## Abstract

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton’s method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.

Original language | English |
---|---|

Pages (from-to) | 23291-23312 |

Number of pages | 22 |

Journal | AIMS Mathematics |

Volume | 9 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2024 |

## Keywords

- incompressible Navier-Stokes equations
- Jacobian
- Newton’s method
- nonlinear initial boundary value problems
- summation-by-parts
- weak boundary conditions

## ASJC Scopus subject areas

- General Mathematics