The shearing box approximation to the compressible unsteady Euler equations in a shearing, rotating, and stratified flow is investigated. A fundamental review is presented, detailing the development of a 2D finite volume method in a cartesian shearing box representing a circular ac- cretion disc. An overall 2nd order method is constructed using the HLLC approximate Riemann solver, a dimensionally split WAF flux-limited reconstruction scheme, and a symplectic source term integrator to conserve energy. The developed algorithm is validated against a comprehen- sive test suite modelled after that of Toro (1999) at each 1D, 2D, and source term construction step. The solver is finally employed to explore and further validate against a number of popular problems from accretion disc literature, looking both in the plane of the disc (r-φ) as well as at azimuthally invariant solutions in the r-z plane.