Prize-winning Research in Computational Science and EngineeringApril 11, 2007
Chi-Wang Shu of Brown University received the 2007 SIAM/ACM Prize in Computational Science and Engineering in February at the SIAM Conference on Computational Science and Engineering, Costa Mesa, California. Shu, shown here with SIAM vice president at large David Keyes, was cited for “the development of numerical methods that have had a great impact on scientific computing, including TVD temporal discretizations, ENO and WENO finite difference schemes, discontinuous Galerkin methods, and spectral methods.”
Philip J. Davis
It was recently my pleasure to step into the office adjacent to mine and congratulate a genial colleague who had just received an important prize: the SIAM/ACM Prize in Computational Science and Engineering. For some time I had wondered what Professor Chi-Wang Shu "did" and with what material he confronted the steady stream of graduate students who have worked with him. By "interviewing" him in preparation for this article, I gained some insight into the work mentioned in the prize citation, along with a tutorial in computational methods for fluids.
Despite coming to maturity in China during the disastrous "cultural revolution," which severely downplayed science and technology, Chi-Wang got some mathematics in high school. Happily, this dark period in China's history came to an end in 1976, and the traditional college curriculum was restored. At the age of twenty Shu entered the University of Science and Technology of China, in Hefei, Anhui Province, where he received a bachelor's degree in mathematics in 1982. (One of his teachers in Hefei was Geng-Zhe Chang, whom I knew and collaborated with on several projects.) Applied mathematics courses were available only in the students' last year.
Shortly thereafter, Shu came to the U.S. as a graduate student in the mathematics department at UCLA. At the end of his first year, he fell in with Stanley Osher. He liked Osher's area of applied mathematics and in 1986 received his PhD under Osher's guidance, with a thesis titled "Numerical Solutions of Conservation Laws." In 1987 Shu accepted a position in the Division of Applied Mathematics at Brown; in rapid succession he achieved professorship and chairmanship, all the while turning out research papers and PhD students.
Over the years his research interests have been consistently focused on numerical methods for the solution of "convection-dominated" hyperbolic and mixed, time-dependent partial differential equations arising in fluid theory and other application areas. He has found collaboration easy and natural, working constantly not only with his students but also with such researchers as David Gottlieb, Bernardo Cockburn, Joseph Jerome, and Irene Gamba.
Asked how he places himself in the spectrum of mathematicians, Shu replied, "I think of myself as half a numerical analyst, while the other half is more applied: designing algorithms for applications and not necessarily involving analysis. I believe I'm a specialist and not a generalist." In performing computational experiments, he and his group write their own codes in Fortran, relying occasionally on Mathematica for arduous algebraic manipulations.
The SIAM/ACM prize (the most recent of a number of honors accorded Shu), was presented in February at the 2007 SIAM Conference on Computational Science and Engineering. The prize recognized his accumulated work, particularly on the TVD temporal discretization, ENO and WENO finite difference schemes, discontinuous Galerkin methods, and spectral methods. In the course of our interview, I learned a bit about each method, including benefits and drawbacks, as well as recent improvements and applications.
TVD (total variation diminishing) refers to an idea from spatial discretization that allows the scheme to maintain stability in the bounded variation semi-norm for discontinuous solutions, thus avoiding the nonphysical oscillations around discontinuities (Gibbs phenomenon) typical of traditional high-order accurate schemes. Even though a scheme is TVD for the first-order time discretization, however, it is not easy to maintain this stability when a higher-order time discretization is applied. Shu began to design schemes for such time discretization, starting from his PhD thesis; this work led to a paper in SIAM Journal on Scientific and Statistical Computing covering multistep methods and a joint paper with Osher in Journal of Computational Physics (JCP) covering Runge–Kutta methods (both in 1988). Shu has continued to work in this area with students, most notably with Sigal Gottlieb, whose thesis is based mainly on this work.
ENO stands for "essentially non-oscillatory"; WENO adds "weighted" to this description. The popular TVD schemes do not increase the bounded variation semi-norm but have the drawback of being limited, basically, to second-order accuracy. ENO schemes were originally developed in a seminal 1987 JCP paper by Ami Harten, Björn Engquist, Osher, and Sukumar Chakravarthy (the first three were among Shu's teachers at UCLA). The idea is essentially to use interpolation stencils adaptively. Traditional schemes typically use a fixed stencil: To interpolate at i, you might use i – 1, i, and i + 1 for a second-order polynomial; of course, you could also use i – 2, i – 1, and i. The idea of ENO is to choose automatically the best stencil to use locally. An ENO scheme can provide high-order accuracy without oscillations around shocks. Shu's main work on ENO schemes, described in two JCP papers with Osher, employs a conservative finite difference framework that allows simple and efficient computation for multidimensional problems---the reason for the popularity of ENO schemes.
WENO schemes automatically pick linear combinations of the stencils with suitable locally determined weights. Xu-Dong Liu, Osher, and Tony Chan designed the first WENO scheme, in third-order finite volume form, in a 1994 JCP paper. Shu, with his then PhD student Guang-Shan Jiang, formulated a general strategy for finite difference WENO schemes of arbitrary accuracy in a 1996 JCP paper; the fifth-order WENO scheme from this paper is the one used most often in later applications.
Finite difference schemes have the ad-vantages of simplicity and low storage requirements; they can be used on domains with regular geometries that can be put onto a very smooth mesh. The finite element method, by contrast, is a bit more complicated to design and harder to code, but it fits arbitrary geometries and accommodates adaptive methods more easily.
Shu has been working with his University of Minnesota colleague and friend Bernardo Cockburn for almost 20 years on the design, analysis, and application of the discontinuous Galerkin (DG) finite element method, which is particularly suitable for convection-dominated partial differential equations. In the past few years, the DG method has become tremendously popular among those working in applications. Shu has also been collaborating with his Brown colleague David Gottlieb on spectral methods for discontinuous problems, mainly on the recovery of uniform spectral accuracy from spectral expansion coefficients of discontinuous but piecewise-smooth functions.
As to applications, Shu works with people in the computational fluid group at NASA, Langley, in particular with Harold Atkins. He mentioned the exterior of an airfoil as an example of the geometries he considers. Emphasizing the experimental nature of his work, he pointed out that he may get "crazy" ideas for many situations. He tries them out; some work and some do not. In the case of hyperbolic problems, theoretical mathematics sometimes provides guidelines, but they are not always sufficient. Many aspects of algorithm design depend on intuition and heuristics, he said. It is rewarding to see that algorithms are playing important roles in applications. While the "digital wind tunnel" may not have completely replaced the physical wind tunnel, computational methods are making a substantial contribution to the craft of engineering.
Shu is delighted by the growing popularity of the high-order, high-resolution schemes for shock calculations and for general convection-dominated problems on which his research is focused. And I, in turn, was delighted to learn of developments in a field in which I was engaged years ago, as a young aerodynamicist at Langley.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island, and can be reached at email@example.com.