My research develops high-order, efficient, and structure-preserving numerical methods for multiscale and multiphysics PDEs arising in wave propagation, charged-particle transport, liquid crystals, and quasiperiodic materials.
A central emphasis is the design of structure-preserving spectral and spectral-element methods for electromagnetic wave problems and plasma computation, together with asymptotic-preserving and geometric integrators that remain accurate across parameter regimes.
How can one design computational methods that are simultaneously mathematically reliable, physically faithful, and practical for genuinely multiscale and multiphysics systems?
Although the physical models I study are diverse, they share a common mathematical core: highly oscillatory and multiscale PDE systems involving transport, wave propagation, self-consistent field interactions, internal microstructure, geometric constraints, and in many cases relaxation or dissipation. My work aims to construct algorithms that remain accurate across regimes while preserving key analytical and physical structures.
Structure-preserving spectral and spectral-element methods with exact curl/divergence information and robust treatment of unbounded domains.
Analytically solvable and asymptotic-preserving schemes for kinetic systems with severe scale separation.
Projection and geometric-discrete-gradient methods for quasiperiodicity, anisotropy, and manifold constraints.
I have developed structure-preserving spectral and spectral-element methods for curl-curl equations and Maxwell systems, with emphasis on divergence constraints, Gauss's law, high-order accuracy, and robust solvers for highly oscillatory regimes.
A major part of my current work concerns multiscale kinetic plasma and charged-particle transport models, especially Vlasov-Poisson, Vlasov-Ampère, and Vlasov-Maxwell systems in regimes with severe scale separation.
In this direction I developed an ASAP-style viewpoint: rather than resolving dominant collective oscillations by brute-force over-discretization, identify the mechanisms producing stiffness and integrate them analytically whenever possible.
I also work on computational methods for condensed-matter systems where microstructure, anisotropy, symmetry, and quasiperiodicity are central. These include liquid crystals and photonic moiré lattices.
Looking ahead, I plan to further develop the ASAP framework into a broader methodology for oscillatory multiscale computation, including quantum transport models and collisional kinetic equations.
A second direction is to combine reduced-order modeling and scientific machine learning with the structure-preserving philosophy, so that reduced models inherit conservation laws, asymptotic limits, and physical constraints.