I propose to numerically address the question of finite time singularity development in the three-dimensional incompressible Euler equations with smooth initial data. These equations model the flow of incompressible ideal fluids. In particular, I will consider the axisymmetric with swirl case which allows for higher resolution than possible in fully three-dimensional experiments. Several different finite difference techniques will be implemented. I will investigate the performance of the different numerical schemes in order to assess the quality of each method. Also, I will study the effects different numerical boundary conditions have on vorticity amplification as well as the fully three-dimensional problem using particle methods. In conjunction with these experiments, I will perform numerical simulations of the Vlasov-Poisson equations in one dimension. These equations model a collisionless plasma of electrons in a uniform background of ions, and serve as a simpler analogue of the two-dimensional incompressible Euler equations. I will numerically study the behavior of weak solutions to the Vlasov-Poisson and Fokker-Planck-Poisson equations arising from non-smooth electron sheet initial data. An electron sheet describes a concentrated beam of electrons. The equations will be regularized by either smoothing the initial condition or by including collisions modeled by the Fokker-Planck-Poisson equations. I propose to use both a a finite difference method developed by Jack Schaeffer as well as particle methods to examine the solution of the Vlasov-Poisson equations obtained in the limit of vanishing regularization. And finally, this proposed research will prepare me to afterward conduct research related to the Boltzmann and Fokker-Planck equations with non-qmooth initial data which has potential medical appliations. These equations are used in the computation of dosage calculations in radiation therapy.
