Numerical solutions to ODE/PDEs with applications

Faculty Mentor: Zhillin Li

Prerequisites: Calculus, linear algebra, differential equations, numerical analysis.

Outline: To solve differential equations either ordinary (ODEs) or partial (PDEs). Our focus will be on two-phase and multi-phase flow problems, free boundary, and moving interface problems.

Research objectives: We will explore accuracy of computing the solution and the derivative for ODE/PDE using finite difference or finite element discretization (1,2). We will also explore traditional and new methods for evolving free boundaries and moving interfaces such as the front tracking method, the level set method, arbitrary Lagrangian, and the machine learning approach. Students will explore parallel and super-computing and carry out a theoretical analysis of the numerical methods.

Outcomes: The students will gain state of art techniques in solving mathematical models described in the proposal. Some of the new numerical methods will appear as software packages for public use. We also expect to have a couple of publications from the research.

References:

• Gong Y, Li B, Li Z. Immersed-interface finite-element methods for elliptic interface problemswith non-homogeneous jump conditions. SIAM J Numer ANal. 2008;46:472-495.
• LeVeque R, Li Z. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J Numer ANal. 1994;31:1019-1044.