Strict upper and lower bounds of quantities in linear second-order systems

Abstract

• A strict upper bound of the time-discretization error for linear second-order systems is obtained. • Upper and lower bounds on quantities of interest are also obtained by dual analysis. • The bounds on quantities of interest are further improved through an optimization procedure. • Numerical examples show good verification of the bounds. • The upper bound of the time-discretization error is well applicable to adaptive time-stepping. This paper aims to present a model verification technique for general numerical computations of linear second-order systems. Strict upper and lower bounds on quantities of interest are eventually obtained. The model verification technique consists of two major steps. The first is the representation of the error in quantities of interest through an adjoint correction; the second is the application of the strict upper bound derived for the time-discretization error. Moreover, the bounds on quantities of interest are further improved through an optimization procedure. Academic examples are studied to verify the proposed bounds and to explore the potential application of these bounds to adaptive time-stepping.