Eitan Tadmor - 50
Professor Eitan Tadmor turned 50 earlier this year. He is one of the most active and influential mathematicians
in the area of numerical analysis, general theory of applied PDEs,
and scientific computing. He has influenced the field of applied mathematics
in many ways: through his deep and broad mathematical research, his strong efforts
in advising, training, and mentoring young scientists, and active participation
in the scientific life of the international mathematical community by holding important
administrative positions and serving on editorial boards of leading mathematical
journals. The aim of this article is to briefly summarize some of his major
scientific achievements and his contributions to various branches of the
field of mathematics.
Eitan Tadmor did his both undergraduate and graduate studies at the Tel Aviv University
in Israel. He earned the M.Sc. degree in mathematics in 1975 under the supervision of
Professors Gideon Zwas and Moshe Goldberg, and the Ph.D. degree in mathematics in 1979
under the supervision of Professor Saul S. Abarbanel. After completing the dissertation,
he held a postdoctoral position at the California Institute of Technology (Cal. Tech.)
and at the Institute for Computer Applications in Science and Engineering (ICASE),
NASA Langley Research Center, and afterward he had regular faculty positions at the
Tel Aviv University (1983-1998) and at the University of California, Los Angeles (UCLA)
(1995-present). Currently, Eitan Tadmor is at the University of Maryland where he
holds a position of a director of the Center for Scientific Computation and Mathematical
Modeling (CSCAMM) and also professor positions at the Institute for Physical Science and
Technology (IPST) and at the Department of Mathematics.
During his fruitful career, Eitan Tadmor has had many scientific collaborators (among
them are G.-Q. Chen, D. Gottlieb, J. Goodman, T. Hou, P.-L. Lions, Y. Maday, S. Osher,
B. Perthame, C.-W. Shu, and other excellent scientists), graduate students (among them are
T. Tassa, H, Nessyahu, A. Kurganov, D. Levy, C.-T. Lin, J. Tanner, S. Nezzar, J. Balbás),
and post-docs. Eitan also actively serves the mathematical community by being an editor in
several leading journals in applied and computational mathematics, including SIAM Journal
on Numerical Analysis, Numerische Mathematik, M
2AN Mathematical Modelling and Numerical Analysis, Journal of Hyperbolic Differential Equations, IMA Journal of Numerical
Analysis, and others. One of Tadmor's biggest administrative achievements was a successful
bid of UCLA for the site of the third national NSF institute for mathematical research -
Institute for Pure and Apllied Mathematics (IPAM), where he served as a director in the
period 2000-2001.
Most of Tadmor's early works are related in one way or another to linear PDEs. His very
frst works were on the numerical radius of linear bounded operators in finite- and infinite-dimensional Hilbert spaces. A main result was the application of the methods to the stability
theory of the Lax-Wendroff scheme for linear multidimensional hyperbolic problems [6]. In
his remarkable work [34], Tadmor sharpened the known results regarding the equivalence
between the resolvent condition and the
L2-stability of a family
of
N x
N matrices as stated
by the Kreiss matrix theorem. Specifically, he showed that the power-boundedness depends,
at most, linearly on the dimension N. This result improved the best known bound at the
time which scaled like
N2 (compared with the original (
r(A))NN ).
The optimal result was proved by Spijker conferming the linear bound and placing the constant
e
in front of
N.
The work that emerged from Tadmor's Ph.D. dissertation can be characterized as
"beyond GKS". In 1972, Gustafsson, Kreiss, and Sundström published a stability criterion for
numerical approximations of initial-boundary value problems. In a series of works, most of
which were co-authored with Moshe Goldberg, Eitan has gradually improved the stability
criterion for general classes of approximations (see [7, 8] and the references therein). An
important account of stability results for various approximations of solutions to linear initial
value hyperbolic problems is presented in [37]. There, Tadmor provided a uni.ed framework
for the stability analysis of finite-diference, pseudo-spectral, and Fourier-Galerkin methods.
