### Abstract

This paper considers the stability of explicit, implicit and Crank-Nicolson schemes for the one-dimensional heat equation on a staggered grid. Furthermore, we consider the cases when both explicit and implicit approximations of the boundary conditions are employed. Why we choose to do this is clearly motivated and arises from solving fluid flow equations with free surfaces when the Reynolds number can be very small, in at least parts of the spatial domain. A comprehensive stability analysis is supplied: a novel result is the precise stability restriction on the Crank-Nicolson method when the boundary conditions are approximated explicitly, that is, at t=nt rather than t=(n+1)t. The two-dimensional Navier-Stokes equations were then solved by a marker and cell approach for two simple problems that had analytic solutions. It was found that the stability results provided in this paper were qualitatively very similar, thereby providing insight as to why a Crank-Nicolson approximation of the momentum equations is only conditionally stable.

Original language | English |
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Pages (from-to) | 945-967 |

Number of pages | 22 |

Journal | Numerical Linear Algebra with Applications |

Volume | 15 |

Issue number | 10 |

DOIs | |

Publication status | Published - Dec 2008 |

### Keywords

- stability analysis
- implicit schemes
- staggered grids
- boundary conditions
- Navier-Stokes equations

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## Cite this

Oishi, C. M., Cuminato, J. A., Yuan, J. Y., & McKee, S. (2008). Stability of numerical schemes on staggered grids.

*Numerical Linear Algebra with Applications*,*15*(10), 945-967. https://doi.org/10.1002/nla.597