Jie Shen

TL;DR: In this paper, a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed, and the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners.

TL;DR: In this article, a unified framework for designing and analyzing spectral algorithms for a variety of problems, including in particular high-order differential equations and problems in unbounded domains, is presented.

TL;DR: In this paper, an efficient and accurate numerical method is implemented for solving the time-dependent Ginzburg-Landau equation and the Cahn-Hilliard equation, where the time variable is discretized by using semi-implicit schemes which allow much larger time step sizes than explicit schemes; the space variables are discretised by using a Fourier-spectral method whose convergence rate is exponential in contrast to second order by a usual finite-difference method.

TL;DR: In this paper, the authors proposed a diffuse-interface approach to simulating the flow of two-phase systems of microstructured complex fluids, where the energy law of the system guarantees the existence of a solution.

TL;DR: In this paper, stability analyses and error estimates for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations were carried out and it was shown that all the schemes were either unconditionally energy stable, or reasonably stable with reasonable stability conditions in the semi-discretized versions.