A benchmark problem for the two- and three-dimensional Cahn–Hilliard equations

Jeong, Darae; Choi, Yongho; Kim, Junseok

doi:10.1016/j.cnsns.2018.02.006

Cited by 15 publications

(17 citation statements)

References 41 publications

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“…However, these problems lack concrete numerical targets to assess accuracy. Another set of benchmark problems in [26] with radial symmetry is posed in an infinite domain, not suitable for comparison with many approaches in the literature. In this work, we propose four benchmark problems, three for Cahn-Hilliard and one for the second order Allen-Cahn equation.…”

Section: Introductionmentioning

confidence: 99%

High Accuracy Benchmark Problems for Allen-Cahn and Cahn-Hilliard Dynamics

Church¹,

Guo²,

Jimack³

et al. 2019

CICP

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There is a large literature of numerical methods for phase field models from materials science. The prototype models are the Allen-Cahn and Cahn-Hilliard equations. We present four benchmark problems for these equations, with numerical results validated using several computational methods with different spatial and temporal discretizations. Our goal is to provide the scientific community with a reliable reference point for assessing the accuracy and reliability of future software for this important class of problem.

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Section: Introductionmentioning

confidence: 99%

High Accuracy Benchmark Problems for Allen-Cahn and Cahn-Hilliard Dynamics

Church¹,

Guo²,

Jimack³

et al. 2019

CICP

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“…Some scientific journal considers manuscripts only if accuracy and convergence of numerical solutions are established by discussion of results on multiple grids. Recently, there have been many research studies on the second-order convergence schemes for parabolictype partial differential equations [1][2][3][4][5][6][7][8][9][10][11][12]. To demonstrate the second-order convergence, some authors [1][2][3] showed the convergence by refining the spatial and temporal steps at the same time; some authors in [4][5][6][7][8][9][10] showed the convergence by refining the spatial and temporal steps separately.…”

Section: Introductionmentioning

confidence: 99%

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

Jeong

Lee

et al. 2019

Mathematical Problems in Engineering

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In this paper, we propose a verification method for the convergence rates of the numerical solutions for parabolic equations. Specifically, we consider the numerical convergence rates of the heat equation, the Allen–Cahn equation, and the Cahn–Hilliard equation. Convergence test results show that if we refine the spatial and temporal steps at the same time, then we have the second-order convergence rate for the second-order scheme. However, in the case of the first-order in time and the second-order in space scheme, we may have the first-order or the second-order convergence rates depending on starting spatial and temporal step sizes. Therefore, for a rigorous numerical convergence test, we need to perform the spatial and the temporal convergence tests separately.

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“…Let d be the space dimension. We present numerical solutions of the CH equation in symmetric (d = 1), radially symmetric (d = 2), and spherically symmetric (d = 3) coordinates [14,15]:…”

Section: Symmetric Coordinatesmentioning

confidence: 99%

“…Let φ n i and µ n i be the numerical approximations of φ(r i , t n ) and µ(r i , t n ), respectively, where t n = n∆t and ∆t is the temporal step size. In [14], the authors used the explicit Euler's scheme. On the other hand, we adopt an auxiliary variable q = φ(φ − 1) and the second-order backward difference formula (BDF2) for temporal discretization [36] to achieve second-order temporal accuracy.…”

Section: Symmetric Coordinatesmentioning

confidence: 99%

On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source

Lee¹

2021

NMTMA

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We investigate the effect of cutoff logistic source on evolutionary dynamics of a generalized Cahn-Hilliard (CH) equation in this paper. It is a well-known fact that the maximum principle does not hold for the CH equation. Therefore, a generalized CH equation with logistic source may cause the negative concentration blow-up problem in finite time. To overcome this drawback, we propose the cutoff logistic source such that only the positive value greater than a given critical concentration can grow. We consider the temporal profiles of numerical results in the one-, two-, and three-dimensional spaces to examine the effect of extra mass source. Numerical solutions are obtained using a finite difference multigrid solver. Moreover, we perform numerical tests for tumor growth simulation, which is a typical application of generalized CH equations in biology. We apply the proposed cutoff logistic source term and have good results.

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A benchmark problem for the two- and three-dimensional Cahn–Hilliard equations

Cited by 15 publications

References 41 publications

High Accuracy Benchmark Problems for Allen-Cahn and Cahn-Hilliard Dynamics

High Accuracy Benchmark Problems for Allen-Cahn and Cahn-Hilliard Dynamics

Verification of Convergence Rates of Numerical Solutions for Parabolic Equations

On the Evolutionary Dynamics of the Cahn-Hilliard Equation with Cut-Off Mass Source

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