Runge–Kutta Residual Distribution Schemes

Abstract

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

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Acknowledgments

The first author would like to acknowledge EPSRC who funded this work under grant number EP/G003645/1.

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Correspondence to Andrzej Warzyński.

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Warzyński, A., Hubbard, M.E. & Ricchiuto, M. Runge–Kutta Residual Distribution Schemes. J Sci Comput 62, 772–802 (2015). https://doi.org/10.1007/s10915-014-9879-0

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Keywords

  • Hyperbolic conservation laws
  • Time-dependent problems
  • Second order schemes
  • Residual distribution
  • Runge–Kutta time-stepping