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

Haishen Dai, Huan Lei, Bin Zheng

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.

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

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.
Paper Structure (13 sections, 12 theorems, 100 equations, 6 figures, 3 tables)

This paper contains 13 sections, 12 theorems, 100 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Discretization and fixed-point iteration
  3. sBDF1 scheme
  4. sBDF2 scheme
  5. sBDF3 scheme
  6. sBDF4 scheme
  7. Unified formulation for sBDF$k$ schemes
  8. Contractivity of the fixed-point iteration
  9. Error analysis
  10. Discrete maximum bound principle
  11. Discrete energy stability analysis
  12. Numerical experiments
  13. Concluding Remarks

Key Result

Theorem 3.1

Let $\{\phi^{n+1,(m)}\}_{m\ge0}$ be the algorithmic iterates generated by the sBDF$k$ schemes in Section 2, where $k\in\{1,2,3,4\}$. Let $\widetilde{\phi}^{n+1,(m)}$ denote the corresponding auxiliary iterates produced by the fixed-point update (before applying the cut-off when $k\ge2$). Assume that Then the fixed-point iterations are contractive in the discrete $\ell^2$-norm: where the contracti

Figures (6)

  • Figure 1: Evolution of the maximum norms of the numerical solutions in Example 7.1 on a fixed $512\times512$ grid.
  • Figure 2: Energy evolution of the numerical solutions in Example 7.1 on a fixed $512\times512$ spatial grid.
  • Figure 3: Tumor growth (first row), nutrient distribution (second row), and PSA distribution (third row) for an aggressive tumor under baseline conditions with (I) pure cytotoxic drug therapy, with $S_c = 2.75$ and $\gamma_c = 17$.
  • Figure 4: Tumor growth (the first row), Nutrient distribution (the second row) and PSA distribution (the third row) of an aggressive tumor with (II) pure anti-angiogenic drug therapy under baseline, with $S_c = 2.75$ and $\gamma_c = 17$.
  • Figure 5: Tumor growth (the first row), Nutrient distribution (the second row) and PSA distribution (the third row) of an aggressive tumor with (III) combined therapy under baseline, with $S_c = 2.75$ and $\gamma_c = 17$.
  • ...and 1 more figures

Theorems & Definitions (27)

  • Remark 2.1
  • Theorem 3.1: Unconditional contractivity of the fixed-point iterations
  • proof
  • Remark 3.1
  • Remark 3.2
  • Lemma 4.1: Local truncation error
  • proof
  • Lemma 4.2: Error recursion via an energy argument
  • proof
  • Lemma 4.3: Termination error of the fixed-point iteration
  • ...and 17 more