Research on
Long Time Behavior of Numerical Methods
1977-2007

William Layton
W J L at P I T T dot E D U

I've had a great interest in the long time behavior of numerical methods for evolution equations since my PhD thesis research. This page describes some of the previous work on the topic.

The dissipative case.

One of the topics of my thesis

The Galerkin Method for First Order Hyperbolic Equations , Ph.D.
dissertation ,1980 University of Tennessee

was error estimates for semi and fully discrete Galerkin FEMs over long
time intervals for problems of the general form

(1) d/dt u + Au = f(x,t) , t > 0 , u(x,0) given , where

A: semi-bounded operator (e.g., hyperbolic systems, parabolic equations ... )
on appropriate spaces:

for some real number a:

(2) ( Av , v ) <= a ( v , v ) , for all v in D(A) in a Hilbert space,

and where f is a body force that is NOT asymptotically negligible, e.g.
including the case f=f(x),

(3) ess sup {||f(.,t)||, 0 less than t less than infinity} is finite.

When (1) is dissipative, i.e., a < 0, it was proven in  that the numerical
methods converged uniformly in t over the whole time interval, 0< t <
infinity.

Some of these estimates appear in

Stable Galerkin Methods for Hyperbolic Systems , SIAM J. Numer. Anal., 20, 1983, 221--233 , MR 95c: 65120

This has developed in interesting directions since then. Very recently Mihai
Anitescu, Faranak Pahlevani and I considered similar questions related
to multi-scale stabilization of evolution equations in

M. Anitescu, W. Layton and F. Pahlevani, Implicit for local effects, explicit
for nonlocal is unconditionally stable, ETNA, 18174-187.

The Conservative Case.

The case a<0 is (or at least should be by now) recognized as the easy case.
When a = 0 the problem can be exactly conservative , such as find
u=u(x,t) satisfying

(4) du/dt + du/dx = 0 , u(x,0)= f(x) , x in IR

(NOTE: Interpret all derivatives as partial derivatives please!)
The difficulty in conservative problems is that

any slight (and unavoidable) phase error will cause linear growth of numerical errors until the error is O(1).

On the other hand, the solution structure is wave like and my intuition was
that error waves would pass through any bounded region (think of the
scattering problem) and IN THAT BOUNDED REGION, the wave like
error structure leads to errors bounded uniformly in time. This was
proven for (4) in

Error Estimates for Finite Difference Approximations to Hyperbolic
Equations for Large Time, P.A.M.S., 92 ,1984 ,425--431, MR 86h: 65135

In the next paper, this research was continued and similar results were
proven for general, multi-dimensional hyperbolic systems. (I am still
quite proud of this paper.)

On the Behavior Over Long Time Intervals of Finite Difference and Finite
Element Approximations to Hyperbolic Equations Comp. and Mth.
with Appls , 11 , 1985,93--112, MR 86g: 65176.

In bounded regions where error waves (we intuit) reflect through the region
indefinitely, the phase error and particularly the dispersion error is
where the battle must be waged to increase the time interval of
accuracy. This was considered and minimal dispersion methods were
developed in

with Quian Du and M. Gunzburger , Low Dispersion, High Accuracy
Finite Element Method for Hyperbolic Systems in Several Space
Variables , V15 , 1988 pages 447--457 Computers and
Mathematics with Applications

This is related to work in

High Accuracy Finite Element Methods for Positive Symmetric Systems
Comp. and Math. W. Appls. , 12 , 1986 pages 565-579

with Q. Du and M. Gunzburger A nonstandard method of higher
accuracy for hyperbolic systems in several space variables
Advances in Computer Methods for P.D.E.'s., IV, IMACS pages 92-97 , 1989

A Theory of Discrete Limiting Equations

One way to study asymptotics of equations like (1) is to translate t
backwards in the equation ands solution, passing to the limiting
equation. The limiting equation is posed over all of IR and the initial
condition (and its influence) disappears. Find u*(x,t) satisfying in an
appropriate sense

(5) d/dt u* + Au* = f*(x) , t in IR .

When || f*(.,t)|| is bounded in time, this is also the appropriate sense and
often (but not always) it can be shown that a unique solution exists.

