Achieving third-order accuracy via recursive time filtering: a seamless extension of the filtered backward Euler method

This paper introduces a novel recursive time-filtering framework designed to elevate the second-order filtered backward Euler (FBE) method to third-order accuracy while preserving its inherent computational simplicity. While FBE remains a staple in computational science due to its robust stability, its second-order convergence often necessitates prohibitively small time steps for high-fidelity simulations. We resolve this by proposing a non-intrusive extension that functions as a modular post-processing step, requiring no additional implicit solves or structural modifications to existing numerical solvers. By leveraging principles from discrete differential geometry, we generalize the framework to variable time-step regimes through a rigorous definition of discrete curvature based on quadratic interpolants. The theoretical foundation involves recasting the FBE scheme into its one-leg equivalent and applying the framework of linear multistep methods (LMM). Through a derivation of the local truncation error (LTE), we identify a unique, step-dependent filtering parameter $ \beta(\tau_n, \tau_{n-1}) $ that ensures a consistent transition to third-order convergence across non-uniform grids. Furthermore, we prove that the filtering operation satisfies a discrete maximum principle for curvature, demonstrating that the updated curvature is a convex combination of previous values. This ensures the scheme is strictly dissipative, effectively dampening high-frequency numerical artifacts without introducing new local extrema. Stability analysis via the boundary locus method confirms that the resulting recursive scheme is $ A(\alpha) $-stable, providing a vast stability region for stiff differential systems. Numerical validations on a suite of oscillatory and quasiperiodic benchmark problems demonstrate that the method recovers the theoretical order of accuracy and significantly mitigates the excessive numerical dissipation characteristic of standard FBE schemes, offering a powerful and low-effort upgrade path for legacy codes in fluid dynamics and structural mechanics.