A tsunami is about to hit Sydney after an earthquake thousands of kilometres south-east of the city. Hundreds of thousands of people are in danger, and the evacuation plan for the city is activated.

Emergency services personnel may soon be able to predict which suburbs would be most at risk, thanks to mathematical modeling at the Australian National University and Geoscience Australia.

ANU mathematician Steve Roberts and colleagues are developing a model to predict the run-up height of tsunamis and how far inland the waves would strike.

Existing models calculate the wave height from the energy released in submarine earthquakes. They are imprecise because they are based on simple linear equations and do not take into account details of the water depth and topography along the coast, factors that determine the wave height.

Roberts' model, ANUGA, solves the shallow water wave equations, more complex non-linear differential equations that use parameters including water depth and wave momentum to calculate the wave height at particular points along the coast.

It uses the finite volume method. The area of interest is broken into a mesh of triangles, usually extending up to 200 kilometres out from the coast. As the virtual wave advances, the model solves the wave equations for each triangle. The output data from each triangle forms the input data for the next ones as the wave approaches the shoreline.

The model would have a resolution of tens of metres as long as good water depth and seabed geometry data were available, Roberts says. For most areas, those figures are not accessible.

Roberts' interest in computational mathematics dates back to earlier work on supersonic gas flow. "Amazingly, those equations were almost the same as the shallow water wave equations," he says.

His focus has remained on applied mathematics. "I love seeing mathematical equations being able to predict the physical world."

He was drawn to the ANU because it has enough computational scientists for modeling and theoretical research to reach critical mass.