I was interested to know if a theory of discrete limiting equations could be
developed and used to give insight into long time behavior of
numerical errors. My work on this work began not for PDEs but rather
for (nonlinear) functional differential equations in

On Nonlinear Difference Approximations to Nonlinear Functional
Differential Equations, with L. Drager , Libertas Math., 3 , 1983,45-65 , MR85j: 34165

and

The Galerkin Method for the Approximation of Almost Periodic Solutions
of Functional Differential Equations Funk. Eva. , 29 , 1986 pages 19-29

Some theory was developed and it was shown that the discrete limiting
equation has a unique appropriate sense solution which approximates,
uniformly over all t in IR, the appropriate sense solution of the
continuous limiting equation.

This work continued in

Asymptotics of Numerical Methods for Nonlinear Evolution Equations
with L. Drager and R. Mattheij in: Proc. VI Int. Conf. on Trends in
Thy. and Practice of Nonlinear Anal. , North-Holland Publishing Co. (1984), 131-136

for PDEs and semi-linear evolution equations

(6) d/dt u + Au = g(u) + f(x,t) , t > 0 , u(x,0) given , where A is semibounded.

Under structural , non-resonance type assumptions on g(.) a theory of
discrete limiting equations was developed and optimal convergence
obtained.

Interestingly,..... it applies to the unstable case as well. In the unstable case it
was proven that there exists an initial condition within approximation
accuracy of the true initial condition and starting here the discrete
pproximation is optimal uniformly in t in IR.

A qualitative theory of errors was also developed. For example, it was
proven that:

The fundamental frequencies of discrete approximate solutions of continuous solutions to (6) converge optimally to the true ones.

To understand this, think of a quasi-periodic behavior in time. This includes
the trivial cases of u(x,t) -> v(x) as t -> infinity and u being periodic in
time. This is a powerful result that deserves to be pursued further..

Non-physical temporal oscillations in discrete Natural Convection problems

This work was driven by the nonlinearity in the continuous and discrete models.

In 1986, thanks to a lot of discussion with Chuck Hall, Tom Porsching and
Hami Melhem, I got interested in a natural convection problem in fluid
mechanics: the double pane window problem. For high Rayleigh
number, their computations showed a nonphysical temporal oscillation
(periodic solution in time with delta-t dependent period) when they
expected convergence to steady state, see

Analytical and Numerical Studies of Natural Convection, with J. Boland,
G.B. Ermentrout, C.A. Hall and H. Melhem, Proc. Int. Conf. on Thy.
and Appls. of Diff. Eqns., 1988.

Jim Boland and I tackled this on several fronts. First the approximation
theory of the equilibrium limit was developed in

with J. Boland. Error Analysis of Finite Element Methods in Steady Natural
Convection Problems, Num. Funct. Anal. and Opt.,11, 1990 449-483

(This also contained a determining modes result for the continuous
problem.) Next, approximation over bounded time intervals considered in

An Analysis of the Finite Element Method for Natural Convection Problems
with J. Boland Num. Meth. For P.D.E's , 2 , 1990 pages 115--126

This still did not explain the non-physical oscillations however. We
simplified to the discrete 1-d Burgers equation. We showed that a
normal (but not best) discretization of the nonlinearity actually induced
non-physical, time periodic oscillations in the discrete approximation.
We also showed that since the boundary data was non-zero,

existence of a equilibrium discrete solution (necessary for a limiting discrete solution) needed one mesh line in the FEM mesh to be within Ra^{-1/5} of either the hot or cold wall.

These results appeared in the conference proceeding above. This is a case
where the secondary results are more interesting (to me at least) than
the generic and more fundamental results. It was also one of the times
where I realized that

than ending with them.

Modeling and Simulation of Turbulence

Turbulence is all about understanding what flow functionals (called
turbulent statistics) are predictable over long time intervals. The
modeling, analysis and numerical analysis of LES thus must be about
understanding what models and numerical methods can really predict
them over long time intervals.

These questions are the focus of my current work. Some papers are below,
for a description please see my LES page. The reports are available on
the mathematics departments technical report server.

