Numerical Methods for Hyperbolic
Conservation Laws
Random Choice Method
Dr. Aamer Haque
http://math.iit.edu/~ahaque6
[email protected]
Illinois Institute of Technology
June 23, 2009
Overview of Random Choice Method

Earliest ideas by Glimm, 1965
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Developed by Chorin, 1976
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Samples solutions to Riemann Problem


Requires accurate Riemann solver

Requires “good” random number generator
Not conservative!
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Conservative asymptotically
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More accurate than Godunov's method
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Does not appear to work beyond 1D
Define Cell Averages

L
R
L
R
xi−1
xi
xi1
Assume piecewise constant data for Riemann solver
(i.e. Cell averages)
x
1
 =
U
∫
x x
n
i
i1 /2
i−1 /2
U  x , t n  dx
Solve Riemann Problem
t n1
tn
xi−1
xi
xi1
U  x , t n1 = RP  x , t n1 ; i−1,i , t n  if
x∈[ xi−1 , x i ]
U  x ,t n1 =RP  x , t n1 ; i ,i1, t n  if
x∈[ x i , x i1 ]
Update Through Random Sampling

t n1
tn
xi−1
xi
xi1
 n1
U
=U  , t n1 
i
x i−1/2 ≤≤x i1/2
1 x
 t≤
2 max S i
Sod-Like Test Problem
Image from Toro