The scramjet is a rather a new technology and there are many issues related to their operation, especially when it comes to the combustion processes. Combustion in high-speed flows causes various problems such as flame instability and poor fuel-air mixing efficiency. One of the methods used to overcome these problems is to recess a cavity in the combustor wall where a secondary flow is generated. In this study, a computational fluid dynamics (CFD) code is developed to analyse the reacting flow passing through the cavity-based scramjet combustor. The developed code is based on three-dimensional coupled Navier-Stokes and finite rate chemistry equations. An ethylene-air reduced chemical reaction model is used as a fuel-air combination. The Spalart-Allmaras model is utilised for turbulence closure. The non-dimensional form of the flow and chemical reaction equations are discretised using a finite volume method. The Jacobian-Free Newton-Krylov (JFNK) method is used to solve the coupled system of non-linear equations. The JFNK is a matrix-free solution method which improves the computational cost of Newton's method. The parameters that affect the performance of the JFNK method are studied in the analysis of a scramjet combustor. The influence of the forcing term on the convergence of the JFNK method is studied in the analysis of scramjet combustor. Different upwind flux vector splitting methods are utilised. Various flux limiter techniques are employed for the calculations of higher order flux vectors. The effects of flux vector splitting and flux limiter methods on the convergence and accuracy of the JFNK method are evaluated. Moreover, the variations of the mixing efficiency with fuel injection angles are studied.