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Numerical Solutions of Hyperbolic Partial Differential Equations [M16]

The objective of this course is to introduce students to numerical methods for partial differential equations, especially those of physical importance. It will be shown that many obvious methods are unsuccessful, and that the majority of the successful methods are guided by the physics and mathematics of the problem at hand. Simple model problems representing several major classifications are studied for the sake of the general messages that they convey. Relevant numerical techniques are discussed.

In detail, the course will cover the following topics:

Course structure:

  1. Classification of partial differential equations &physical behaviour and approximate levels, Scalar hyperbolic equations and the Riemann problem.
  2. Discrete representation of continuous functions and operators.  Finite difference methods for parabolic problems.
  3. Difference schemes for scalar hyperbolic equations (first-order upwind,Lax-Wendroff, Lax-Friedrichs and Warming & Beam) and the modified equation.
  4. The integral form of the equations and conservative, finite volume methods. Use of the Riemann problem for solving scalar linear and nonlinear conservation laws.
  5. Monotonicity, total variation and Godunov's theorem. High-resolution methods, TVD schemes and second-order Riemann problem based methods for scalar conservation laws.
  6. Extension to higher space dimensions and dealing with inhomogeneous laws.

The second part of the course considers in detail nonlinear, multi-dimensional, inhomogeneous systems of partial differential equations and their numerical solution using contemporary numerical schemes. The aim is to cover systems which are employed by major science and technology disciplines (general fluid dynamics, combustion, earth system science, semiconductors, aerospace etc), so that students are well-equipped for research on the topic of their choice.

  1. Nonlinear systems of partial differential equations; the Navier-Stokes equations and reduced systems.
  2. Exact and approximate solutions of the Riemann problem for the compressible unsteady Euler equations and their use in Godunov-type methods.
  3. Higher-order methods for the Euler equations. MUSCL reconstructions.
  4. Other nonlinear hyperbolic systems and systems of mixed character; the pressure correction method.
  5. Approaches for higher dimensions and non-Cartesian geometries.
  6. Multi-component and inhomogeneous systems of equations.