More general questions of stability were dealt with in connection with ODE solvers. The
work of Tadmor and Levy on the stability of Runge-Kutta schemes [20] established the stability
of high-order discretizations in time provided that the first-order method is stable.
This is valid, for example, for coercive operators (that can be viewed as dissipative operators
in the sense of Kreiss). When put into the framework of specific numerical approximations,
this result enables one to extend the results of Tadmor and Gottlieb [12] on the stability
of pseudo-spectral spatial discretizations of linear hyperbolic equations with variable
coefficients with a forward Euler temporal discretization to higher-order discretizations in
time. Strong-stability preserving high-order time discretizations for semidiscrete methods of
lines approximations of PDEs were presented in [13]. These schemes extend the class of schemes
of Shu and Osher, previously termed total variation diminishing (TVD) Runge-Kutta methods.
In the mid-1980s, Tadmor's research was focused on nonlinear finite-difference approximations,
their total variation, and entropy stability. In [35], Tadmor introduced the notions
of a numerical viscosity coefficient and a viscosity form of finite-difference schemes. He proved
that a general conservative (2
p+1)-point scheme is TVD provided its viscosity
coefficients satisfy certain lower and upper bounds (this introduced a new framework for total
variation stability analysis of finite-difference schemes). It was also demonstrated in [35]
that the entropy stability of the schemes can also be expressed in terms of the corresponding
bounds on the numerical viscosity coefficients. In [32], Tadmor and Osher utilized the numerical
viscosity framework to prove convergence of second-order TVD schemes for scalar conservation
laws with convex fluxes by enforcing a single discrete entropy inequality. Second-order entropy
conservative schemes for one-dimensional systems of conservation laws were introduced
by Tadmor in [36]. These schemes were used to show that conservative schemes are entropy
stable if and - for three-points schemes - only if they contain more viscosity than that
present in the entropy conservative schemes. This framework has been recently extended in
Tadmor's milestone work [40], where a detailed study of entropy stability for a host of
first- and second-order schemes for both scalar problems and systems of conservation laws.
Another area of Tadmor's interests is the theory of spectral methods. It is well known
that spectral projections are high order accurate provided the data is globally smooth but
they exhibit spurious Gibbs type oscillations near jump discontinuities of the underlying
function. This deteriorates their efficiency to first order for piecewise smooth data. Tadmor
and his collaborators investigated the stability of spectral and pseudospectral approximation
for linear scalar equations and linear systems of conservation laws in [9-12]. Unfortunately,
the oscillations in spectral methods lead to their instability when applied to nonlinear
conservation laws. To overcome this fundamental difficulty, Tadmor proposed the spectral
viscosity (SV) method and proved the convergence of such SV-approximations via compactness
arguments in [38]. The resulting SV-approximation is stable without sacrificing spectral accuracy
but the method is at best first order as a result of the formation of shock discontinuities
in weak solution. This raised the important questions of accurate edge detection (also of
independent interest) and elimination of the oscillations in the SV-approximation to
underlying piecewise smooth solution. In a sequence of papers with Gelb, Tadmor developed
the theory of edge detectors and applied it to identify discontinuities and post-process the
SV-approximations [3-5]. The result was the enhanced SV-method which recovers the exact
entropy solutions with high resolution and eliminates the oscillations present in the regular
SV-solution. Recently, Tadmor and Tanner developed a new class of spectral mollifiers that
gives optimal recovery of piecewise smooth data from its spectral information [43].
Another important direction of Tadmor's work is related to the development of high
resolution numerical methods for (systems of) multidimensional conservation laws and
Hamilton-Jacobi (HJ) equations. He was the first to develop high-resolution nonoscillatory
central schemes, which became a simple, robust, and universal alternative to more complex and
problem-oriented numerical methods based on upwinding. Since no (approximate) Riemann
problem solvers or characteristic decomposition are involved, central schemes can be used as a
"black-box" solver for various applied problems, including very complicated ones.