This next paper fights the constant in Gronwall inequality. For the NSE with
reasonable regularity assumptions on the true solution, the constants in
typical error estimates typically grow like

exp{ Re^3 t) , think of Re ~ O(1,000,000 ) .

Volker John and I show that discretizations of the Smagorinsky model do
not exhibit this catastrophic error growth:

V. John and W. Layton, Analysis of numerical errors in large eddy
simulation, SIAM JNA,40 995-1020.

Of course, the most interesting case which is most relevant to turbulence and
most intractable is when there is in essence no regularity (so this is for
general weak solutions for which theoretical uniqueness has not even
been proven). Volker John, Carolina Manica and I studied this case in

V John, S Kaya and W Layton, _A two-level variational multiscale method
for convection diffusion equations, _Comp. Meth. Appl. Mech.
Engrg., 195, 4594-4603, 2006.

In cases when Gronwall's inequality cannot even be applied, we show that
long time averages of approximate solutions can converge optimally to
corresponding averages of the (possibly non-unique) weak solution(s)
of the NSE. We also give a mesh dependence result on when computed
ime averaged energy dissipation rates will have the correct physical
scaling with respect to the global flow parameters.

Time averaged velocities of turbulent flows are often observed to have some
slight regularity in accord with Kolmogorov's K41 theory of
homogeneous, isotropic turbulence. Roger Lewandowski and I
considered the time averaged Modeling error in cases where this
observed (and thus expected) time averaged slight regularity holds. We
probe time averaged consistency error convergence like
O( LES model's averaging radius^{1/3} ) which is

uniform in the Reynolds number!

This is in the report:

With R Lewandowski , Residual stress of approximate deconvolution
models of turbulence, Journal of Turbulence, 2  1-21.

and so far it is the only result of its kind that I am aware of.

Predictability and prediction of long time averaged flow statistics.

These topics are studied in papers including

W. Layton, Bounds on energy dissipation rates of large eddies in turbulent
shear flows, Mathematical and computer modeling, 35, 2002, 1445-1451.

W. Layton, C. Manica, M. Neda and L. Rebholz __The joint helicity-energy
cascade for homogeneous, isotropic turbulence generated by
approximate deconvolution models _Submitted: SIASM J Multiscale Modeling and Simulation, 2005.

Vincent J. Ervin, W. Layton and Monika Neda, _Numerical Analysis of a
Higher Order Time Relaxation Model of Fluids, _Accepted: Inter J
Numer Anal & Modeling, 2006.

W. Layton __Bounds on energy and helicity dissipation rates of
approximate deconvolution models of turbulence _Submitted to
SIAM J Mathematical Analysis, 2006.

W. Layton, C. Manica, M. Neda and L. Rebholz, Numerical Analysis and
Computational Testing of a high-order Leray-deconvolution turbulence
model, submitted to Numerical Methods for PDE, 2006.

W. Layton and M. Neda __Truncation of scales by time relaxation _JMAA, 325(2007)788-807.

W. Layton and Monika Neda, _A similarity theory of approximate
deconvolution models of turbulence, _ to appear in JMAA 2007.

Uncertainty

The uncertainly in predictions can only increase with the accumulation of
errors from previous time intervals. Our idea is that

a numerical method for a serious application must include BOTH a method for its approximate solution AND a method to evaluate efficiently and simultaneously uncertainties and sensitivities in the algorithm's and model's predictions.

This was studied by Mihai Anitescu and I in

M. Anitescu and W. Layton, Uncertainties in large eddy simulation and
improved estimates of turbulent flow functionals,
technical report, 2002, to appear in: SIAM J. Scientiific Computing, 2007.

Physical fidelity over long time intervals

One good test of long time physical fidelity is near conservation of
physically important integral invariants. In 2d flows these are energy
and enstrophy and in 3d they are energy and HELICITY, about which
not enough is known. We have considered helicity statistics of LES
models and used helicity conservation as a model and method diagnostic in

W. Layton, C. Manica, M. Neda and L. Rebholz __The joint helicity-energy
cascade for homogeneous, isotropic turbulence generated by
approximate deconvolution models _Submitted: SIASM J Multiscale
Modeling and Simulation, 2005.

