Search for tag: "differential equations"

Eduard Feireisl - Friday 12 November

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…

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Partial Differential Equations Lecture 10 - Results

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.

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Partial Differential Equations Lecture 10 - The Solver

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

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Partial Differential Equations Lecture 10 - 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.

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Partial Differential Equations Lecture 9 - Results

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

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Partial Differential Equations Lecture 9 - MacCormack method in 2D

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

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Partial Differential Equations Lecture 9 - Two and three dimensional solvers

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.

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Partial Differential Equations, Lecture 8c: MUSCL Schemes

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

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Partial Differential Equations, Lecture 8b: TVD MacCormack

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

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Partial Differential Equations, Lecture 8a: Monotone Methods for Conservation Laws

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

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Partial Differential Equations, Lecture 7a: Numerical solution of hyperbolic PDEs.

A lecture introducing numerical methods for hyperbolic PDEs

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Lecture 6 Part 3: Driven, damped oscillators

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.

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Lecture 6 Part 2: Damped oscillators

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…

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Lecture 6 Part 1: Applications of Differential Equations - undamped oscillators

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.

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Lecture 5 Part 3: Oscillators

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.

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Lecture 5 Part 2: Phase portraits

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

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