Implicit and Explicit Treatments of Model Error in Numerical Simulation

Numerical simulations of physical systems exhibit discrepancies arising from unmodeled physics and idealizations, as well as numerical approximation errors stemming from discretization and solver tolerances. This article reviews techniques developed in the past several decades to approximate and account for model errors, both implicitly and explicitly. Beginning from fundamentals, we frame model error in inverse problems, data assimilation, and predictive modeling contexts. We then survey major approaches: the Bayesian approximation error framework, embedded internal error models for structural uncertainty, probabilistic numerical methods for discretization uncertainty, model discrepancy modeling in Bayesian calibration and its recent extensions, machine-learning-based discrepancy correction, multi-fidelity and hybrid modeling strategies, as well as residual-based, variational, and adjoint-driven error estimators. Throughout, we emphasize the conceptual underpinnings of implicit versus explicit error treatment and highlight how these methods improve predictive performance and uncertainty quantification in practical applications ranging from engineering design to Earth-system science. Each section provides an overview of key developments with an extensive list of references to facilitate further reading. The review is written for practitioners of large-scale computational physics and engineering simulation, emphasizing how these methods can be incorporated into PDE solvers, inverse problem workflows, and data assimilation systems.