Fourth-Order Spatial and Second-Order Temporal Accurate Compact Scheme for Cahn–Hilliard Equation

2019 ◽  
Vol 20 (2) ◽  
pp. 137-143
Author(s):  
Seunggyu Lee
Keyword(s):  
Fourth Order ◽  
Mass Conservation ◽  
Second Order ◽  
High Order Accuracy ◽  
Step Size ◽  
Order Accuracy ◽  
Central Difference ◽  
Hilliard Equation ◽  
Temporal And Spatial ◽  
Cahn Hilliard Equation

AbstractWe propose a fourth-order spatial and second-order temporal accurate and unconditionally stable compact finite-difference scheme for the Cahn–Hilliard equation. The proposed scheme has a higher-order accuracy in space than conventional central difference schemes even though both methods use a three-point stencil. Its compactness may be useful when applying the scheme to numerical implementation. In a temporal discretization, the secant-type algorithm, which is known as the second-order accurate scheme, is applied. Furthermore, the unique solvability regardless of the temporal and spatial step size, unconditionally gradient stability, and discrete mass conservation are proven. It guarantees that large temporal and spatial step sizes could be used with the high-order accuracy and the original properties of the CH equation. Then, numerical results are presented to confirm the efficiency and accuracy of the proposed scheme. The efficiency of the proposed scheme is better than other low order accurate stable schemes.

scholarly journals On a fractional step-splitting scheme for the Cahn-Hilliard equation

2014 ◽  
Vol 31 (7) ◽  
pp. 1151-1168 ◽  
Author(s):  
A.A. Aderogba ◽  
M. Chapwanya ◽  
J.K. Djoko
Keyword(s):  
Fourth Order ◽  
Second Order ◽  
Computational Time ◽  
Fractional Step ◽  
Step Method ◽  
Fractional Step Method ◽  
Order Accuracy ◽  
Content Type ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Purpose – For a partial differential equation with a fourth-order derivative such as the Cahn-Hilliard equation, it is always a challenge to design numerical schemes that can handle the restrictive time step introduced by this higher order term. The purpose of this paper is to employ a fractional splitting method to isolate the convective, the nonlinear second-order and the fourth-order differential terms. Design/methodology/approach – The full equation is then solved by consistent schemes for each differential term independently. In addition to validating the second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Findings – The scheme is second-order accuracy, the authors will demonstrate the efficiency of the proposed method by validating the dissipation of the Ginzberg-Lindau energy and the coarsening properties of the solution. Originality/value – The authors believe that this is the first time the equation is handled numerically using the fractional step method. Apart from the fact that the fractional step method substantially reduces computational time, it has the advantage of simplifying a complex process efficiently. This method permits the treatment of each segment of the original equation separately and piece them together, in a way that will be explained shortly, without destroying the properties of the equation.

scholarly journals A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Makoto Okumura ◽  
Takeshi Fukao ◽  
Daisuke Furihata ◽  
Shuji Yoshikawa
Keyword(s):  
Boundary Condition ◽  
Numerical Solutions ◽  
Normal Derivative ◽  
Second Order ◽  
Dynamic Boundary Condition ◽  
Central Difference ◽  
Structure Preserving ◽  
Hilliard Equation ◽  
Dynamic Boundary ◽  
Cahn Hilliard Equation

<p style='text-indent:20px;'>We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [<xref ref-type="bibr" rid="b14">14</xref>]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [<xref ref-type="bibr" rid="b13">13</xref>] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme.</p>

A Second Order BDF Numerical Scheme with Variable Steps for the Cahn--Hilliard Equation

2019 ◽  
Vol 57 (1) ◽  
pp. 495-525 ◽  
Author(s):  
Wenbin Chen ◽  
Xiaoming Wang ◽  
Yue Yan ◽  
Zhuying Zhang
Keyword(s):  
Numerical Scheme ◽  
Second Order ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

scholarly journals Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Zhifeng Weng ◽  
Langyang Huang ◽  
Rong Wu
Keyword(s):  
Numerical Approximation ◽  
Second Order ◽  
Differentiation Formula ◽  
Numerical Examples ◽  
Backward Differentiation Formula ◽  
Hilliard Equation ◽  
Order Case ◽  
Energy Stable ◽  
Cahn Hilliard Equation ◽  
Fourier Spectral Methods

In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

On the accuracy of some explicit and implicit methods for the inviscid GRLW equation subject to initial Gaussian conditions

2016 ◽  
Vol 26 (3/4) ◽  
pp. 698-721 ◽  
Author(s):  
J I Ramos
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Finite Difference Methods ◽  
Second Order ◽  
Time Step ◽  
Content Type ◽  
Difference Methods ◽  
Temporal And Spatial ◽  
Runge Kutta ◽  
Grlw Equation

Purpose – The purpose of this paper is to both determine the effects of the nonlinearity on the wave dynamics and assess the temporal and spatial accuracy of five finite difference methods for the solution of the inviscid generalized regularized long-wave (GRLW) equation subject to initial Gaussian conditions. Design/methodology/approach – Two implicit second- and fourth-order accurate finite difference methods and three Runge-Kutta procedures are introduced. The methods employ a new dependent variable which contains the wave amplitude and its second-order spatial derivative. Numerical experiments are reported for several temporal and spatial step sizes in order to assess their accuracy and the preservation of the first two invariants of the inviscid GRLW equation as functions of the spatial and temporal orders of accuracy, and thus determine the conditions under which grid-independent results are obtained. Findings – It has been found that the steepening of the wave increase as the nonlinearity exponent is increased and that the accuracy of the fourth-order Runge-Kutta method is comparable to that of a second-order implicit procedure for time steps smaller than 100th, and that only the fourth-order compact method is almost grid-independent if the time step is on the order of 1,000th and more than 5,000 grid points are used, because of the initial steepening of the initial profile, wave breakup and solitary wave propagation. Originality/value – This is the first study where an accuracy assessment of wave breakup of the inviscid GRLW equation subject to initial Gaussian conditions is reported.

scholarly journals Numerical Simulation for a Multidimensional Fourth-Order Nonlinear Fractional Subdiffusion Model with Time Delay

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3050
Author(s):  
Sarita Nandal ◽  
Mahmoud A. Zaky ◽  
Rob H. De Staelen ◽  
Ahmed S. Hendy
Keyword(s):  
Fourth Order ◽  
Numerical Scheme ◽  
Source Term ◽  
Variable Coefficients ◽  
Second Order ◽  
Order Accuracy ◽  
Nonlinear Source ◽  
Second Order In Time ◽  
Convergence Orders ◽  
Temporal Direction

The purpose of this paper is to develop a numerical scheme for the two-dimensional fourth-order fractional subdiffusion equation with variable coefficients and delay. Using the L2−1σ approximation of the time Caputo derivative, a finite difference method with second-order accuracy in the temporal direction is achieved. The novelty of this paper is to introduce a numerical scheme for the problem under consideration with variable coefficients, nonlinear source term, and delay time constant. The numerical results show that the global convergence orders for spatial and time dimensions are approximately fourth order in space and second-order in time.

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