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Rosenbrock Methods

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Encyclopedia of Applied and Computational Mathematics

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Synonyms

Generalized Runge-Kutta methods; Linear-implicit Runge-Kutta methods; Rosenbrock methods; SDIRK methods

Definition

Rosenbrock methods are suitable for the numerical solution of stiff initial value problems

$$\displaystyle{ y^{\prime} = f(x,y),\quad y(x_{0}) = y_{0},\quad y \in \mathbb{R}^{n}. }$$

Using the Jacobian \(J = \frac{\partial f} {\partial y}\) a fixed number of linear equation systems must be solved in every integration step.

Description

Rosenbrock [14] originally studied stabilization problems arising from the one-dimensional heat equation when he applied the method of lines approach. Within this context he defined a new class of methods, which he characterized as follows:

Some general implicit processes are given for the solution of simultaneous first-order differential equations. These processes, which use successive substitution, are implicit analogues of the (explicit) Runge-Kutta processes. They require the solution in each time step of one or more sets of...

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References

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Author information

Authors and Affiliations

  1. Technische Universität München, Fakultät Mathematik, Munich, Germany

    Florian Augustin & Peter Rentrop

Authors
  1. Florian Augustin
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  2. Peter Rentrop
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Editor information

Editors and Affiliations

  1. University of Texas at Austin, Austin, TX, USA

    Björn Engquist

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Augustin, F., Rentrop, P. (2015). Rosenbrock Methods. In: Engquist, B. (eds) Encyclopedia of Applied and Computational Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70529-1_143

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