MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 7:
Higher-order methods and challenges
In the limit of the method we know that the error behaves as O h . Writing the error
Eaf ter haf ter and Ebef ore hbef ore with
MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 6:
Roe's matrix and TVD methods
Exercise 6.1 (a)
By the denition z = 1/2 u, we have
1/2
=
m1/2
(1)
,
where z = [, ]T . We solve for and m to get
2
u (z ) =
(2)
.
Subst
MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 5:
Systems and Approximate Riemann Solvers
Exercise 5.1 (a)
to get
To write the scheme in explicit form we simply insert the ux in the conservation law and reduce
n
n
MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 4:
Godunov's method
The method rst proposed by Godunov can be outlined in 3 steps. (1) reconstruct a piecewise
polynomial function un (x, tn+1 ) dened for all x, from
MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 3:
Conservative methods
See gures generated by the code attached at the end of the solution manual. (b) The numerical
solution converges to the function u (x, t) = u (
MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 2:
Finite Dierence Schemes
Exercise 2.1
Consistency A method is consistent if its local truncation error Tk satises
where
Tk (x, t) = O (k p ) + O (hq )
p, q > 0 .
(1)
MATH-459 Numerical Methods for Conservation Laws by Prof. Jan S. Hesthaven
Solution set 1:
Scalar Conservation Laws
Exercise 1.1
The integral form of the scalar conservation law ut + f (u)x = 0 is given in Eq. 1 below.
x2
x1
x2
u (x, t2 ) dx =
x1
t2
t2
f