The Generalized Rosenbrock formulation

Following the methodologies outlined in Hairer & Wanner ( Solving Ordinary Differential Equations, II), the generalized formulation for the Rosenbrock methods of higher order can be stated as follows:

 

Once the ki’s are determined, the integration to the next step is achieved using

 

The error at each time step is given by

 

The above formulation accounts for the scaling factor by dividing the identity matrix by hγ. (ref: Shampine, Implementation of Rosenbrock methods). The Jacobian is assumed to be approximately equal to the partial derivative of the function with respect to the dependent variables (ref: Shampine & Reichelt, The Matlab ODE Suite). Much like the Generalized Runge Kutta methods, the Rosenbrock methods are also explicit single-step formulations that require an inversion of the A matrix at each time step. As indicated earlier, the LU-decomposition & LU-back substitution routines come in handy for these operations.

Similar to the RK methods, a variety of coefficients exist in the open literature for the 4^th order Rosenbrock methods.