NUMERICAL ANALYSIS PRELIMINARY EXAM
August 2006
1. Consider the ODE boundary value problem
d2 u
= f (x), 0 < x < 1,
dx2
du
u(0) + (0) = 0, u(1) = 0.
dx
−
Recall that the variational, or weak, formulation for this problem is to find u ∈ V for which
a(u, v) = L(v),
for all v ∈ V.
a. Specify the appropriate vector space V and derive the functionals a(u, v) and L(v) in
the variational form.
b. Verify that if u is a weak solution, and u ∈ C 2 [0, 1], then u solves the boundary value
problem.
c. Explain how to implement the Ritz-Galerkin method for this problem.
2. Consider the parameterized family of time-marching methods
un+1
− unj =
j
∆t
n+1
[α(unj−1 − 2unj + unj+1 ) + (1 − α)(un+1
+ un+1
j−1 − 2uj
j+1 )]
h2
for the PDE ut = uxx , 0 < x < 1, t > 0, with homogeneous Dirichlet boundary conditions,
where unj ≈ u(jh, n∆t), and α is real-valued. Determine the ranges of parameters α for which
the time-marching method is unconditionally stable and for which it is always unstable.
For remaining values of α (where the method is conditionally stable) find conditions on
h, ∆t, and α which guarantee stability.
3. Consider the difference method
n
uni−1,j + uni+1,j + uni,j−1 + uni,j+1 − 4uni,j
un+1
i,j − ui,j
=
∆t
h2
for the 2-d heat equation,
ut = uxx + uyy .
a. Perform a Von Neumann stability analysis to determine the range of values of ∆t and
h for which this method is stable.
b. Provide a general definition of local truncation error for evolution equations, and then
compute the local truncation error for this method.
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4. Derive the Gauss-Legendre quadrature formula,
Z
1
−1
f (x) dx ≈
n
X
f (xk ) wk ,
k=1
which is exact for all polynomials of degree ≤ 2n − 1, i.e., explain how to obtain the xk ’s and
wk ’s, and prove that your formula does indeed have the desired exactness. You may use the
Hermite polynomial interpolation formula: If f ∈ C 2n [a, b] and {xk }nk=1 are distinct points
in [a, b], then
f (x) =
n
X
k=1
f (xk )hk (x) +
n
X
f 0 (xk )h̄k (x) +
k=1
n
f (2n) (ξ) Y
[ (x − xk )]2 ,
(2n − 1)! k=1
where for k = 1, . . . , n,
hk (x) = [1 − 2(x − xk )`0k (xk )][`k (x)]2 ,
h̄k (x) = (x − xk )[`k (x)]2 ,
n
Y
(x − xi )
.
`k (x) =
i=1,i6=k (xk − xi )
5. The leapfrog method for the 1-d linear avection equation
ut + aux = 0
is
un − uni−1
− uin−1
un+1
i
+ a i+1
= 0.
2∆t
2h
Analyze the stability and accuracy of this method.
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