {D}^{\left( 1\right) } ,\ldots ,\mathbf {D}^{\left( j\right) }\right] g \end{aligned}$$and used (93).
but not explicitly on 
is independent of 
instead of References
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Acknowledgments
Most part of the present paper was developed while the first author was affiliated at the Institute of Mathematics of the University of Zurich. The first author was also partially supported by the Spanish grants MTM2012-31298 and MTM2014-54710-P.
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Lopez-Fernandez, M., Sauter, S. Generalized convolution quadrature based on Runge-Kutta methods. Numer. Math. 133, 743–779 (2016). https://doi.org/10.1007/s00211-015-0761-2
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DOI: https://doi.org/10.1007/s00211-015-0761-2