Abstract
The stabilization approach has been known to permit large time-step sizes while maintaining stability. However, it may “slow down the convergence rate” or cause “delayed convergence” if the time-step rescaling is not well resolved. By considering a fourth-order-in-space viscous Cahn–Hilliard (VCH) equation, we propose a class of up to the fourth-order single-step methods that are able to capture the correct physical behaviors with high-order accuracy and without time delay. By reformulating the VCH as a system consisting of a second-order diffusion term and a nonlinear term involving the operator \(({I} - \nu \Delta )^{-1}\), we first develop a general approach to estimate the maximum bound for the VCH equation equipped with either the Ginzburg–Landau or Flory–Huggins potential. Then, by taking advantage of new recursive approximations and adopting a time-step-dependent stabilization, we propose a class of stabilization Runge–Kutta methods that preserve the maximum principle for any time-step size without harming the convergence. Finally, we transform the stabilization method into a parametric Runge–Kutta formulation, estimate the rescaled time-step, and remove the time delay by means of a relaxation technique. When the stabilization parameter is chosen suitably, the proposed parametric relaxation integrators are rigorously proven to be mass-conserving, maximum-principle-preserving, and the convergence in the \(l^\infty \)-norm is estimated with pth-order accuracy under mild regularity assumption. Numerical experiments on multi-dimensional benchmark problems are carried out to demonstrate the stability, accuracy, and structure-preserving properties of the proposed schemes.
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Funding
This work was supported by the National Natural Science Foundation of China (12271523, 12071481), Defense Science Foundation of China (2021-JCJQ-JJ-0538), National Key R &D Program of China (SQ2020YFA0709803), Science & Technology Innovation Program of Hunan Province (2021RC3082, 2022RC1192), and Research fund from College of Science, National University of Defense Technology (2023-lxy-fhjj-002).
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Appendices
Appendix 1. Proof of Theorem 2.6
Proof
Consider the Ginzburg–Landau function (10) as an example. For a given \(u^0 \in X\) and \(t_0 > 0\), we denote \(X_{\beta } = \{ v \in X | \Vert v\Vert _{L^\infty } \le \beta \}\) and \(C([0, t_0]; X_{\beta }) = \{v: [0, t_0] \rightarrow X_{\beta } | v \text {~is~continuous}\}\). Note that the forward Euler condition (17) is equivalent to the circle condition,
Letting \(\kappa \ge \frac{1}{\tilde{\tau }_{FE}}\), for a given \(v \in C([0, t_0]; X_{\beta })\), we define \(w: [0, t_0] \rightarrow X\) as the solution to the system
Then, w is uniquely defined because of the linearity of (70). By Duhamel’s formula, we have
Taking \(\Vert \cdot \Vert _{L^\infty }\)-norm on both sides of (71) and applying the circle condition (69) yields
Therefore, \(w \in C([0, t_0]; X_{\beta })\). Next by defining a mapping \({M}: C([0, t_0]; X_{\beta }) \rightarrow C([0, t_0]; X_{\beta })\) as \({M}(v) = w\) through (70), we show that \({M}\) is a contraction for sufficiently small \(t_0\). Assuming that \(v_1, v_2 \in C([0, t_0]; X_{\beta })\), \(w_1 = {M}(v_1)\) and \( w_2 = {M} (v_2)\), we then obtain
Since \({N}(u)\) satisfies the Lipschitz condition (20), we derive
Noting that \({L} = \frac{\epsilon ^2}{\nu } \Delta \) is the generator of a contraction semigroup with respect to the supremum norm on X [20], then it holds that
If \(t_0 < \frac{1}{\kappa }\ln \frac{l_N + \kappa }{l_N}\), we have \(\frac{l_N + \kappa }{\kappa }(1 - \text {e}^{-\kappa t_0}) < 1\), and
and then \({M}\) is a contraction. Since \(X_{\beta }\) is closed in X, \(C([0,t_0]; X_{\beta })\) is thus complete with respect to the metric induced by the norm \(\Vert \cdot \Vert _{C([0, t_0]; X)}\), and then Banach’s fixed point theorem gives a unique fixed point \(u \in C([0, t_0]; X_{\beta })\) of \({M}(u) = u\), which is the unique solution to Eq. (8). Continuing the process gives global existence of the unique solution \(u \in C([0, T]; X_{\beta })\). \(\square \)
Appendix 2. Some non-negative RK Butcher tableaux
RK(5, 4)[65], \({C} \approx 1.508\), \(c \approx [0, 0.3918, 0.5861, 0.4745, 0.9350]^T\):
Appendix 3. Third- and fourth-order RK Butcher tableaux with non-decreasing parametric abscissas
RK\(^+(3, 3)\) [36], \({C} = \frac{3}{4}\), \(c = [0, \frac{2}{3}, \frac{2}{3}]^T\):
RK\(^+(5, 4)\)[36], \({C} \!=\! 1.346586417284006\), \(c \approx \! [0, 0.4549, 0.5165, 0.5165, 0.9903]^T\):
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Zhang, H., Zhang, G., Liu, Z. et al. On the maximum principle and high-order, delay-free integrators for the viscous Cahn–Hilliard equation. Adv Comput Math 50, 41 (2024). https://doi.org/10.1007/s10444-024-10143-6
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DOI: https://doi.org/10.1007/s10444-024-10143-6
Keywords
- Viscous Cahn–Hilliard equation
- Maximum principle
- Forward Euler condition
- Parametric Runge–Kutta method
- Time-step relaxation