This chapter focuses on conservative numerical methods for solving systems of hyperbolic balance laws in conservation-law form. All methods are characterised by a numerical flux. First, we introduce the finite volume framework, which accommodates both upwind and centred methods. We then present the Godunov method, along with several Riemann solvers for determining the numerical flux. The Riemann solvers explored include the exact solver, a two-rarefaction approximation, the HLL solver, the HLLC solver, the DOT solver, a TV flux vector splitting solver, and the simple Rusanov solver. Next, we discuss classical finite difference methods formulated in conservative form with a numerical flux. These methods include the Lax-Friedrichs method, the FORCE method, the centred Godunov method, and the Lax-Wendroff method. The numerical schemes are applied to the one-dimensional blood flow equations, augmented by an advection equation for a concentration variable. A systematic evaluation of the performance of these methods is conducted through a carefully selected suite of test problems for arteries and veins, with comparisons between the numerical solutions and exact solutions. Finally, suggested exercises are provided at the end of the chapter.