A one-sided view

P. L. Roe, Royal Aircraft Establishment Bedford, England.
R. LeVeque, University of California, Los Angeles
B. van Leer, Delft University of Technology, Delft, The Netherlands

Abstract: A critical study is made of the tricks of the upwinding trade. Five lines, it would seem, can describe any scheme of the class that the authors surveyed.

1 Introduction

I said to my darling: “I may go
to meet at U. C. San Diego
the full upwind clan
invited by Stan,
for a boost of the mutual ego.

By often escaping detection
the sun caused no marked defection.
To swim the Pacific
ain’t all that terrific
if you must get dry by convection.

2 Basic numerical techniques

2.1 Conservative difference schemes

Conventional difference equations
give shocks that induce oscillations.
By adding some logic
we get monotonic
numerical representations.

A characteristic equation
when differentiated defies conservation,
which so badly we need.
But at last we were freed
by the grace of Roe’s linearization.

To the podium many will hustle
to enter their claim in the tussle.
For the issue is fame
and it seems such a shame
we can’t all take credit for MUSCL.

It’s really not easy outsmartin’
the TVD schemes of A. Harten.
The name of the game
is they’re all the same
so you’d better give up before startin’.

The sight of the slides of the Colella
turns all his competitors yella.
Where others may fail
he’s got the detail,
cause the grids are paid for by Ed Teller.

In spite of the entropy glitch
those contracts are making Stan’rich.
He claims Engquist-Osher
is totally kosher
and runs like a son-of-a-bitch.

2.2 The state of the art in related areas

The conference could not have been better
except for the following matter:
that out of those listed
some speakers insisted
that they’d give a talk on the weather.

“We all know the problem of Riemann,
the basis of all our schemin’,”
This assertion will get
uninitiates upset
and the meteorologists steamin’.

So, medium-term weather prediction
turns out to be merely a fiction.
It’s just anyone’s guess,
if you ask how to dress
it will offer no useful restriction.

2.3 Auxilliary techniques

To sort out a boundary procedure
just talk to this elegant Swede here.
Your results will look nice
in the sense of Heinz Kreiss
and your program may even be speedier.

When exhausted by over-refinin’
don’t throw up your hands and start whinin’.
No sense in postponin’,
just go for rezonin’
(for details please talk to Mac Hyman).

Approximate factorizations
applied to the Euler equations,
are not all that fast,
in fact, they’re surpassed
by classical point relaxations.

Now listen and please do not mock
the spectral technique I’ll unlock.
A hundred harmonics
make quite good transonics,
though fifty must die for the shock.

3 Theoretical results

Full proofs are exceedingly rare
except in the simple case
the f is convex
in u_t + f_x = 0
such as f = (1/2)u²

At dinner on day number three
the cook said to my friend and me:
“You’ve had burgers enough,
try more variable stuff:
may I offer you eggs and Roe tea?”

Note: The full limericks can be found in Innovative methods for numerical solution of partial differential equations, Hafez and Chattot (eds.), World Scientific, 2002.