MATH0033 Numerical Methods, 2022-2023,
Theoretical exercise sheet 3
Differential equations
Exercises 2,3 and 5 (marked *) to be submitted via *Crowdmark* in pdf
format (either handwritten and scanned, or typeset using LaTeX).
A subset of these will be marked.
Deadline: 23:59hrs Sunday 18th December
EXERCISE 1 Consider the linear Cauchy problem:{
y′(t) = ?ety(t), t ∈ [0, 1],
y(0) = 2.
(a) Write down the forward Euler method for the approximation of the solution y(t).
(b) Let h = 110 . Compute the approximate solution at time t1 = t0 + h (where t0 = 0) using
the forward Euler method.
(c) Now consider the nonlinear Cauchy problem:{
y′(t) = ?ety2(t), t ∈ [0, 1],
y(0) = 2.
Write down the backward Euler method for the approximation of the solution y(t).
(d) Rewrite the backward Euler method in the form
F (un+1; tn, un, h) = 0,
and write down Newton’s method for solving this nonlinear equation.
EXERCISE 2(*) Consider the following differential equation:{
y′(t) = ?C arctan(ky), t > 0
y(0) = y0 ,
(1)
where C and k are given real positive constants.
(a) Write the backward Euler scheme for solving (1) in the form
un+1 = g(un, un+1, h), (2)
specifying the function g, where h is the timestep and un the approximation of y(tn).
(b) For each timestep one has to solve the nonlinear equation (2). Interpret this equation
as a fixed point problem for the computation of un+1, and determine a condition on h
which guarantees that there exists a unique fixed point to which the fixed point iteration
xk+1 = ?(xk) converges for any initial guess.
(c) Write down the Newton iteration for the solution of the nonlinear equation (2).
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EXERCISE 3(*) For the numerical solution of the Cauchy problem{
y′(t) = f(t, y(t)), t ∈ (0, T ], 0 < T < +∞,
y(0) = y0,
where f is assumed to be uniformly Lipschitz continuous with respect to its second argument,
consider the following method for n ≥ 0, with u0 = y0, and uniform step size h = T/N for
some N ∈ N:
(a) Is the method given by (3) explicit or implicit ?
(b) Write the method in the form
un+1 = un + hΨ(tn, un, un+1;h)
for an appropriate increment function Ψ.
(c) Prove that the method is zero-stable. Reference carefully any theorems you use from
lectures.
(d) Write down the formula for the truncation error of the method, and show that the method
is consistent, with Tn = O(h
2) as h → 0 for y and f sufficiently smooth. Under what
smoothness assumptions on y and f is your analysis valid?
Hint: You may find it helpful to recall the “generalised chain rule”
(T (h), Y (h)).
(e) By quoting an appropriate theorem from lectures, combine the results of parts (c) and
(d) to prove that the method is second order convergent.
EXERCISE 4 Consider the Cauchy problem{
y′(t) = 1? y2, t > 0,
y(1) = (e? 1)/(e+ 1).
(a) By deriving an exact solution and considering the behaviour of ?f/?y on the solution
trajectory (or otherwise), determine an estimate for the critical value of h below which
perturbations due to roundoff errors are controlled when the forward Euler method is
used.
(b) Now do the same for Heun’s method.
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EXERCISE 5(*) Consider a two-by-two system of differential equations{
w′(t) = Aw(t),
w(0) = w0,
w(t) =
[
w1(t)
w2(t)
]
, (4)
where A is a given 2× 2 matrix.
(a) Write down the forward Euler method, the backward Euler method and the Crank-
Nicolson method for the system (4).
(b) Suppose that you are given the diagonalization A = V DV ?1, whereD is a diagonal matrix
containing the eigenvalues d1 and d2 of A, and V is an invertible matrix whose columns are
the corresponding eigenvectors. Show how the problem (4) and the three schemes in part
(a) can be rewritten in terms of the diagonal matrix D and the transformed unknowns
x = V ?1w and xn = V ?1wn.
(c) Using the transformations in (b), determine for which step sizes h > 0 the three schemes
are absolutely stable, under the assumption that the eigenvalues satisfy d1, d2 < 0.
(d) Now consider the particular system of differential equations:
w′1(t) = w2(t), t > 0,
w′2(t) = λw1(t) μw2(t), t > 0,
w1(0) = w1,0,
w2(0) = w2,0,
(5)
where λ and μ are two positive real numbers such that μ2 ? 4λ > 0.
Write the system (5) in the form (4), specifying the matrix A. Use your results in (c) to
determine the stability of the three schemes in this case. What is the stability condition
for the forward Euler method in the special case λ = 6 and μ = 5?
EXERCISE 6 Let A be the (N ? 1)-by-(N ? 1) matrix defined by
where h > 0 and rn > 0 for all n = 1, . . . , N ? 1.
Show that, for any 0 ?= x = (x1, . . . , xN )T ∈ RN?1,
xTAx ≥ x21 + x2N?1 +
N?1∑
n=2
(xn ? xn?1)2,
and explain why this proves that the matrix A is invertible.
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EXERCISE 7 Prove the following discrete maximum principle:
Let an, bn, cn, n = 1, . . . , N ? 1 be positive real numbers such that
bn ≥ an + cn, (6)
and let Un, n = 0, . . . , N , be real numbers such that
anUn1 + bnUn cnUn+1 ≤ 0, n = 0, . . . , N. (7)
Then
Un ≤ max{U0, UN , 0}, n = 0, . . . , N.
Hint: Let Ur = maxn=0,...,N |Un|. If r = 0 or r = N , or Ur ≤ 0, then we are done. Otherwise,
suppose that 1 ≤ r ≤ N1 and that Ur > 0. Use (6) and (7) to show that Ur = Ur?1 = Ur+1.
Then show by repeating this argument that we must have either Ur = U0 or Ur = UN .