Rosenbrock-Wanner method

The Rosenbrock-Wanner methods (or ROW methods, often just referred to as Rosenbrock methods ) are special one-step methods in numerics for the approximate solution of common differential equations . They are named after Howard H. Rosenbrock and Gerhard Wanner .

In the one-step method, certain implicit Runge-Kutta methods for stiff initial value problems have very good stability properties, but their practical implementation requires a high computational effort because of the solution of non-linear equations. For this reason, linear-implicit methods such as the Rosenbrock-Wanner method are considered.

Procedure structure

As with the Runge-Kutta method , the methods have different stages, which approximate the solution of the system at intermediate points of a time step of the step size . In contrast to the Runge-Kutta method , only linear systems of equations can be solved. The method has sets of coefficients , the method shape is ${\displaystyle s}$ ${\displaystyle y(t)}$ ${\displaystyle y'(t)=f{\big (}t,y(t){\big )}}$ ${\displaystyle t_{n}+h_{n}c_{i},\,i=1,\ldots ,s,}$ ${\displaystyle h=h_{n}}$ ${\displaystyle a_{ij},\gamma _{ij},b_{i}}$

${\displaystyle {{\big (}I-h\gamma _{ii}T{\big )}k_{i}=f{\Bigl (}t_{n}+hc_{i},y_{n}+h\sum _{j=1}^{i-1}a_{ij}k_{j}{\Bigr )}+hT\sum _{j=1}^{i-1}\gamma _{ij}k_{j}+hd_{i}f_{t}(t_{n},y_{n}),\ i=1,\ldots ,s}}$

${\displaystyle y_{n+1}=y_{n}+h\sum _{i=1}^{s}b_{i}k_{i}}$

So in every level a linear system has to be solved if is. The matrix in the step systems is the Jacobian matrix at the beginning of the time step, and the relationships between the process coefficients are required ${\displaystyle d\times d}$ ${\displaystyle f:\,\mathbb {R} \times \mathbb {R} ^{d}\to \mathbb {R} ^{d}}$ ${\displaystyle T}$ ${\displaystyle T=f_{y}(t_{n},y_{n})}$

${\displaystyle c_{i}=\sum _{j=1}^{i-1}a_{ij},\quad d_{i}=\sum _{j=1}^{i}\gamma _{ij}.}$

If all are the same, the expensive LR decomposition only needs to be calculated once with the Gaussian algorithm . The procedures can also be carried out using an (extended) Butcher panel ${\displaystyle \gamma _{ii}\equiv \gamma }$

${\displaystyle {\begin{array}{c|c|c}c&A&\Gamma \\\hline &b&\end{array}}}$

where and are lower triangular matrices. An original form of the procedure without the additional terms with goes back to HH Rosenbrock (1963); the complete form was introduced in 1977 by G. Wanner. ${\displaystyle A=(a_{ij})}$ ${\displaystyle \Gamma =(\gamma _{ij})}$ ${\displaystyle \gamma _{ij},j<i}$

Consistency and stability

The ROW methods can be interpreted in such a way that exactly one step of the Newton method is carried out in a diagonal-implicit Runge-Kutta method . Therefore, at least stages are required for a process of order . With a suitable choice of the diagonal value , A-stable methods exist . ${\displaystyle p}$ ${\displaystyle s\geq p-1}$ ${\displaystyle \gamma \equiv \gamma _{ii}}$

Example procedure

The two-step process with the panel

${\displaystyle {\begin{array}{c|cc|cc}0&0&&\gamma &0\\{\frac {2}{3}}&{\frac {2}{3}}&0&-{\frac {4}{3}}\gamma &\gamma \\\hline &{\frac {1}{4}}&{\frac {3}{4}}&\end{array}}}$

and is of order 3 and is A-stable . There is an efficient method ROW GRK4T of capes and Rentrop with steps and order , in which an embedded process a step control is possible. ${\displaystyle \gamma =(1+1/{\sqrt {3}})/2}$ ${\displaystyle s=4}$ ${\displaystyle p=4}$

Generalizations

If you drop the condition , you get so-called W-methods, in which you can use a rough approximation of the Jacobian matrix of , for example by not recalculating the LR decomposition of in every time step. For this type, however, there are only low order methods. ${\displaystyle T=f_{y}(t_{n},y_{n})}$ ${\displaystyle T}$ ${\displaystyle f}$ ${\displaystyle I-h\gamma T}$

literature

• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff problems, Springer Verlag.
• E. Hairer, G. Wanner: Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems . Second Revised Edition, Springer Verlag.
• K. Strehmel, R. Weiner, H. Podhaisky: Numerics of ordinary differential equations - non -rigid, rigid and differential-algebraic equations . Springer Spectrum, 2012.