Matrix-Free Stabilized BDF Schemes for Semilinear Parabolic Equations with Unconditional Maximum Bound Principle Preservation and Energy Stability

TL;DR

This paper introduces stabilized high-order BDF schemes for semilinear parabolic equations that preserve maximum principles and energy stability unconditionally, while being matrix-free and efficient for complex domains.

Contribution

The paper presents a novel family of stabilized BDF schemes that achieve maximum bound principle preservation, energy stability, and matrix-free implementation simultaneously for the first time.

Findings

Schemes are unconditionally stable and preserve maximum bounds.

Numerical results confirm theoretical convergence and robustness.

Methods outperform ETD approaches on mixed boundary condition problems.

Abstract

We develop a family of stabilized backward differentiation formula (sBDF) schemes of orders one through four for semilinear parabolic equations. The proposed methods are designed to achieve three properties that are rarely available simultaneously in high-order time discretizations: unconditional preservation of the maximum bound principle (MBP), unconditional discrete energy stability, and practical matrix-free implementation. The construction integrates carefully designed stabilization terms, fixed-point iterations, and a pointwise cut-off strategy. The nonlinear algebraic systems arising from the implicit sBDF discretizations are solved by fixed-point iteration, resulting in fully matrix-free algorithms. This makes the approach particularly attractive for practical computations on general domains and under mixed boundary conditions, where FFT-based exponential time differencing methods are often unavailable or inefficient. We further present a unified analysis for the fully implemented schemes, explicitly incorporating the interplay among time discretization, nonlinear iteration, and cut-off. Unconditional contractivity of the fixed-point iterations and error estimates are established. For the Allen-Cahn equation, we additionally prove an unconditional discrete energy dissipation law. Numerical experiments confirm the theoretical convergence rates and demonstrate the robustness and efficiency of the proposed methods, particularly relative to ETD-based approaches for problems with mixed boundary conditions.