Reliable and accurate computations for fluid flows

Although fluid flows are commonplace, we still have not found a way to simulate them accurately. One example is cyclone forecasting. If we can determine the accurate location of a cyclone’s centre at every instant, we can forecast the path of the cyclone accurately. The Navier-Stokes (NS) equations model fluid flows accurately, based on various demonstrations performed during the past 150 years. However, the analytical solutions of the NS equations are not yet available (it is one of the seven millennium mathematical problems). We currently calculate the numerical solutions of the NS equations using computers. A limitation of using computers is that we can only perform finite operations. However, there are infinite points in a domain of interest. Therefore, there are two main parts involved in computing accurate numerical solutions of a mathematical model:
1. Selection of the finite points in a domain that provide sufficient information about the characteristics of flows in that domain.
2. Computing accurate velocity fields of the flows at the selected finite points.
The selection of the finite points is normally completed by specialised software. Since we do not know the details of a flow when selecting the points, the selection only reflects the characteristics of the geometry of the domain. The characteristics of velocity fields are hidden in the fields themselves. Therefore, if we want to include the characteristics of a velocity field in selecting the points, we must select the points based on the information from both the geometry of the domain and the velocity field.

In this talk, I will briefly introduce methods I proposed about selecting the finite points and where these methods can be applied to solve practical problems.