08 Nov- 12 Nov 2021ICMS hosted the Convex Integration and Nonlinear Partial Differential Equations workshopOn strong continuity of weak solutions to the compressible Euler equations Eduard Feireisl…

In the last part of this lecture we look at the results we obtain to the test problem and look forward to solving real fluid dynamics problems using commercial solvers in CFD5.

We now introduce a family of test problems and look at the Matlab code needed to solver the shallow water equations.

In this lecture we extend our solver to apply to a system of equations in one spatial dimension. In this case we are solving the St Vennant, or Shallow Water Equations.

Here we look in detail at the results from the test problem and discuss dispersive and dissipative errors.

This part of the lecture develops a 2D solver using the method of Fractional Steps and the MacCormack scheme and applies it to a test problem

In this lecture we look at operator splitting and the method of fractional steps which can be used to solve 2 and 3D problems with 1D schemes.

We look briefly at how MUSCL schemes use weighted interpolation of fluxes as an alternative to the TVD schemes.

Here we implement the MacCormack scheme with a TVD correction in Matlab and apply it to a test problem

In this lecture we look at schemes which are not prone to producing the spurious "Gibbs" oscillations we saw in the last lecture. We start b talking about TVD schemes

A lecture introducing numerical methods for hyperbolic PDEs

Here we add a driving force to the oscillator and consider how the standard form of the equation is modified. We look at the amplification factor and see what happens under resonant conditions.

We extend the ideas from the first part of the lecture to oscillators which include damping effects. Writing the governing equation in a standard form and looking at the meaning of the damping…

Here we look at how the the same (or a very similar) differential equation applies to several different engineering problems. We consider cantilevers, pendulums and a floating buoy.

Here we look at some more general oscillators and their phase portraits and consider how plotting vector plots can be useful for understanding autonomous systems.

How do understand the behaviour of the oscillator, why is plotting velocity against displacement useful? What are stationary points and limit cycles?