# Faster computation times for numerical solutions

Shock waves are difficult to simulate on a computer. Due to the large, rapid changes, often oscillations occur in the mathematical solutions. Anna Motsartova, researcher at the Centrum Wiskunde & Informatica (CWI) in Amsterdam, developed a method to find sharp boundaries for time steps in the calculations that can prevent the occurrence of such fluctuations. On 26 November Motsartova defends her PhD thesis ‘Boundedness and monotonicity properties in numerical initial value problems’ at the Radboud Universiteit Nijmegen. Her research might speed up the computation time for important problems in physics, chemistry and many other fields.

Numerical simulations with differential equations are often plagued by the occurrence of unwanted numerical oscillations, in particular when the exact solutions possess steep gradients or shocks. This can lead to non-physical numerical approximations, for example, with negative densities or concentrations, which leads to unreliable results.

In Motsartova’s PhD thesis sharp boundaries are given for the allowable discrete time steps that can be taken in a simulation without the occurrence of undesirable oscillations. With these results, it is possible to determine in advance which numerical time-step methods are suitable for such problems. This can thus give a rapid acceleration in the required computation time. This research was funded by NWO.

Supervisors: Prof. W.H. Hundsdorfer (CWI and RUN) and Prof. M.N. Spijker (UL)

*Picture: Shock waves can also occur in thunderstorms - image: Shutterstock*