Tadmor's pioneering joint work with Nessyahu had been a corner stone in the design of such
schemes for conservation laws, where the approximate Riemann problem solvers are avoided
by integrating over the local Riemann fans. The one-dimensional second-order Nessyahu-Tadmor
scheme [29] was extended to third order in [25] and to higher number of spatial dimensions
in [14]. A more accurate estimate of the local speed of propagation of the waves generated at
the local Riemann fans helps to reduce the amount of numerical dissipation present in central
schemes. Tadmor and Kurganov utilized this idea in [16], where a new class of semidiscrete
central schemes for multidimensional (systems of) conservation laws was introduced. This
type of central schemes is especially advantageous when small time steps are enforced or
when a long time integration or steady state computations are to be performed. The central
schemes have been successfully applied to a number of complex models, including compress-
ible and incompressible Euler equations and the MHD equations in [1, 15, 18, 19]. Further,
Tadmor and Lin extended the Nessyahu-Tadmor scheme to multidimensional HJ equations
[21]. They developed the second-order Godunov-type central scheme based on global
projection operators and also studied its
L1 stability and error estimates for the
Cauchy problem associated with the multidimensional HJ equation [22]. The semidiscrete
central schemes for multidimensional HJ equations with reduced numerical dissipation
have been developed by Tadmor and Kurganov in [17].
Another avenue of Tadmor's research is the convergence rate estimates for approximate
solutions of scalar conservation laws. Tadmor was the .rst to recognize the importance of
the one-sided Lipschitz condition (OSLC) in deriving convergence and error estimates for
conservation laws. In his pioneering work [39], Tadmor developed the theory of the
Lip+
stability and used the dual
Lip´ approach to derive local error estimates
for viscosity approximations of scalar conservation laws. This duality approach was later
used by Tadmor and his collaborators to derive error estimates, including pointwise estimates
away from the discontinuities, for Godunov-type schemes, SV-approximations and relaxation
approximations to genuinely nonlinear scalar conservation laws in [30, 31, 41, 42] and for
approximate solutions of multidimensional HJ equations in [22].
One of the directions of Tadmor's theoretical work is the investigation of the link between
the continuum mechanics models (macroscopic equations, such as Navier-Stokes or Euler
equations) and kinetic models arising at a more detailed description of the evolution. More
precisely, Tadmor and Perthame [33] studied the hydrodynamical limit of BGK-type models
and their relation to the multidimensional scalar conservation laws. Further, in [23] Tadmor,
Lions, and Perthame introduced a new kinetic formulation of a general multidimensional
scalar conservation law, coupled with the corresponding entropy inequalities, which they have
later extended to the case of isentropic gas dynamics and
p-systems [24]. This
formulation is called kinetic because of its analogy with the classical kinetic models such
as the Boltzmann or Vlasov models. It allows the derivation of completely new estimates,
compactness and regularity results for solutions of conservation laws in the multidimensional
case and new, or extensions of existing, results in the one-dimensional case, such as the
celebrated Tartar's result obtained by the study of oscillations and the use of the compensated
compactness theory.
Another area of Tadmor's recent interest is critical threshold phenomena in the
Euler dynamics. In the break-through paper [2], Tadmor with Engelberg and Liu studied
hyperbolic-elliptic Euler-Poisson equations ranging from the one-dimensional case with or
without forcing mechanisms to multidimensional isentropic models with geometrical sym-
metry. A new phenomenon of a critical threshold for global smoothness vs. finite time
breakdown was discovered - only waves with sufficiently large initial gradients break down.
However, the long time behavior of the solutions is independent of these initial thresholds.
These results were extended by Tadmor and Liu to Burgers-type equations with a nonlocal
viscous term ([26]) and to the semiclassical limit of the nonlinear SchrAodinger-Poisson
equations ([27]). For the models with the rotational Coriolis forces, Tadmor and Liu [28]
showed that rotation prevents finite-time breakdown. Namely, the rotating two-dimensional
Euler equations admit global smooth solutions for a subset of generic initial configurations.
For other initial data, the breakdown depends on a critical threshold for the initial
vorticity.
