47  Final Exam - ideas

Published

May 3, 2026

Things we’ve done:

47.1 General numerical analysis

  • Taylor expansion,
  • How to see algebraic/exponential convergence? Consider the following plot, is it algebraic or exponential decay?, …
  • If you increase the constant prefactor in y = C e^{-n}, how does this affect the log-y plot?
  • Sparse arrays
  • Why do we stop the iteration when the residual is small rather than the error?

47.2 Chapter 1: IVP

  • T/F what can happen in the exact solution? The solution may exist for all time. The solution may exist on [0,T) but blow up as t\to T. There may be multiple solutions. There may be infinitely many solutions. There may be no solution, …
  • What is the solution to the differential equation u'(t) = \mu u(t) with u(0)=u_0? Write down the first 3 iterations of the midpoint method,
  • Picard theorem: show that equations have a unique solution on some time interval,
  • Well-posedness,
  • Euler: O(h) convergence
  • General one-step: local truncation error (LTE) and consistency + stability = convergence,
  • Why are the higher order Taylor methods not used in practice?
  • Runge-Kutta: e.g. f(t+\alpha,u+\beta) and try to make LTE as small as possible (this leads to midpoint method), order of accuracy, Butcher tableau (write general s-stage methods),
  • Show that, if \sum_{i=1}^s b_i = 1, then we have consistency,
  • Adaptive RK methods: basic idea, embedded RK methods, exercise 6.2,
  • Linear multistep methods: AB methods, LTE, show e.g. AB2 is order-2 accurate, AM methods (implicit), show that the sum of the \alpha’s is 1 for any consistent method,
  • Zero stability: T/F if a linear multistep method is consistent, then it is convergent? Show that AB2 is zero stable. Write down the characteristic polynomial, does it satisfy the root condition?
  • Absolute stability
  • Systems of IVP e.g. higher order equations (e.g. shooting method), method of lines, …
  • Not seen: nonlinear IVP, …

47.3 Chapter 2: Iterative Methods

  • Why are iterative methods useful?
  • Recap linear algebra: e.g. …, spectral radius, …
  • Splitting methods: Jacobi method, Gauss-Siedel, SOR (SOR not on the exam)
  • General convergence result for methods of the form x^{(k+1)} = Tx^{(k)} + c,
  • Here’s a matrix and its eigenvalues. Do you expect Jacobi or Gauss-Siedel to converge? If so, what is the rate of convergence?
  • Steepest descent,
  • Conjugate gradient,

47.4 Chapter 3: BVP

  • Forwards/backwards differences,
  • Order of accuracy of finite differences
  • Differentiation matrices,
  • Discrete Laplacian,
  • Consistency + stability = convergence
  • Stability for SDD matrices and stability of ADR in L^\infty for P_e := \frac{|p| h}{2} < 1 and q > 0 (the stability constant we founds was 1/q),
  • M matrices and stability when p=q=0,
  • Discrete Fourier transform and stability in L^2 norm (with periodic BCs)
  • Galerkin and finite elements & error estimates,
  • Not seen: other boundary conditions e.g. Neumann or Robin,

47.5 Chapter 4: PDE

  • Heat equation: consistency and (von Neumann) stability conditions for finite difference approximations (FTCS) and (BTCS)
  • Transport & wave equations: finite difference approximations, CFL conditions, and von Neumann stability

47.6 Shooting method

Suppose that we want to solve the following boundary value problem

\begin{align} &u''(x) = f\big( x, u(x), u'(x) \big) \quad \text{on } (0,1)\nonumber\\ &u(0) = A, \quad u(1) = B \tag{BVP} \end{align}

The shooting method tries to use our knowledge of initial value problems to obtain the solution to (BVP). Consider the family of initial value problems

\begin{align} &v''(x) = f\big(x, u(x), u'(x) \big) \quad \text{on } (0,1) \nonumber\\ &v(0) = s, \quad v'(0) = t \tag{IVP$_{s,t}$} \end{align}

Let the solution of this problem be denoted v_{s,t}(x).

  • (i) Explain why we are interested in finding the roots of the equation F(t) := v_{A,t}(1) - B,

Now take the linear boundary value problem (where f is a linear function of u and u':

\begin{align} -&u''(x) + p(x) u'(x) + q(x) u(x) = g(x) \quad \text{on } (0,1)\nonumber\\ &u(0) = A, \quad u(1) = B \tag{linear BVP} \end{align}

  • Write down the corresponding initial value problem and suppose v_{s,t;g}(x) is the unique solution to this problem (we can use Chapter 1 to approximate v_{s,t;g})

  • (ii) Explain why one can write the solution to (BVP) as a linear combination of v_{A,0;g} and v_{0,1;0}.

Therefore, we are left with studying the initial value problems (in this case we have a second order equation)

  • (iii) Write your IVP as a system of first order equations. Hint: you may want to write w = v'.

  • (iv) Explain how one may implement the midpoint method from Chapter 1 to this system of initial value problems.

47.7 FEM: boundary conditions

Suppose we apply Galerkin approximation to the equation

\begin{align} -&(\alpha u')' + \beta u = f \qquad \text{on }(a,b)\nonumber\\ &u(a) = A, \quad u(b) = B \end{align}

In lectures, we wrote down a method fem(α, β, f) for approximating the solution to the homogeneous version (with A = B = 0) of this problem.

  • Explain how one could adapt the code to implement the non-homogeneous boundary conditions,
  • Alternatively, one can use fem(α, β, F) (with a different function F) and modify the solution in some way to approximate the solution to the non-homogeneous problem. Explain how you can do this. Hint: what happens if you add a linear function to the solution of the homogeneous problem?

We are also interested in solving the same problem but with Neumann boundary conditions:

\begin{align} -&(\alpha u')' + \beta u = f \qquad \text{on }(a,b)\nonumber\\ &\alpha(a) u'(a) = g_a, \quad \alpha(b) u'(b) = g_b \end{align}

  • Explain how you would implement this type of boundary condition in your FEM code.

47.8 Method of lines

Another general approach to solve (time dependent) PDE is semi-discretisation: we again take x_i := a + \frac{b-a}{m} i and h := \frac{b-a}{m} and now approximate \{u(x_i, t)\}_{i=0}^m with the (time-dependent) vector \bm u(t). (This is known as the semi-discretisation step because we have discretised in space but not time).

Then, if one is interested in solving the heat equation (u_t = \alpha^2 u_{xx}), we can replace this with

\begin{align} \bm u'(t) &= \alpha^2 D\bm u(t) \end{align}

where D is a differentiation matrix approximating the second derivative in space. This is a linear, constant coefficient system of ODEs that we can solve directly using techniques from Chapter 1 (also see A2).

  • Approximate the heat equation \square u := u_{tt} - c^2 u_{xx} = 0 with a linear system of ODEs using the method of lines. Hint: define u_t = y
  • What is the resulting equation if you instead consider the choice u_t = z_x and z_t = c^2 u_{x} (this leads to Maxwell’s equations)