Research Projects

Towards a new-generation numerical scheme based on Active flux method for the compressible Navier-Stokes equations

• We introduced a novel treatment of the discrete conservation law for the Active Flux (AF) method.

• We extended the original 3rd-order Active Flux method to a family of fully-discrete numerical schemes of abitray order of accuracy for multidimensional space.

• We introduced a novel hyperbolic reformulation of the Navier-Stokes equations, which allows us to develop an AF-based solver for the Navier-Stokes equations.

Advisor: Philip Roe (Aerospace Engineering) & Smadar Karni (Math), University of Michigan

Recovery-based discontinuous Galerkin method for the Cahn-Hilliard equation (2018)

The Recovery-based discontinuous Galerkin (Recovery DG) method is a promising approach for computing second-order partial differential equations (PDEs), e.g., the heat equation. The high-order accuracy of Recovery DG makes it a good candidate for computing yet higher-order PDEs, such as the Cahn-Hilliard equation, which includes fourth-order spatial derivatives. In this article, we develop a Recovery-based discontinuous Galerkin method for the Cahn-Hilliard equation. We additionally suggest a new approach for analysis of Recovery DG schemes via Taylor expansion, which we use to confirm the accuracy of the Recovery DG method and guide further application. Numerical experiments demonstrate the accuracy and superconvergence of our proposed Recovery scheme for the Cahn-Hilliard equations. Compared to the established Local discontinuous Galerkin (LDG) approach, our proposed method is more accurate and allows for larger time steps.

Advisor: Eric Johnsen (Mechanical Engineering), University of Michigan

Application of modified Cahn-Hilliard model for binary image inpainting in grayscale/color image inpainting (2015)

In this project, we extended the work by Andrea Bertozzi, Selim Esedoglu and Alan Gillette on a method for binary image inpainting based on modified Cahn-Hilliard Equations for general color image inpainting. In our new algorithm, we transformed the color image inpainting problem into grayscale image inpainting of each RGB component and then decomposed a grayscale inpainting problem to several binary image inpainting problems.

Advisor: Wenbin Chen (Math), Fudan University

Computing the stationary distribution of a Markov Chain (2014)

In the project, I analyzed and compared the performance of two classic general methods: Power Method and Gauss Elimination in computing the stationary distribution of a Markov Chain. Then I developed new algorithm with time and space complexity of O(N) for Markov Chains which describe random walk on trees, where N is the number of nodes. I also illustrated how to use this idea to improve the implementation of Gauss Elimination for large sparse Markov Chains.

Advisor: Jiangang Ying (Math), Fudan University