It is impossible in such a brief article as this one to comment on all achievements of
Professor Eitan Tadmor during his outstanding carrier as a mathematician. We hope that
this short note will make some of his major results known to a wider audience and thus,
popularize this very important area of applied mathematics.
Many happy returns of the day, good health and all the best!
A. Kurganov, R. Lazarov, D. Levy, G. Petrova, and B. Popov
References
[1] J. Balbas, E. Tadmor, and C.-C. Wu, Non-oscillatory central schemes for one- and
two-dimensional MHD equations,
J. Comput. Phys., to appear.
[2] S. Engelberg, H. Liu, and E. Tadmor, Critical thresholds in Euler-Poisson equations,
Indiana Univ. Math. J., 50 (2001), pp. 109-157.
[3] A. Gelb and E. Tadmor, Detection of edges in spectral data,
Appl. Comput. Harmon. Anal., 7 (1999), pp. 101-135.
[4] A. Gelb and E. Tadmor, Detection of edges in spectral data II. Nonlinear enhancement,
SIAM J. Numer. Anal., 38 (2000), pp. 1389-1408.
[5] A. Gelb and E. Tadmor, Spectral reconstruction of one- and two-dimensional piecewise
smooth functions from their discrete data,
M
2AN Math. Model. Numer. Anal., 36 (2002), pp. 155-175.
[6] M. Goldberg and E. Tadmor, On the numerical radius and its applications,
Linear Algebra Appl., 42 (1982), pp. 263-284.
[7] M. Goldberg and E. Tadmor, Convenient stability criteria for difference approximations
of hyperbolic initial-boundary value problems,
Math. Comp., 44 (1985), pp. 361-377.
[8] M. Goldberg and E. Tadmor, Convenient stability criteria for difference approximations
of hyperbolic initial-boundary value problems. II,
Math. Comp., 48 (1987), pp. 503-520.
[9] J. Goodman, T. Hou, and E. Tadmor, On the stability of the unsmoothed Fourier method
for hyperbolic equations,
Numer. Math., 67 (1994), pp. 93-129.
[10] D. Gottlieb, L. Lustman, and E. Tadmor, Convergence of spectral methods for hyperbolic
initial-boundary value systems,
SIAM J. Numer. Anal., 24 (1987), pp. 532-537.
[11] D. Gottlieb, L. Lustman, and E. Tadmor, Stability analysis of spectral methods for
hyperbolic initial-boundary value systems,
SIAM J. Numer. Anal., 24 (1987), pp. 241-256.
[12] D. Gottlieb and E. Tadmor, The CFL condition for spectral approximations to hyperbolic
initial-boundary value problems,
Math. Comp., 56 (1991), pp. 565-588.
[13] S. Gottlieb, C.-W. Shu, and E. Tadmor, Strong stability-preserving high order time
discretization methods,
SIAM Rev., 43 (2001), pp. 89-112.
[14] G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensional
hyperbolic conservation laws,
SIAM J. Sci. Comput., 19 (1998), pp. 1892-1917.
[15] R. Kupferman and E. Tadmor, A fast high-resolution second-order central scheme for
incompressible flows,
Proc. Natl. Acad. Sci. USA, 94 (1997), pp. 4848-4852.
[16] A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear
conservation laws and convection-diffusion equations,
J. Comput. Phys., 160 (2000), pp. 214-282.
[17] A. Kurganov and E. Tadmor, New high-resolution semi-discrete central schemes for
Hamilton-Jacobi equations,
J. Comput. Phys., 160 (2000), pp. 720-742.
[18] A. Kurganov and E. Tadmor, Solution of two-dimensional Riemann problems for gas
dynamics without Riemann problem solvers,
Numer. Methods Partial Differential Equations, 18 (2002), pp. 584-608.
[19] D. Levy and E. Tadmor, Non-oscillatory central schemes for the incompressible
2-D Euler equations,
Math. Res. Lett., 4 (1997), pp. 1-20.
