Aziz Takhirov

Aziz Takhirov, Driss Yakoubi

We develop a novel iterative approach for solving the incompressible magnetohydrodynamics problem. The main idea is to split the velocity-momentum and magnetic induction equations with respect to the diffusive terms. As a result, we get a smaller system that is iteration-level-dependent, along with two Stokes systems that need to be assembled only once. We also extended the scheme to the Els{ä}sser variables reformulation of the equations. For both schemes, we established boundedness and convergence. Several numerical experiments are presented to show the effectiveness of the schemes.

Aziz Takhirov, Driss Yakoubi

We develop a novel and efficient iterative scheme for solving incompressible steady Navier-Stokes equations. The method is an adaptation of the Incremental Viscosity Splitting approximation for unsteady flows to steady equations. At each nonlinear iteration, the scheme requires solving an elliptic PDE for the velocity variable and a system with an SPD matrix for the pressure variable, which remains the same across all nonlinear iterations. The method can also be interpreted as an algebraic splitting approach. We prove boundedness and geometric convergence. Numerical tests illustrate the efficiency of the proposed algorithm.

Aziz Takhirov, Jiajia Waters

We propose novel ensemble calculation methods for Navier-Stokes equations subject to various initial conditions, forcing terms and viscosity coefficients. We establish the stability of the schemes under a CFL condition involving velocity fluctuations. Similar to related works, the schemes require solution of a single system with multiple right-hand sides. Moreover, we extend the ensemble calculation method to problems with open boundary conditions, with provable energy stability.