A three level linearized compact difference scheme for the Cahn-Hilliard equation

A high-order finite difference method for the two-dimensional complex Ginzburg-Landau equation is considered. It is proved that the proposed difference scheme is uniquely solvable and unconditionally convergent. The convergent order in maximum norm is two in temporal direction and four in spatial direction. In addition, an efficient alternating direction implicit scheme is proposed. Some numerical examples are given to confirm the theoretical results.

show abstract

“…Lemma (). For any grid functions u, v on

Ω_{τ},

the following equality holds,

\sum_{l = 1}^{n} u^{l} Δ_{t} v^{l} = true{\begin{matrix} \frac{1}{2 τ} (u^{1} v^{2} - u^{1} v^{0}), n = 1 \\ \frac{1}{2 τ} (u^{n} v^{n + 1} + u^{n - 1} v^{n} - u^{2} v^{1} - u^{1} v^{0}) - \sum_{l = 2}^{n - 1} (Δ_{t} u^{l}) v^{l}, 2 \leq n \leq N - 1. \end{matrix}

…”

Section: The Solvability and Convergence Of The Difference Schemementioning

confidence: 99%

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

Hao

Sun

Cao

2014

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…Our analysis method may be applicable to the modified phase field crystal model. Li et al [10] proposed a three level linearized difference scheme for the Cahn-Hilliard equation and showed that the difference scheme converges in maximum norm. The phase crystal equation is a sixth order nonlinear partial differential equation.…”

Section: Resultsmentioning

confidence: 99%

Two finite difference schemes for the phase field crystal equation

Cao

Sun

2015

Sci. China Math.

Self Cite

View full text Add to dashboard Cite

The phase field crystal (PFC) model is a nonlinear evolutionary equation that is of sixth order in space. In the first part of this work, we derive a three level linearized difference scheme, which is then proved to be energy stable, unique solvable and second order convergent in L 2 norm by the energy method combining with the inductive method. In the second part of the work, we analyze the unique solvability and convergence of a two level nonlinear difference scheme, which was developed by Zhang et al. in 2013. Some numerical results with comparisons are provided. Keywordsphase field crystal model, nonlinear evolutionary equation, finite difference scheme, solvability, convergence MSC(2010) 65M06, 65M12, 65M15Citation: Cao H Y, Sun Z Z. Two finite difference schemes for the phase field crystal equation.

show abstract

“…Suppose

f (x) \in C^{6} [x_{i - 1}, x_{i + 1}], θ (s) = {(1 - s)}^{3} [5 - 3 {(1 - s)}^{2}]

then we have

\begin{matrix} \frac{1}{12} [f^{''} (x_{i - 1}) + 10 f^{''} (x_{i}) + f^{''} (x_{i + 1})] = \frac{1}{h_{1}^{2}} [f (x_{i - 1}) - 2 f (x_{i}) + f (x_{i + 1})] + \frac{h_{1}^{4}}{360} \int_{0}^{1} true[f^{true(6 true)} true(x_{i} - s h_{1} true) + f^{true(6 true)} true(x_{i} + s h_{1} true) true] θ (s) d s . \end{matrix}

Lemma (). Suppose

f (x) \in C^{5} [x_{0}, x_{1}], θ (s) = {(1 - s)}^{2} [1 - {(1 - s)}^{2}]

…”

Section: Notations and Lemmasmentioning

confidence: 99%

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Sun

Dai

et al. 2015

Numerical Methods Partial

View full text Add to dashboard Cite

Dual-phase-lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher-order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher-order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher-order accurate and unconditionally stable compact finite difference scheme for solving one-dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two-dimensional case and develop a fourth-order accurate compact finite difference method in space coupled with the Crank-Nicolson method in time, where the Robin's boundary condition is approximated using a third-order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth-order in space and second-order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis.

show abstract

A three level linearized compact difference scheme for the Cahn-Hilliard equation

Cited by 54 publications

References 37 publications

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

A three‐level linearized compact difference scheme for the Ginzburg–Landau equation

Two finite difference schemes for the phase field crystal equation

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Contact Info

Product

Resources

About