[20] D. Levy and E. Tadmor, Non-oscillatory central schemes for the incompressible
2-D Euler equations,
Math. Res. Lett., 21 (2001), pp. 2163-2186.
[21] C.-T. Lin and E. Tadmor, High-resolution non-oscillatory central scheme for
Hamilton-Jacobi equations,
SIAM J. Sci. Comput., 21 (2000), pp. 2163-2186.
[22] C.-T. Lin and E. Tadmor,
L1-stability and error estimates for
approximate Hamilton-Jacobi solutions,
Numer. Math., 87 (2001), pp. 701-735.
[23] P.-L. Lions, P. Perthame, and E. Tadmor, A kinetic formulation of multidimensional
scalar conservation laws and related equations,
J. Amer. Math. Soc., 7 (1994), pp. 169-191.
[24] P.-L. Lions, P. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas
dynamics and
p-systems,
Comm. Math. Phys., 163 (1994), pp. 415-431.
[25] H. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic
conservation laws,
Numer. Math., 79 (1998), pp. 397-425.
[26] H. Liu and E. Tadmor, Critical thresholds in a convolution model for nonlinear
conservation laws,
SIAM J. Math. Anal., 33 (2001), pp. 930-945.
[27] H. Liu and E. Tadmor, Semi-classical limit of the nonlinear
Schrödinger-Poisson equation with subcritical initial data,
Methods Appl. Anal., 9 (2002), pp. 517-532.
[28] H. Liu and E. Tadmor, Rotation prevents finite-time breakdown,
Phys. D, 188 (2004), pp. 262-276.
[29] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic
conservation laws,
J. Comput. Phys., 87 (1990), pp. 408-463.
[30] H. Nessyahu and E. Tadmor, The convergence rate of nonlinear scalar conservation laws,
SIAM J. Numer. Anal., 29 (1992), pp. 1505-1519.
[31] H. Nessyahu, E. Tadmor, and T. Tassa, The convergence rate of Godunov type schemes,
SIAM J. Numer. Anal., 31 (1994), pp. 1-16.
[32] S. Osher and E. Tadmor, On the convergence of difference approximations to scalar
conservation laws,
Math. Comp., 50 (1988), pp. 19-51.
[33] B. Perthame and E. Tadmor, kinetic equation with kinetic entropy functions for scalar
conservation laws,
Comm. Math. Phys., 136 (1991), pp. 501-517.
[34] E. Tadmor, The equivalence of
L2-stability, the resolvent condition
and strict
H-stability,
Linear Algebra Appl., 41 (1981), pp. 151-159.
[35] E. Tadmor, Numerical viscosity and the entropy condition for conservative difference
schemes,
Math. Comp., 43 (1984), pp. 369-381.
[36] E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation
laws. I,
Math. Comp., 49 (1987), pp. 91-103.
[37] E. Tadmor, tability analysis of finite-difference, pseudospectral and Fourier-Galerkin
approximations for time-dependent problems,
SIAM Rev., 29 (1987), pp. 525-555.
[38] E. Tadmor, Convergence of spectral methods for nonlinear conservation laws,
SIAM J. Numer. Anal., 2 (1989), pp. 30-44.
[39] E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic
equations,
SIAM J. Numer. Anal., 28 (1991), pp. 891-906.
[40] E. Tadmor, Entropy stability theory for difference approximations of nonlinear
conservation laws and related time dependent problems,
Acta Numer., (2003), pp. 451-512.
[41] E. Tadmor and T. Tang, Pointwise error estimates for scalar conservation laws with
piecewise smooth solutions,
SIAM J. Numer. Anal., 36 (1999), pp. 1739-1756.
[42] E. Tadmor and T. Tang, Pointwise error estimates for relaxation approximations to
conservation laws,
SIAM J. Numer. Anal., 32 (2001), pp. 870-886.
[43] E. Tadmor and J. Tanner, Adaptive mollifiers - high resolution recovery of piecewise
smooth data from its spectral information,
Found. Comput. Math., 2 (2002), pp. 155-189.
