Damped Multistep Formulas for Solving Stiff Dynamic Problems

A novel series of damped multistep formulas of matrix coefficients are developed to find the solutions of dynamic problems with negligible stiff components or eigenmodes. They are derived from the concept of eigenmode and the development details are presented. A total of six multistep formulas is developed and they have an order of accuracy from 1 to 6 corresponding to the 1-step to 6-step multistep formula. Each formula has damping to suppress or remove the stiff components or eigenmodes that are of no importance or of no interest while the slow eigenmode solutions can be still accurately calculated. The 1-step and 2-step formulas of the series are L-stable. Besides, each formula can have a non-iterative solution procedure and hence it is of high computational efficiency in contrast to the corresponding conventional backward differentiation method. This series can give a better quality of numerical solutions for dynamic problems with negligible stiff components or eigenmodes when compared to the methods with no damping. Numerical tests reveal that this series performs well like the backward differentiation method and it can save many efforts in computing if a non-iterative solution procedure is used when compared to the backward differentiation method.