Construction of Rosenbrock–Wanner method Rodas5P and numerical benchmarks within the Julia Differential Equations package

Rosenbrock–Wanner methods for systems of stiff ordinary differential equations are well known since the seventies. They have been continuously developed and are efficient for differential-algebraic equations of index-1, as well. Their disadvantage that the Jacobian matrix has to be updated in every time step becomes more and more obsolete when automatic differentiation is used. Especially the family of Rodas methods has proven to be a standard in the Julia package DifferentialEquations. However, the fifth-order Rodas5 method undergoes order reduction for certain problem classes. Therefore, the goal of this paper is to compute a new set of coefficients for Rodas5 such that this order reduction is reduced. The procedure is similar to the derivation of the methods Rodas4P and Rodas4P2. In addition, it is possible to provide new dense output formulas for Rodas5 and the new method Rodas5P. Numerical tests show that for higher accuracy requirements Rodas5P always belongs to the best methods within the Rodas family