A 1D Macroscopic Phase Field Model for Dislocations and a Second Order $\Gamma$-Limit

2008 ◽  
Vol 6 (4) ◽  
pp. 1098-1124 ◽  
Author(s):  
Matteo Focardi ◽  
Adriana Garroni
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Second Order

On a SAV-MAC scheme for the Cahn–Hilliard–Navier–Stokes phase-field model and its error analysis for the corresponding Cahn–Hilliard–Stokes case

2020 ◽  
Vol 30 (12) ◽  
pp. 2263-2297
Author(s):  
Xiaoli Li ◽  
Jie Shen
Keyword(s):  
Error Analysis ◽  
Phase Field ◽  
Field Model ◽  
Chemical Potential ◽  
Phase Field Model ◽  
Auxiliary Variable ◽  
Field Variable ◽  
Second Order ◽  
Navier Stokes ◽  
Order Error

We construct a numerical scheme based on the scalar auxiliary variable (SAV) approach in time and the MAC discretization in space for the Cahn–Hilliard–Navier–Stokes phase- field model, prove its energy stability, and carry out error analysis for the corresponding Cahn–Hilliard–Stokes model only. The scheme is linear, second-order, unconditionally energy stable and can be implemented very efficiently. We establish second-order error estimates both in time and space for phase-field variable, chemical potential, velocity and pressure in different discrete norms for the Cahn–Hilliard–Stokes phase-field model. We also provide numerical experiments to verify our theoretical results and demonstrate the robustness and accuracy of our scheme.

Numerical Tests of a Phase Field Model with Second Order Accuracy

2008 ◽  
Vol 68 (6) ◽  
pp. 1518-1534 ◽  
Author(s):  
Gunduz Caginalp ◽  
Xinfu Chen ◽  
Christof Eck
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Phase Field Model ◽  
Second Order ◽  
Order Accuracy ◽  
Numerical Tests
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