scholarly journals A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1242 ◽  
Author(s):  
Hyun Geun Lee ◽  
Jaemin Shin ◽  
June-Yub Lee
Keyword(s):  
Splitting Method ◽  
Partition Of Unity ◽  
Energy Functional ◽  
High Order ◽  
Third Order ◽  
Energy Functionals ◽  
Runge Kutta Method ◽  
Highly Nonlinear ◽  
Convex Splitting ◽  
Energy Stable

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

A High Order Time Advancement Scheme for Prediction of Solidification Processes

2010 ◽  
Vol 297-301 ◽  
pp. 779-784 ◽  
Author(s):  
A. Abbasnejad ◽  
M.J. Maghrebi ◽  
H. Basirat Tabrizi
Keyword(s):  
Analytical Solutions ◽  
Second Order ◽  
High Order ◽  
Kutta Method ◽  
Third Order ◽  
Runge Kutta Method ◽  
Solidification Processes ◽  
Order Scheme ◽  
High Order Time ◽  
Good Agreement

The aim of this study is the simulation of alloys and pure materials solidification. A third order compact Runge-Kutta method and second order scheme are used for time advancement and space derivative modeling. The results are compared with analytical and semi-analytical solutions and show very good agreement.

The modified high-order Haar wavelet scheme with Runge–Kutta method in the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation

2021 ◽  
pp. 2150419
Author(s):  
Ming Zhong ◽  
Qi-Jun Yang ◽  
Shou-Fu Tian
Keyword(s):  
High Efficiency ◽  
Haar Wavelet ◽  
High Order ◽  
Fisher Equation ◽  
Kutta Method ◽  
Third Order ◽  
Time Layer ◽  
Runge Kutta Method ◽  
Huxley Equation ◽  
Runge Kutta

In this work, we focus on the modified high-order Haar wavelet numerical method, which introduces the third-order Runge–Kutta method in the time layer to improve the original numerical format. We apply the above scheme to two types of strong nonlinear solitary wave differential equations named as the generalized Burgers–Fisher equation and the generalized Burgers–Huxley equation. Numerical experiments verify the correctness of the scheme, which improves the speed of convergence while ensuring stability. We also compare the CPU time, and conclude that our scheme has high efficiency. Compared with the traditional wavelets method, the numerical results reflect the superiority of our format.

scholarly journals Unconditionally energy stable second-order numerical scheme for the Allen–Cahn equation with a high-order polynomial free energy

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Junseok Kim ◽  
Hyun Geun Lee
Keyword(s):  
Free Energy ◽  
Strong Stability ◽  
Second Order ◽  
High Order ◽  
Computational Method ◽  
Strong Stability Preserving ◽  
Energy Potential ◽  
Convex Splitting ◽  
Order Polynomial ◽  
Energy Stable

AbstractIn this article, we consider a temporally second-order unconditionally energy stable computational method for the Allen–Cahn (AC) equation with a high-order polynomial free energy potential. By modifying the nonlinear parts in the governing equation, we have a linear convex splitting scheme of the energy for the high-order AC equation. In addition, by combining the linear convex splitting with a strong-stability-preserving implicit–explicit Runge–Kutta (RK) method, the proposed method is linear, temporally second-order accurate, and unconditionally energy stable. Computational tests are performed to demonstrate that the proposed method is accurate, efficient, and energy stable.

scholarly journals Integrating Semilinear Wave Problems with Time-Dependent Boundary Values Using Arbitrarily High-Order Splitting Methods

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1113
Author(s):  
Isaías Alonso-Mallo ◽  
Ana M. Portillo
Keyword(s):  
Numerical Experiments ◽  
Splitting Method ◽  
Method Of Lines ◽  
Time Integration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
High Order ◽  
Time Dependent ◽  
Boundary Values ◽  
Lipschitz Continuous

The initial boundary-value problem associated to a semilinear wave equation with time-dependent boundary values was approximated by using the method of lines. Time integration is achieved by means of an explicit time method obtained from an arbitrarily high-order splitting scheme. We propose a technique to incorporate the boundary values that is more accurate than the one obtained in the standard way, which is clearly seen in the numerical experiments. We prove the consistency and convergence, with the same order of the splitting method, of the full discretization carried out with this technique. Although we performed mathematical analysis under the hypothesis that the source term was Lipschitz-continuous, numerical experiments show that this technique works in more general cases.

Natural Frequencies of Parallelogrammic Plates Using Characteristic Orthogonal Polynomials in Rayleigh-Ritz Method

1992 ◽  
Author(s):  
L. T. Lee ◽  
W. F. Pon
Keyword(s):  
Boundary Conditions ◽  
Coordinate System ◽  
Orthogonal Polynomials ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Energy Functional ◽  
Comprehensive Treatment ◽  
Energy Functionals ◽  
Simply Supported ◽  
Cartesian Coordinate

Abstract Natural frequencies of parallelogrammic plates are obtained by employing a set of beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. The orthogonal polynomials are generalted by using a Gram-Schmidt process, after the first member is constructed so as to satisfy all the boundary conditions of the corresponding beam problems accompanying the plate problems. The strain energy functional and kinetic energy functionals are transformed from Cartesian coordinate system to a skew coordinate system. The natural frequencies obtained by using the orthogonal polynomial functions are compared with those obtained by other methods with all four edges clamped boundary conditions and greet agreements are found between them. The natural frequencies for parallelogrammic plates with other boundary conditions, such as four edges simply supported, clamped-free and simply supported-free, are also obtained. This method is considered as a better and accurate comprehensive treatment for this type of problems.

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