The Magnetohydrodynamics (MHD) model is an important model to describe the behavior of the plasma phenomenon. The MHD equations are challenging because one needs to maintain the divergence free condition, 2207 · B = 0. Many numerical methods have been developed to enforce this condition. In this thesis we propose a hybrid MHD solver via the constrained transport method which is based on kernel based approximations to correct the solution of magnetic field equation so that it satisfies the divergence free condition. The main focus of this work is on the constrained transport methodology that intends to eliminate the need of artificial diffusion limiters (introduced in the previous work [29]) for the magnetic vector potential equation for ideal MHD and extend the proposed scheme to resistive MHD, which includes a term that takes the form of diffusion.In the first part of the thesis, we focus on ideal MHD model and we further our work on mesh aligned constrained transport [29] by developing a new kernel based approach for the vector potential in 2D and 3D. The approach for solving the vector potential is based on the method of lines transpose and is A-stable, eliminating the need for diffusion limiters needed in our previous work in 3D. The work presented here is an improvement over the previous method in the context of problems with strong shocks due to the fact that we could eliminate the diffusion limiter that was needed in our previous version of constrained transport. The method is robust and has been tested on the 2D and 3D cloud shock, blast wave and field loop problems.In the second part of the thesis, we turn our focus to resistive MHD model, which allows magnetic diffusion terms in the magnetic field induction equation. We derive magnetic potential equation with diffusion terms and develop our kernel based method for the second order derivative terms appearing in the magnetic potential equation. We update the magnetic field solutions using the constrained transport framework based on our novel kernel based scheme. Our novel scheme is robust and unconditionally stable. Numerical examples in 2D are provided to verify the order of accuracy and to test the performance of the proposed scheme.
Read
