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


**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<sub>t</sub> + f<sub>x</sub> = 0

such as f = (1/2)u<sup>2</sup>

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.