A novel series of solution methods for solving nonlinear stiff dynamic problems

A novel type of linear multi-step formulas is proposed for solving initial value problems, such as the problems of multi-body systems and vibration systems and a variety of dynamic problems in engineering. This type of formulas is derived from an eigen-based theory and is characterized by problem dependency since it has problem-dependent coefficients, which are functions of physical properties for defining the problem under analysis and applied step size. These coefficients are no longer limited to be scalar constants but can be constant matrices. The detailed development of a set of problem-dependent formulas is presented, and their numerical properties are intensively explored. These formulas can be explicitly or implicitly implemented in the solution of initial value problems. One of these problem-dependent methods can combine A-stability and explicit formulation together, and thus, it is best suited to solve nonlinear stiff problems, such as problems in chemical kinetics and vibrations, the analysis of control systems and the study of dynamical systems. It is validated that A-stability in conjunction with explicit formulation can save many computational efforts for solving nonlinear stiff dynamic problems when compared to conventional implicit methods. Notice that there exists no conventional method that can simultaneously combine A-stability and explicit formulation.