A Noniterative Problem-Dependent Formula for Stiff Dynamic Problems

Abstract

A novel one-step formula is proposed for solving initial value problems based on a concept of eigenmode. It is characterized by problem dependency since it has problem-dependent coefficients, which are functions of the product of the step size and the initial physical properties to define the problem under analysis. It can simultaneously combine A-stability, explicit formulation, and second-order accuracy. A-stability implies no limitation on step size based on stability consideration. An explicit formulation implies no nonlinear iterations for each step. The second-order accuracy with an appropriate step size can have good accuracy in numerical solutions. Thus, it seems promising for solving stiff dynamic problems. Numerical tests affirm that it can have the same performance as that of the trapezoidal rule for solving linear and nonlinear dynamic problems. It is evident that the most important advantage is of high computational efficiency in contrast to the trapezoidal rule due to no nonlinear iterations of each step.