There are two commonly used numerical approaches to solve the gradient flow problem while preserving the desired energy stability: the Convex Splitting approach and the Stabilized approach. The Convex Splitting approach is energy stable, however, it produces a nonlinear scheme at most cases, thus the implementation is complicated and the computational cost is high. Stabilized approach generates linear scheme that is extremely easy to implement, however, the magnitude of the stabilizing term depends on the upper bound of the second order derivative of the nonlinear potential. Therefore, such method is particularly reliable for those models with maximum principle. For many nonlinear models, both of the two methods are not optimal choices. We introduce a novel, so called Compulsory-Quasi-Lagrange-Multiplier approach, that can possess the advantages of both the convex splitting approach and the stabilized approach, but avoid their imperfections mentioned above.

More precisely, the schemes (i) are accurate (up to second order in time); (ii) are stable (unconditional energy dissipation law holds); and (iii) are efficient and easy to implement (only need to solve some linear equations at each time step.

Biographical Sketch: Dr. Yang got his Ph.D from Purdue University at 2007.  He finished his postdoc at University of North Carolina at Chapel Hill at 2009, and started the assistant professorship at University of South Carolina. He is now an associate professor of USC. His research is about the modeling, numerical analysis and simulations for Complex fluids. He published more than 40 peer reviewed journal papers and was invited to give more than 60 talks and seminars in many conferences, universities and institutes around the world.