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Dr ST Millmore (12 hours)

Lectures: (Note that lectures 1 and 2 are given in a single two-hour slot)

  1. Computational Continuum Modelling
    Course overview; an overview equations for solving continuum systems and the mathematical properties of these equations; introduction to conservation laws
  2. Scalar hyperbolic equations
    Mathematical properties of hyperbolic equations; properties of the advection (transport) equation; an introduction to numerical discretisation; a first numerical method for solving the advection equation
  3. Scalar hyperbolic equations - stable solutions
    Techniques required to ensure a numerical method produces stable solutions; brief overview of von Neumann stability analysis; numerical methods for linear hyperbolic equations; physical and mathematical properties of Burgers’ equation
  4. Finite volume methods
    Conservation form for numerical schemes; an overview of the finite volume approach; centred finite volume schemes for non-linear hyperbolic equations; Godunov’s method for non-linear hyperbolic equations
  5. Systems of hyperbolic equations
    Mathematical properties of the compressible Euler equations; definition of the speed of sound; primitive and characteristic variable forms of the Euler equations; the Riemann problem for the Euler equations
  6. Approximate Riemann problem solutions and second order methods
    Mathematics of approximate Riemann problem solvers; total variation diminishing methods for non-linear equations; high-resolution shock-capturing numerical methods; flux and slope limiters
  7. Going beyond one dimension
    Mathematics of the two- and three-dimensional Euler equations; numerical techniques for multi-dimensional systems of equations

Practicals:

  1. The advection equation
    Implementation of simple numerical schemes for the advection equation to investigate how choices of scheme affect the results
  2. The advection equation (again)
    Implementation and investigation of a range of stable schemes for the advection equation, including convergence properties
  3. Finite volume methods for scalar equations
    Implementation and investigation of finite volume schemes applied to Burgers’ equation
  4. FORCE for the Euler equations
    Implementation of a simple first order centred numerical scheme for a non-linear system of equations, solving solutions of shock tube problems
  5. SLIC for the Euler equations
    Implementation of a 2nd-order centred shock capturing scheme for the Euler equations, including the effects of slope limiting
  6. Exact Riemann solver for the Euler equations
    Calculation of the exact solution to the Euler equations for certain physical initial data