Numerical weather prediction requires the use of high performance computing. At the core of this we must solve the Navier-Stokes equations, which are a non-linear hyperbolic system of PDES. Simpler forms can be used to demonstrate some of the behaviour of the Navier-Stokes equations, here the advection equation is studied.

Unlike a conformal finite element method, the constraint of continuity between mesh cells is relaxed for a DG method. This is replaced with a numerical flux between cells.

For a higher order DG method, a higher degree polynomial basis is used to approximate the solution, which leads to higher accuracy.

The PDE we solve is the linear advection equation: $$\partial_t u + \beta\cdot\nabla{u} + au = f$$

Where $\partial_t u$ is the rate of change in the solution $u$ in time.

$\beta$ is the advection vector describing which way the solution moves.

$au$ is a reaction term that depends on the value of the solution and the right hand side, $f$, is an external forcing.

We can solve this as a time dependent problem in one dimension, here we look at two different initial conditions.

By setting $\partial_t u = 0$ we can solve a stationary problem.

Removing this restriction we can solve the problem in time.

Using a suitable grid we can also solve the advection problem on the surface of a sphere. Such problems are essential for NWP.

- Kronbichler, Martin and Kormann, Katharina

A generic interface for parallel cell-based finite element operator application*Computers & Fluids*, 63:135--147, 2012. - Vos, Peter EJ and Sherwin, Spencer J and Kirby, Robert M

From h to p efficiently: Implementing finite and spectral/hp element methods to achieve optimal performance for low-and high-order discretisations*Journal of Computational Physics*, 229(13):5161--5181, 2010. - ...

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