Advanced numerical methods for dissipative and non-dissipative relativistic hydrodynamics

High-energy physical phenomena such as astrophysical events and heavy-ion collisions contain a hydrodynamic aspect in which a branch of fluid dynamics called relativistic hydrodynamics (RHD) is required for its mathematical description. The resulting equations must be, more often than not, solved numerically for scientists to ascertain useful information regarding the fluid system in question. This thesis describes and presents a twodimensional computational fluid dynamics (CFD) solver for dissipative and non-dissipative relativistic hydrodynamics, i.e. in the presence and absence of physically resolved viscosity and heat conduction. The solver is based on a finite volume, Godunov-type, HighResolution Shock-Capturing (HRSC) framework, containing a plethora of numerical implementations such as high-order Weighted-Essentially Non-Oscillatory (WENO) spatial reconstruction, approximate Riemann solvers and a third-order Total Variation Diminishing (TVD) Runge–Kutta method. The base numerical solver for the solution of non-dissipative RHD is extensively tested using a series of one-dimensional test cases, namely, a smooth flow problem and shock-tube configurations as well as the two-dimensional vortex sheet and Riemann problem test cases. For the case of non-dissipative relativistic hydrodynamics the relativistic CFD solver is found to perform well in terms of the orders of accuracy achieved and its ability to resolve shock wave patterns. Numerical pathologies have been identified when the relativistic HLLC Riemann solver is used in multi-dimensions for problems exhibiting strong shock waves. This is attributed to the so-called Carbuncle problem which is shown to occur because of pressure differencing within the process of restoring the missing contact discontinuity of its predecessor, the HLL Riemann solver. To avoid this numerical pathology and improve the robustness of numerical solutions that make use of the HLLC Riemann solver, the development of a rotated-hybrid Riemann solver arising from the hybridisation of the HLL and HLLC (or Rusanov and HLLC) approximate Riemann solvers is presented. A standalone application of the HLLC Riemann solver can produce spurious numerical artefacts when it is employed in conjunction with Godunov-type high-order methods in the presence of discontinuities. It has been found that a rotated-hybrid Riemann solver with the proposed HLL/HLLC (Rusanov/HLLC) scheme could overcome the difficulty of the spurious numerical artefacts and presents a robust solution for the Carbuncle problem. The proposed rotated-hybrid Riemann solver provides sufficient numerical dissipation to capture the behaviour of strong shock waves for relativistic hydrodynamics. Therefore, focus is placed on two benchmark test cases (odd-even decoupling and double-Mach reflection problems) and the investigation of two astrophysical phenomena, the relativistic Richtmyer– Meshkov instability and the propagation of a relativistic jet. In all presented test cases, the Carbuncle problem is shown to be eliminated by employing the proposed rotated-hybrid Riemann solver. This strategy is problem-independent, straightforward to implement and provides a consistent robust numerical solution when combined with Godunov-type highorder schemes for relativistic hydrodynamics...[cont.]