Estimating Global Errors in Time Stepping
For researchers and practitioners in numerical analysis and scientific computing, this offers a novel approach to global error estimation in time integration, though the impact is incremental as it builds on existing concepts.
This work introduces new time-stepping strategies that provide built-in global error estimators for solving differential equations, generalizing classical methods and enabling overlapped internal computations. The proposed explicit self-starting schemes are demonstrated on several examples.
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.