A-stable implicit rapid scheme and software solution for electromagnetic wave propagation

"A robust and rapid scheme to solve electromagnetics (EM) is an important requirement in the scientific computing environment in which there are several useful methods used to solve tasks in EM. Our research study is motivated by this need and is targeted to develop a fast A-stable implicit numerical scheme and scalable software solution for EM wave propagation. Our scheme is based on the Method Of Lines Transpose (MOLT) approach which discretizes time first and then solves boundary value problems. By applying the free-space Green's function, the solution is derived by decomposing particular and homogeneous solutions. The compact Simpson's quadrature based, O(N) fast convolution, a recursive algorithm, is used to solve the particular solution for N number of grid points. The homogeneous solution is obtained using a particular solution at the boundary points and the applied boundary conditions. The multi-dimensional scheme is developed using the ADI splitting approach and an arbitrary order accuracy in time is achieved by switching the time derivation to a spatial derivation using the Lax-Wendroff approach.The focus of the work in this thesis has been to overcome the limitations in Neumann and outflow boundary conditions to get high-order accuracy by using special treatments that deal with a choice of the interpolation, finite difference stencil, and the initial conditions. In addition, we have extended these ideas to construct perfectly electrically conducting boundary conditions in 2D for the MOLT.In addition to introducing higher-order boundary conditions, an embedded boundary method is employed to deal with complex geometries. As the method is A-stable, it does not suffer from small-time step limitations that are found in explicit finite difference time domain methods when using either embedded boundary or cut cell methods to capture geometry. Further, we are developing an open source code MOLTN (Method Of Lines Transpose, Nth order) which is intended to be a hardware-independent, scalable software tool, using multi-node MPI, multi-core OpenMP, and GPU CUDA implementation. As a test case of the method, we implement and study the A6 magnetron with our embedded boundary method using point sources inside of the domain. The eventual goal is to combine this method with a novel particle method for the simulations of plasma. The particle method would treat particles as point particles that generate fields that are tracked on the mesh. No density or current will be mapped to the mesh. The consistency and performance of the scheme are evaluated for EM wave propagation and scattering using different shaped objects including curved boundaries and the introduction of true point sources that demonstrate how we handle particles. Stable solutions result for a wide range of mesh sizes and potential to leverage novel computing architectures, such as GPU, have been demonstrated."--Pages ii-iii.