6/28 Week
3: Systems I
Ahat, P-System, Elastic
String
With the tools in hand to study systems
of partial differential equations, we familiarized ourselves with
many classical systems. The process consisted of writing down the
system, finding the appropriate flux matrix and taking the eigensystem
of said matrix. As this was computationally intensive, we used
Mathematica and Maple to speed calculations. What we did for each
system below is contained in a mathematica notebook.
The “Ahat” system consists of
a decoupled scalar equation and a system of two other equations
which depend on the solution
to the first. It appears in one form or another in the works of Professor
Young and in several papers we’ve read. It is mainly of theoretical
interest.
1) Ahat system: a, b, c, d, and F are functions
of W
| Ut + (aU+bV)x = 0 |
| Vt + (cU+dV)x = 0 |
| Wt + (F)x = 0 |
The “P” System
consists of two coupled equations. Under certain conditions, many
other systems
of conservation laws reduce
to the P-system. It models Isentropic gas dynamics and the one dimensional
elastic string.
2)P System: U is velocity. F (pressure) is
a function of V (reciprocal of density)
| Ut + (F)x = 0 |
| Vt - (U)x = 0 |
The “Elastic String” system consists
of six coupled equations. It models the motion of a string in three
dimensional
space.
3) Elastic
String: The Fi are functions of U1,
U2, U3.
| U1t + (V1)x = 0 |
| U2t + (V2)x = 0 |
| U3t + (V3)x = 0 |
| V1t + (F1)x = 0 |
| V2t + (F2)x = 0 |
| V3t + (F3)x = 0 |
|
