Relativistic neutron stars in general relativity and fourth order gravity - Masetlwa, Nkosinathi (Department of Mathematics and Applied Mathematics, 2021)
This thesis investigates numerical instabilities arising from stiffness in the models of nonrotating, spherically symmetric single neutron star systems. The work deals with two distinct problems, each of which involves a stiff system of differential equations. In each case, we deal with stiffness by employing an IMEX Runge-Kutta scheme as opposed to the more computationally intensive fully implicit schemes or other adaptive Runge Kutta methods that may be impractical for partial differential equations. The first problem is focused on the mass-radius relation of a neutron star under a quadratic f(R) = R+αR2 theory for various realistic equations of state. This results in a coupled system of ODEs with stiff source terms which we discretize using an IMEX scheme. The observed maximum masses for different values of α, were consistent with the current neutron star maximum mass limit for some equations of state in both GR and beyond. In the second problem, we compute the frequencies of radial oscillations of neutron stars in the context of general relativity. This is achieved by linearly perturbing the ADM equations coupled to a matter source term. We discretize the resulting coupled system of PDEs with a third order WENO scheme in space and an IMEX scheme in time. We obtained 18 frequencies from the Fast Fourier Transform (FFT) of the evolved perturbation equations, which were consistent with the frequencies of the neutron star's Sturm-Liouville problem. The efficiency of the IMEX scheme as compared to other methods such as fully implicit schemes or adaptive methods makes it ideal for implementation in fully 3D numerical relativity codes for modified gravity.