Central Weighted Non-Oscillatory (CWENO) and Operator Splitting Schemes in Computational Astrophysics

Abstract

High-resolution shock-capturing schemes (HRSC) are known to be the most adequate and advanced technique used for numerical approximation to the solution of hyperbolic systems of conservation laws. Since most of the astrophysical phenomena can be described by means of system of (M)HD conservation equations, nding most accurate, computationally not expensive and robust numerical approaches for their solution is a task of great importance for numerical astrophysics. Based on the Central Weighted Non-Oscillatory (CWENO) reconstruction approach, which relies on the adaptive choice of the smoothest stencil for resolving strong shocks and discontinuities in central framework on staggered grid, we present a new algorithm for systems of conservation laws using the key idea of evolving the intermediate stages in the Runge Kutta time discretization in primitive variables . In this thesis, we introduce a new so-called conservative-primitive variables strategy (CPVS) by integrating the latter into the earlier proposed Central Runge Kutta schemes (Pareschi et al., 2005). The advantages of the new shock-capturing algorithm with respect to the state-of-the-art HRSC schemes used in astrophysics like upwind Godunov-type schemes can be summarized as follows: (i) Riemann-solver-free central approach; (ii) favoring dissipation (especially needed for multidimensional applications in astrophysics) owing to the di ffusivity coming from the design of the scheme; (iii) high accuracy and speed of the method. The latter stems from the fact that the advancing in time in the predictor step does not need inversion between the primitive and conservative variables and is essential in applications where the conservative variables are neither trivial to compute nor to invert in the set of primitive ones as it is in relativistic hydrodynamics. The main objective of the research adopted in the thesis is to outline the promising application of the CWENO (with CPVS) in the problems of the computational astrophysics. We tested the method for one dimensional Euler hydrodynamics equations and we assessed the advantages against the operator splitting and finite-volume Godunov-type approaches implemented in the widely used astrophysical codes ZEUSMP/ 2 (Stone and Norman, 1992) and ATHENA (Stone et al., 2008), respectively. We extended the application of the scheme to one dimensional relativistic hydrodynamics (RHD), which (to the author's knowledge) is the fi rst successful attempt to approximate the special relativistic hydrodynamics with CWENO method. We demonstrate that strong discontinuities can be captured within two numerical zones and prevent the onset of numerical oscillations. In the second part of the present thesis, the astrophysical operator-splitting MHD code ZEUS-MP/2 has been used to perform three dimensional nonlinear simulations of MHD instabilities. First, we present global 3D nonlinear simulations of the Tayler instability in the presence of vertical elds. The initial con guration is in equilibrium, which is achieved by balancing a pressure gradient with the Lorentz force. The nonlinear evolution of the system leads to stable equilibrium with current free toroidal eld. We nd that the presence of a vertical poloidal eld stabilizes the system in the range from B phi approximately of order of Bz to higher values of Bz (Ivanovski and Bonanno, 2009). Second, the dynamics of the expansion of two colliding plasma plumes in ambient gas has been investigated via hydrodynamical simulations. Experimental observations of a single plume, generated by high power pulsed laser ablation of a solid target in ambient gas with pressure of about 10^-1 Torr, show possible Rayleigh-Taylor (RT) instability. Our numerical simulations with two plumes show RT instability even in low pressure gas, where single-plume expansion cannot cause instability. In addition, we nd that the RT instability is developed for about ten nanoseconds, while the instability in the case of a single plume typically takes thousand of nanoseconds. We show that the theoretically derived density condition for stability, Rho_plume < Rho_gas, is satis ed in all our simulations (Ivanovski et al., 2010). In the present thesis, we con rm the promising behavior of the conservative-primitive variables strategy with CWENO approach in computational astrophysics. We demonstrated high accuracy and robustness of the method in the essential one dimensional applications, sod-shock tubes and slow-moving shocks. Extending the method to higher dimensions and using the knowledge accumulated by means of direct numerical operator splitting simulations of MHD instabilities motivates building a modern accurate astrophysical code which will be able to resolve a wide range of problems, from ideal (magneto)hydrodynamics to relativistic (magneto)hydrodynamics.