Sparse identification of truncation errors

Highlights: • Sparse regression based framework to automatically identify truncation error terms. • Proving modified differential equations can be identified from simulation data. • Preconditioning at multiple stages attenuates multicollinearity effects. • Highly accurate results for test cases with main limit being machine precision. -- Abstract: This work presents a data-driven approach to the identification of spatial and temporal truncation errors for linear and nonlinear discretization schemes of Partial Differential Equations (PDEs). Motivated by the central role of truncation errors, for example in the creation of implicit Large Eddy schemes, we introduce the Sparse Identification of Truncation Errors (SITE) framework to automatically identify the terms of the modified differential equation from simulation data. We build on recent advances in the field of data-driven discovery and control of complex systems and combine it with classical work on modified differential equation analysis of Warming, Hyett, Lerat and Peyret. We augment a sparse regression-rooted approach with appropriate preconditioning routines to aid in the identification of the individual modified differential equation terms. The construction of such a custom algorithm pipeline allows attenuating of multicollinearity effects as well as automatic tuning of the sparse regression hyperparameters using the Bayesian information criterion (BIC). As proof of concept, we constrain the analysis to finite difference schemes and leave other numerical schemes open for future inquiry. Test cases include the linear advection equation with a forward-time, backward-space discretization, the Burgers' equation with a MacCormack predictor-corrector scheme and the Korteweg-de Vries equation with a Zabusky and Kruska discretization scheme. Based on variation studies, we derive guidelines for the selection of discretization parameters, preconditioning approaches and sparse regression algorithms. The results showcase highly accurate predictions underlining the promise of SITE for the analysis and optimization of discretization schemes, where analytic derivation of modified differential equations is infeasible.