Estimating Global Errors in Time Stepping

Abstract

This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky's procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with global error estimation. The resulting algorithms can be effectively represented as general linear methods. We present a few explicit self-starting schemes akin to Runge-Kutta methods with global error estimation and illustrate the theoretical considerations on several examples.

PETSc/TAO Users Manual: Revision 3.18

“…In this section, we describe explicit and implicit time stepping methods with global error estimation that are introduced in [Con16]. The solution vector for a GLEE method is either [y, ỹ] or [y,ε], where y is the solution, ỹ is the "auxiliary solution," and ε is the error.…”