An efficient two-grid fourth-order compact difference scheme with variable-step BDF2 method for the semilinear parabolic equation

TL;DR

The paper develops an efficient high-order two-grid scheme for semilinear parabolic equations by coupling a nonlinear compact-difference scheme on a coarse grid with a linearized, high-order scheme on a fine grid, connected through a piecewise bi-cubic interpolation operator. Variable-step BDF2 time stepping, together with discrete orthogonal convolution kernels, yields robust stability and error results under step-ratio constraints $0<r_k<4.8645$ and time-step bounds $\tau=o(H^{1/2})$, with a weaker $\tau=o(H^{1/2})$ condition for the two-grid setting when $H=O(h^{4/7})$. A cut-off Lipschitz extension of the nonlinear term and the DOC framework enable rigorous solvability and convergence analyses, while the interpolation operator ensures high-order accuracy transfer between grids. The authors extend the method to periodic boundaries, provide comprehensive numerical experiments (including adaptive temporal stepping and Allen–Cahn applications), and demonstrate substantial efficiency gains without compromising accuracy. Overall, the work offers a practical, provably accurate, high-order framework for stiff nonlinear parabolic PDEs with potential broad applicability in phase-field and related simulations.

Abstract

Due to the lack of corresponding analysis on appropriate mapping operator between two grids, high-order two-grid difference algorithms are rarely studied. In this paper, we firstly discuss the boundedness of a local bi-cubic Lagrange interpolation operator. And then, taking the semilinear parabolic equation as an example, we first construct a variable-step high-order nonlinear difference algorithm using compact difference technique in space and the second-order backward differentiation formula (BDF2) with variable temporal stepsize in time. With the help of discrete orthogonal convolution (DOC) kernels and a cut-off numerical technique, the unique solvability and corresponding error estimates of the high-order nonlinear difference scheme are established under assumptions that the temporal stepsize ratio satisfies rk < 4.8645 and the maximum temporal stepsize satisfies tau = o(h^1/2 ). Then, an efficient two-grid high-order difference algorithm is developed by combining a small-scale variable-step high-order nonlinear difference algorithm on the coarse grid and a large-scale variable-step high-order linearized difference algorithm on the fine grid, in which the constructed piecewise bi-cubic Lagrange interpolation mapping operator is adopted to project the coarse-grid solution to the fine grid. Under the same temporal stepsize ratio restriction rk < 4.8645 and a weaker maximum temporal stepsize condition tau = o(H^1.2 ), optimal fourth-order in space and second-order in time error estimates of the two-grid difference scheme is established if the coarse-fine grid stepsizes satisfy H = O(h^4/7). Finally, several numerical experiments are carried out to demonstrate the effectiveness and efficiency of the proposed scheme.

Figures (6)

• Figure 1: Evolution of exact solution with time at point $(0.25,0.25)$
• Figure 2: Evolution of solution with time (left) at $(0.25,0.25)$ and temporal stepsize (right) of the two-grid scheme and adaptive two-grid scheme
• Figure 4: Evolutions of energy (left) and time steps (right) for the nonlinear scheme, two-grid scheme and adaptive two-grid scheme until time $T = 30$
• Figure 6: Evolutions of energy (left) and time steps (right) for the nonlinear scheme, two-grid scheme and adaptive two-grid scheme until time $T = 100$
• Figure A: Nonlinear scheme with fixed temporal stepsize $\tau = 0.1$

Theorems & Definitions (23)

• Lemma 2.1: Compact_S12
• Lemma 2.2: Compact_S12
• Lemma 2.3: Cubic
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 3.1: JSC_Zhang_2022JSC_Liao_2022
• Lemma 3.2: 2021_MOC_LiaoJSC_Zhang_2022
• Lemma 3.3: 2021_MOC_Liao
• Theorem 3.4