TY - JOUR

T1 - Operator-splitting methods for the 2D convective Cahn–Hilliard equation

AU - Gidey, H. H.

AU - Reddy, B. D.

N1 - Funding Information:
The authors acknowledge the support by the National Research Foundation, through the South African Research Chair in Computational Mechanics. The authors are grateful to the reviewers for their useful comments.
Publisher Copyright:
© 2019 Elsevier Ltd

PY - 2019/6/15

Y1 - 2019/6/15

N2 - In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.

AB - In this work, we present operator-splitting methods for the two-dimensional nonlinear fourth-order convective Cahn–Hilliard equation with specified initial condition and periodic boundary conditions. The full problem is split into hyperbolic, nonlinear diffusion and linear fourth-order problems. We prove that the semi-discrete approximate solution obtained from the operator-splitting method converges to the weak solution. Numerical methods are then constructed to solve each sub equations sequentially. The hyperbolic conservation law is solved by efficient finite volume methods and dimensional splitting method, while the one-dimensional hyperbolic conservation laws are solved using front tracking algorithm. The front tracking method is based on the exact solution and hence has no stability restriction on the size of the time step. The nonlinear diffusion problem is solved by a linearized implicit finite volume method, which is unconditionally stable. The linear fourth-order equation is solved using a pseudo-spectral method, which is based on an exact solution. Finally, some numerical experiments are carried out to test the performance of the proposed numerical methods.

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U2 - 10.1016/j.camwa.2019.01.023

DO - 10.1016/j.camwa.2019.01.023

M3 - Article

AN - SCOPUS:85061604569

VL - 77

SP - 3128

EP - 3153

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 12

ER -