Evaluation of implicit time-stepping schemes by CG-approach

Time-domain solvers for problems in structural dynamics with some thousands of DOF’s are characterized by implicit transition equations since only such codes are unconditionally numerical stable. In addition, efficiency and accuracy demand for proper time-step adaptation procedures in order to control at least the local error. However, each new time-step causes a new coefficient-matrix A and thus a new decomposition. Updating the stiffness for nonlinear systems, too, causes varying coefficient matrices. Here these continuously updated linearized transition equations are replaced by corresponding quadratic forms. This paper is restricted to symmetric positive-definite matrices A; their minimum value is approached by a straightforward iteration starting with the solution of a corresponding explicit time-stepping scheme. Implicit and explicit scheme are characterized by the same local error properties; however, they are different in principle with respect to their stability properties. Here the most popular implicit Padé-P11-Newmark scheme will be treated in an explicit manner.