## Abstract

Finite difference approximations of the second derivative in space appearing in, parabolic, incompletely parabolic systems of, and 2nd-order hyperbolic, partial differential equations are considered. If the solution is pointwise bounded, we prove that finite difference approximations of those classes of equations can be closed with two orders less accuracy at the boundary without reducing the global order of accuracy. This result is generalised to initial-boundary value problems with an mth-order principal part. Then, the boundary accuracy can be lowered m orders. Further, it is shown that schemes using summation-by-parts operators that approximate second derivatives are pointwise bounded. Linear and nonlinear computations, including the two-dimensional Navier-Stokes equations, corroborate the theoretical results.

Original language | English |
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Pages (from-to) | 333-352 |

Number of pages | 20 |

Journal | Journal of Computational Physics |

Volume | 218 |

Issue number | 1 |

DOIs | |

Publication status | Published - 10 Oct 2006 |

Externally published | Yes |

## Keywords

- Boundary closure
- Boundary conditions
- Finite difference methods
- Navier-Stokes equations
- Order of accuracy
- Parabolic partial differential equations
- Stability
- Summation-by-parts

## ASJC Scopus subject areas

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- General Physics and Astronomy
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics