Professor Shu has a variety of research interests including: numerical solutions of conservation laws; convection dominated problems using finite difference (essentially non-oscillatory (ENO) methods and weighted ENO (WENO) methods); finite element discontinuous Galerkin methods and spectral methods; numerical solution of Hamilton-Jacobi type equations; computational fluid dynamics; and numerical solution of equations appearing in semi-conductor device simulations.

Professor Shu's research activities are concentrated on the design,
analysis, implementation and application of high order nonlinearly
stable numerical schemes for solving convection dominated partial
differential equations.

One major type of such schemes is the class of WENO (weighted essentially non-oscillatory) finite difference and finite volume
schemes. These are schemes with solutions which are high order
accurate in smooth regions and at the same time produce sharp and
non-oscillatory shock transitions. They are especially suitable
for solving problems with both strong discontinuities and complex
smooth regions structures, such as the problem of shock interacting with turbulence. Recent development in WENO schemes include the technique to handle negative linear weights, the design of WENO schemes for conservation laws and Hamilton-Jacobi equations on arbitrary unstructured meshes, multi-domain finite difference WENO schemes, Hermite type WENO schemes, anti-diffusive corrections to WENO schemes to sharpen contact discontinuities, well-balanced WENO schemes, and residual distribution finite difference WENO schemes for steady state problems on nonsmooth meshes. Recent applications of WENO schemes include the simulation for the Boltzmann-Poisson system in semiconductor device simulations in one and two spatial dimensions and in two and three phase space dimensions, particle-fluid two phase flow problems, cosmology in astrophysics, high Mach number astrophysical jets with radiative cooling, nearly-incompressible, inviscid Taylor-Green vortex flow, shallow water equations, hyperbolic models for chemosensitive movement, multi-stage interaction of a shock wave and a strong vortex, multi-class traffic flow models, shock mitigation and drag reduction by pulsed energy lines, low-redshift cosmic baryon fluid on large scales, and Rayleigh-Taylor instability.

Another major type of such schemes is the class of discontinuous
Galerkin (DG) methods. These are finite element methods but with
important features borrowed from high resolution finite volume
methodology, such as exact or approximate Riemann solvers as
numerical fluxes, limiters to control spurious oscillations, and
strong stability, preserving high order Runge-Kutta time
discretizations. The major advantages of the discontinuous Galerkin
method include its local conservation, compact data communication
and subsequent high parallel efficiency, flexibility of arbitrary
h-p adaptivity while keeping nonlinear stability, and provable
stability and error estimates for many complicated nonlinear
convection dominated PDEs. Recent development in DG methods
include the design of local discontinuous Galerkin methods for
nonlinear wave equations involving higher spatial derivatives,
such as the KdV type equations, the nonlinear Schroedinger
equations, the Kuramoto-Sivashinsky equations and the Ito-type
coupled KdV equations, and the Kadomtsev-Petviashvili and the
Zakharov-Kuznetsov equations, the design of WENO and Hermite
WENO limiters for discontinuous Galerkin method, the post-processing
technique to enhance the order of accuracy, the design of
discontinuous Galerkin method for Hamilton-Jacobi equations,
the analysis of the discontinuous Galerkin and related methods
such as the spectral finite volume method, the design of locally divergence-free discontinuous Galerkin methods for the Maxwell and MHD equations and in general structure preserving discontinuous Galerkin methods, error estimates for fully discrete Runge-Kutta discontinuous Galerkin method for nonlinear conservation law systems and for other nonlinear wave equations for smooth solutions, and the design of discontinuous Galerkin method based on non-polynomial approximation spaces. Recent applications of the discontinuous Galerkin method include aeroacoustic simulations, Maxwell equations and MHD, compactons, moment models in semiconductor device simulations, and shallow water equations.

Related research has also been performed in multi-scale methods,
spectral methods for discontinuous problems with the emphasis on
accuracy recovery techniques, Lax-Wendroff type time discretizations,
and stable high resolution schemes for a hierarchical size-structured
model in computational biology.

PI continuously on grants from the Army Research Office and National Science Foundations and a co-PI on grants from the Air Force Office of Scientific Research since 1988.

Fellow (inaugural class), American Mathematical Society (AMS), November 2012.

Fellow (inaugural class), Society for Industrial and Applied Mathematics (SIAM), May 2009.

SIAM/ACM Prize in Computational Science and Engineering (SIAM/ACM CSE Prize), February 2007.

ISI Highly Cited Author in Mathematics (http://isihighlycited.com/), ISI Web of Knowledge, Thomson Scientific Company, 2004- .

First Feng Kang Prize of Scientific Computing, Chinese Academy of Sciences, Beijing,October 1995.

NASA Public Service Group Achievement Award for pioneering work in Computational Fluid Dynamics, NASA Langley Research Center, March 1992.

Member (since 1983), Fellow (since 2012) and Council member (2002-2012), American Mathematical Society (AMS)

Member (since 1987) and Fellow (since 2009), Society for Industrial and Applied Mathematics (SIAM)

Managing Editor, Mathematics of Computation, 2/02-1/12. Associate Editor, 93-01; 12- .

Chief Editor, Journal of Scientific Computing, 2/09- . Co-Chief Editor, 3/00-1/09.

Co-Chief Editor, Methods and Applications of Analysis, 1/08- . Member of Editorial Board, 00-07