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1 – 3 of 3Yunxia Sun, Xufeng Xiao, Zhiming Gao and Xinlong Feng
The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the…
Abstract
Purpose
The purpose of this paper is to propose an efficient space-time operator-splitting method for the high-dimensional vector-valued Allen–Cahn (AC) equations. The key of the space-time operator-splitting is to devide the complex partial differential equations into simple heat equations and nolinear ordinary differential equations.
Design/methodology/approach
Each component of high-dimensional heat equations is split into a series of one-dimensional heat equations in different spatial directions. The nonlinear ordinary differential equations are solved by a stabilized semi-implicit scheme to preserve the upper bound of the solution. The algorithm greatly reduces the computational complexity and storage requirement.
Findings
The theoretical analyses of stability in terms of upper bound preservation and mass conservation are shown. The numerical results of phase separation, evolution of the total free energy and total mass conservation show the effectiveness and accuracy of the space-time operator-splitting method.
Practical implications
Extensive 2D/3D numerical tests demonstrated the efficacy and accuracy of the proposed method.
Originality/value
The space-time operator-splitting method reduces the complexity of the problem and reduces the storage space by turning the high-dimensional problem into a series of 1D problems. We give the theoretical analyses of upper bound preservation and mass conservation for the proposed method.
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Keywords
Yasir Khan, Ishtiaq Ali, S. Islam and Qing Biao Wu
The purpose of this paper is to present some numerical methods based on different time stepping and space discretization methods for the Allen‐Cahn equation with non‐periodic…
Abstract
Purpose
The purpose of this paper is to present some numerical methods based on different time stepping and space discretization methods for the Allen‐Cahn equation with non‐periodic boundary conditions.
Design/methodology/approach
In space the equation is discretized by the Chebyshev spectral method, while in time the exponential time differencing fourth‐order Runge‐Kutta (ETDRK4) and implicit‐explicit scheme are used. Also, for comparison the finite difference scheme in both space and time is used.
Findings
It is found that the use of implicit‐explicit scheme allows use of a large time‐step, since an explicit method has less order of accuracy as compared to implicit‐explicit method. In time‐stepping the proposed ETDRK4 does not behave well for this special kind of partial differential equation.
Originality/value
The paper presents some numerical methods based on different time stepping and space discretization methods for the Allen‐Cahn equation with non‐periodic boundary conditions.
Details
Keywords
Gang Peng, Zhiming Gao, Wenjing Yan and Xinlong Feng
This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository.
Abstract
Purpose
This paper aims to consider numerical simulation for radionuclide transport calculations in geological radioactive waste repository.
Design/methodology/approach
The nonlinear two-point flux approximation is used to discretize the diffusion flux and has a fixed stencil. The cell-vertex unknowns are applied to define the auxiliary unknowns and can be interpolated by the cell-centered unknowns. The approximation of convection flux is based on the second-order upwind method with a slope limiter.
Findings
Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.
Originality/value
A new positivity-preserving nonlinear finite volume scheme is proposed to simulate the far-field model used in the geological radioactive waste repository. Numerical results illustrate that the positivity-preserving is satisfied in solving this convection-diffusion system and has a second-order convergence rate on the distorted meshes.
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