An energy-constrained implicit step-and-truncate algorithm and its robust error estimates

Abstract

Deriving mesh-robust error bounds is an open problem for low-rank methods, including dynamical low-rank (DLR) and step-and-truncate (SAT) approaches, as highlighted in a recent review. Existing bounds can depend on Lipschitz constants that grow with inverse mesh size, making them impractical, particularly for kinetic equations, a major application of low-rank methods. We develop a unified framework for implicit SAT that provides a discrete counterpart of energy-localization analysis for nonlinear partial differential equations. By restricting numerical solutions to energy-localized regions, we obtain one-sided Lipschitz stability with constants independent of spatial resolution. The resulting error estimates and time-step restrictions are mesh robust, answering this open problem for the class of implicit SAT schemes covered by our assumptions. We emphasize that energy localization provides a general mechanism for mesh-independent robust error estimates in low-rank approximation, extending beyond the SAT methods considered here. Analogous results for DLR methods will be presented in subsequent work. Returning to implicit SAT, we demonstrate the broad applicability of the framework through error estimates for different classes of equations. Specifically, for semilinear parabolic and Vlasov--Poisson equations, we establish robust first-order error bounds for backward-Euler SAT under suitable regularity assumptions. Energy localization also has practical implications. We design an energy-constrained implicit SAT algorithm. Implicit SAT methods without energy constraints can select incorrect solution branches even when their algebraic residuals are small. Moreover, time-step reduction can be essential for continuation into larger energy regions. Numerical experiments support these conclusions.