New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index.

*(English)*Zbl 1093.65097New Rosenbrock methods for ordinary differential equations, differential-algebraic equations; partial differential equations (PDEs) and partial differential-algebraic equations (PDAEs) of index 1 are presented. These solvers are of order 3, have 3 or 4 internal stages, and fulfil certain order conditions to obtain a better convergence if inexact Jacobians and approximations of \(\frac{\partial f}{\partial t}\) are used. A comparison of the five new methods with five other Rosenbrock solvers shows the advantages of the new methods. They are applied to several differential equations such as a PDE, an index-1 PDAE and the Navier-Stokes equations with different right-hand sides. The results are presented in pleasant figures. The numerically observed temporal order of convergence are also given for the different examples.

Reviewer: Rudolf Scherer (Karlsruhe)

##### MSC:

65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |

65L80 | Numerical methods for differential-algebraic equations |

34A09 | Implicit ordinary differential equations, differential-algebraic equations |

35K55 | Nonlinear parabolic equations |

35R10 | Partial functional-differential equations |

65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

35Q30 | Navier-Stokes equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

##### Keywords:

nonlinear parabolic equations; Rosenbrock methods; order reduction; numerical examples; convergence; Navier-Stokes equations; differential-algebraic equations; partial differential-algebraic equations
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\textit{J. Rang} and \textit{L. Angermann}, BIT 45, No. 4, 761--787 (2005; Zbl 1093.65097)

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