Reproducing the result of a scientific experiment is necessary to establish trust, and reproducibility has long been a key part of the scientific method.
Traditionally, an experiment could be repeated by following the method documented by the original scientists: setting up apparatus, taking measurements, and so on. If the method was sufficiently well documented then it was, perhaps, likely that the original results could be reproduced.
These ‘wet lab’ experiments continue today, but many experiments are now performed entirely on computers.
Such computational experiments involve no physical apparatus, but merely the processing of input data files through some scientific software before writing more data files for later analysis and plotting.
If we know which way the wind is blowing then we can predict a lot about the weather.
We can easily observe the wind moving clouds across the sky, but the wind also moves air pollution and greenhouse gases.
This process is called transport or advection.
Accurately simulating the advection process is important for forecasting the weather and predicting climate change.
Over the last year of my PhD I have been working to create a finite volume transport scheme, called ‘cubicFit’, that is second-order convergent on arbitrary meshes.
Recently, I have tried modifying the cubicFit transport scheme to obtain high-order convergence: that is, convergence greater than second-order.
Here I'll introduce a one-dimensional version of the cubicFit scheme and explain how it can be modified to obtain high-order convergence.
There are many ways to represent terrain in atmospheric models. Terrain following coordinates are in widespread operational use but can suffer from numerical errors near steep slopes. Advection errors, and errors calculating pressure gradients both reduce model accuracy. The cut cell method is often put forward as an alternative which can reduce pressure gradient errors, but the technique may still suffer from advection errors where flow is misaligned with the mesh. Without special treatment, the cut cell method can create very small cells which constrain the timestep when explicit methods are used.
This demonstration shows quadratic and cubic polynomials fitted through a set of points using a least squares approach. This technique generalises to higher dimensions and can be used as an interpolation function for solving the flux form advection equation on arbitrary meshes.