Energy Stable Discontinuous Galerkin Finite Element Method for the Allen–Cahn Equation

In this paper, we investigate numerical solution of Allen–Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen–Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical results reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed, and the ripening time is detected correctly.

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Energy Stable Interior Penalty Discontinuous Galerkin Finite Element Method for Cahn–Hilliard Equation

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2016-0024 ◽

2017 ◽

Vol 18 (5) ◽

pp. 303-314 ◽

Cited By ~ 3

Author(s):

Ayşe Sarıaydın-Filibelioğlu ◽

Bülent Karasözen ◽

Murat Uzunca

Keyword(s):

Discontinuous Galerkin ◽

Convergence Rates ◽

Logarithmic Potential ◽

Galerkin Finite Element Method ◽

Interior Penalty ◽

Galerkin Finite Element ◽

Degenerate Mobility ◽

Discontinuous Galerkin Finite Element ◽

Energy Stable ◽

Cahn Hilliard Equation

AbstractAn energy stable conservative method is developed for the Cahn–Hilliard (CH) equation with the degenerate mobility. The CH equation is discretized in space with the mass conserving symmetric interior penalty discontinuous Galerkin (SIPG) method. The resulting semi-discrete nonlinear system of ordinary differential equations are solved in time by the unconditionally energy stable average vector field (AVF) method. We prove that the AVF method preserves the energy decreasing property of the fully discretized CH equation. Numerical results for the quartic double-well and the logarithmic potential functions with constant and degenerate mobility confirm the theoretical convergence rates, accuracy and the performance of the proposed approach.

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Analysis of a second order discontinuous Galerkin finite element method for the Allen-Cahn equation

10.31274/etd-180810-4586 ◽

2016 ◽

Author(s):

Junzhao Hu

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Discontinuous Galerkin ◽

Second Order ◽

Galerkin Finite Element Method ◽

Galerkin Finite Element ◽

Cahn Equation ◽

Discontinuous Galerkin Finite Element ◽

Element Method

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Numerical analysis of fully discrete energy stable weak Galerkin finite element Scheme for a coupled Cahn-Hilliard-Navier-Stokes phase-field model

Applied Mathematics and Computation ◽

10.1016/j.amc.2021.126487 ◽

2021 ◽

Vol 410 ◽

pp. 126487

Author(s):

Mehdi Dehghan ◽

Zeinab Gharibi

Keyword(s):

Field Model ◽

Phase Field Model ◽

Navier Stokes ◽

Weak Galerkin ◽

Finite Element Scheme ◽

Discrete Energy ◽

Galerkin Finite Element ◽

Element Scheme ◽

Fully Discrete ◽

Energy Stable

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Performance of Adaptive Unstructured Mesh Modelling in Idealized Advection Cases over Steep Terrains

Atmosphere ◽

10.3390/atmos9110444 ◽

2018 ◽

Vol 9 (11) ◽

pp. 444 ◽

Cited By ~ 2

Author(s):

Jinxi Li ◽

Jie Zheng ◽

Jiang Zhu ◽

Fangxin Fang ◽

Christopher. Pain ◽

...

Keyword(s):

Unstructured Mesh ◽

Computational Cost ◽

Adaptive Mesh ◽

Galerkin Finite Element Method ◽

Adaptive Meshes ◽

Galerkin Finite Element ◽

Computational Costs ◽

Cut Cell ◽

Discontinuous Galerkin Finite Element ◽

Terrain Following

Advection errors are common in basic terrain-following (TF) coordinates. Numerous methods, including the hybrid TF coordinate and smoothing vertical layers, have been proposed to reduce the advection errors. Advection errors are affected by the directions of velocity fields and the complexity of the terrain. In this study, an unstructured adaptive mesh together with the discontinuous Galerkin finite element method is employed to reduce advection errors over steep terrains. To test the capability of adaptive meshes, five two-dimensional (2D) idealized tests are conducted. Then, the results of adaptive meshes are compared with those of cut-cell and TF meshes. The results show that using adaptive meshes reduces the advection errors by one to two orders of magnitude compared to the cut-cell and TF meshes regardless of variations in velocity directions or terrain complexity. Furthermore, adaptive meshes can reduce the advection errors when the tracer moves tangentially along the terrain surface and allows the terrain to be represented without incurring in severe dispersion. Finally, the computational cost is analyzed. To achieve a given tagging criterion level, the adaptive mesh requires fewer nodes, smaller minimum mesh sizes, less runtime and lower proportion between the node numbers used for resolving the tracer and each wavelength than cut-cell and TF meshes, thus reducing the computational costs.

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