Numerical Simulations
Our primary focus is on developing highorder shockfitting methods for problems of multidimensional detonation propagation. The phenomenon of detonation poses significant challenges for numerical methods due to very complex instabilities that are inherent to detonation shocks. Very careful resolution of shock propagation is necessary to capture these instabilities. In addition, we also work on algorithms and numerical simulations of systems of reactiondiffusion equations and dispersive PDE.

Shockfitting simulation of twodimensional detonation instability
We use highorder WENO schemes combined with highorder RungeKutta method to integrate the reactive Euler equations in our shockfitting method. The movie below shows the evolution of twodimensional detonation in an ideal gas in a channel from a steady steady 1D ZND solution subject to a small initial perturbation. Due to inherent instability of the steady solution, we observe the growth of instability and subsequent formation of transverse shock waves propagating between the walls of the channel along the lead shock. The simulation algorithm is described in Taylor, Kasimov, and Stewart, Comb. Theo. Modelling (2009).

Obstaclestabilized detonation in radial outflow
A. Kasimov, S. Korneev, Detonation in supersonic radial outflow, J. Fluid Mech., 760, 313341, 2014 .

Numerical solution of dispersive PDE
With the goal of accurate numerical solution of the multidimensional nonlinear Schroedinger equation arising in the study of BoseEinstein condensate, we develop algorithms for obtaining the steadystate solutions, calculating linear stability properties, and for full timeintegration of the PDE. The movie below shows formation of vortices in a numerical simulation of the complex GrossPitaevskii equation for excitonpolariton condensate starting from a steadystate radially symmetric solution. This result is part of an MS thesis by Jesus Sierra, KAUST, 2011. For details, see J. Sierra, A. Kasimov, P. Markowich, R.M. Weishäupl, On the GrossPitaevskii equation with pumping and decay: stationary states and their stability, submitted, 2013, arXiv preprint.