W. Layton __Bounds on energy and helicity dissipation rates of
approximate deconvolution models of turbulence _Submitted to
SIAM J Mathematical Analysis, 2006.

W. Layton, A. Labovschii, C. C. Manica, Monika Neda and L. G. Rebholz, _
The stabilized, extrapolated trapezoidal-Galerkin finite element method, _Technical report

Further papers on long time behavior and long time statistics of LES models
and their numerical approximations include:

with G.P. Galdi Approximating the larger eddies in fluid motion II: A
model for space filtered flow Math. Methods and Models in Appl.
Sci. v. 10, no. 3 , 2000 pages 1-8

with T. Ilieiscu Approximating the larger eddies in fluid motion III: The
Boussinesq Model for Turbulent Diffusion, Analele Stiintifice ale
Universitatii ``Al. 1 Cuza'', Series Mathematics, tomul XLIV,245-261.

Approximating the Larger Eddies in Fluid Motion V: A New Scale
Similarity Model , Mathematical and Computer Modeling 31,1-7.

Analysis of a scale-similarity model of the motion of large eddies
in turbulent flows , JMAA 264,546-559.

T. Iliescu, V. John, W. Layton, G. Matthies and L. Tobiska, A numerical
study of a class of LES models, International Journal computational
fluid dynamics,17  75-85.

M. Kaya and W. Layton, On verifiability of models of the motion of large
eddies in turbulent flows, Differential and integrals equations, 15  1395-1407.

W. Layton and R. Lewandowski, Analysis of an eddy viscosity model for
large eddy simulation of turbulent flows, Journal of mathematical fluid
mechanics, 2  374-399.

A. Dunca, v. John, W. Layton and N. Sahin, Numerical analysis of large
eddy simulation,
359-364 in: DNS/LES progress and challenges (editors: C. Liu, L.
Sakeland and T. Beutner) Greyden press, Columbus, 2001.

A. Dunca, V. John, W.J. Layton, "The Commutation Error of the Space
Averaged Navier-Stokes Equations on a Bounded Domain", in G.P.
Galdi, J.G. Heywood, R. Rannacher (Eds.), Contributions to Current
Challenges in Mathematical Fluid Mechanics, Advances in
Mathematical Fluid Mechanics 3, BirkhŠuser Verlag Basel, 53 - 78, 2004

LC. Berselli, G. P. Galdi, T. Iliescu and W. Layton, Existence of weak
solutions for a rational LES model of turbulent flow, Mathematical
methods and models in the applied sciences, 12 (2002), 1131-1152.

V. John, W. Layton and N. Sahin, Derivation and analysis of near wall
models for channel and recirculating flows, Computers and
mathematics with applications,481135-1151.

S. Kaya and W. Layton, Subgrid scale eddy viscosity is a variational
multiscale method, Preprint, 2002.

W. Layton, _A Mathematical Introduction to Large Eddy Simulation , in:
Computational Fluid Dynamics-Multiscale Methods_(H. Deconinck,
editor) , Von Karman Institute for Fluid Dynamics,_Rhode-Saint-
Gen\`ese, Belgium, 2002.

W. Layton, _Advanced models for large eddy simulation, in:
Computational Fluid Dynamics-Multiscale Methods_(H. Deconinck,
editor) , Von Karman Institute for Fluid Dynamics,_Rhode-Saint-Gen\`ese, Belgium, 2002.

W. Layton, _Variational Multiscale Methods annd Subgrid Scale Eddy
Viscosity, in: Computational Fluid Dynamics-Multiscale Methods_(H.
Deconinck, editor) , Von Karman Institute for Fluid Dynamics,_Rhode-Saint-Gen\`ese, Belgium, 2002.

W. Layton and R . Lewandowski,
A simple, accurate and stable scale similarity model for large eddy
simulation: energy balance and existence of weak solutions, Applied
math letters , 16 1205-1209.

With R. Lewandowski, On a well posed turbulence model, Discrete and
Continuous Dynamical Systems Series B, Vol 6, nb 1, pp 111-128, 2006