Problem-dependent formulas for stiff problems

A novel family of problem-dependent formulas is derived from a concept of eigenmode for the solution of ordinary differential equations. It has problem-dependent coefficients although constant coefficients are generally found for the currently available first order solvers. In general, it is a free parameter control formula. It generally possesses a first order accuracy and one formula can have a second order accuracy in addition to A-stability. It possesses a dual implementation since it can be non-iteratively and iteratively implemented. It is natural that a non-iterative solution procedure is preferred since it is computationally inexpensive for each step. It is substantiated that the second order accurate explicit formula can have the same performance as that of the trapezoidal formula for solving linear ODEs while for nonlinear ODEs it is roughly the same. Since this explicit formula can combine A-stability and second order accuracy, it is best suited to solve stiff ODEs. In contrast to the trapezoidal formula, the most important improvement of this explicit formula is that it can be non-iteratively implemented. Consequently, it is very efficient in computing the solution of large systems of stiff ODEs.