RESEARCH

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Mathematical Modeling and Numerical Simulation

In various fields such as biology, medicine, financial engineering, material science, image science, fluid dynamics, data science, etc., many of problems can be represented by mathematical concepts and ideas to make better comparison, prediction, and improvement. The goal of our research group is to understand topics arising in a real-word (not limited to specific fields) for mathematical description and to perform numerical simulations to mimic the phenomena in silico environment.

· Interfacial problems using phase-field and immersed-boundary methods

· Mathematical biology and medicine such as pattern formation, cell division, eye movement, enzyme kinetics, circadian clock, liquid-liquid phase separation, and etc

· Industrial problems including autonomous driving, gait analysis, bone registration, and etc

· Finance engineering especially volatility and relationship between stochastic and deterministic PDEs

· Data-based approaches including supervised/unsupervised/reinforce learnings and generative models

FIGURE 1. Cell division driven by contractile ring using phase-field and immersed-boundary methods

FIGURE 2. Numerical simulation of shallow water equation on sphere using closest point method and WENO scheme

FIGURE 3. Data classification of biological pattern generated by reaction-diffusion equation

Numerical Analysis and Scientific Computing

From a mathematical description to a computer-based simulation, most of cases need approximation with mathematical theory of stability, unique solvability, convergence, consistency, less computational cost and complexity, etc. We aim to prove and analyze such topics in a discrete or semi-discrete sense using linear algebra, numerical analysis, functional analysis, etc.

· High-order stable numerical schemes for gradient flows including Allen-Cahn, Cahn-Hilliard, Swift-Hohenberg, phase-field crystal, Keller-Siegel, and other related equations

· High-order stable numerical schemes for fluid dynamics including Navier-Stokes and shallow water equations

· Numerical approximation of nonlinear parabolic (fractional) PDEs

· Convex splitting method, effective time step analysis, spectral method, compact scheme, and etc

· Moving overset grid method for a fluid-solid interaction

· Finite difference method for PDEs on a curved surface

FIGURE 4. Temporal convergence order; first-, second-, and third-order accuracy

FIGURE 5. Unconditionally gradient energy dissipation

FIGURE 6. Consistency between analytic and numerical solutions

WORKING PAPERS

· Seunggyu Lee and Yong-Jung Kim, Numerical simulations of a shallow water equation using the finite difference WENO scheme on curved surfaces

· Young Ho Lee and Seunggyu Lee, Hypothesis on fingerprint patterning related with sweat gland