Our primary focus is on developing high-order shock-fitting methods for problems of multi-dimensional 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 reaction-diffusion equations and dispersive PDE.
Shock-fitting simulation of two-dimensional detonation instability
We use high-order WENO schemes combined with high-order Runge-Kutta method to integrate the reactive Euler equations in our shock-fitting method. The movie below shows the evolution of two-dimensional 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).
Obstacle-stabilized detonation in radial outflow
A. Kasimov, S. Korneev, Detonation in supersonic radial outflow, J. Fluid Mech., 760, 313-341, 2014 .
Numerical solution of dispersive PDE
With the goal of accurate numerical solution of the multi-dimensional nonlinear Schroedinger equation arising in the study of Bose-Einstein condensate, we develop algorithms for obtaining the steady-state solutions, calculating linear stability properties, and for full time-integration of the PDE. The movie below shows formation of vortices in a numerical simulation of the complex Gross-Pitaevskii equation for exciton-polariton condensate starting from a steady-state 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 Gross-Pitaevskii equation with pumping and decay: stationary states and their stability, submitted, 2013, arXiv preprint.