This paper discusses a class of compact second order accurate finite difference equations for mixed initial-boundary value problems for hyperbolic and convective-diffusion equations. Convergence is proved by means of energy arguments and both types of equations are solved by similar algorithms. For hyperbolic equations an extension of the Lax–Wendroff method is described which incorporates dissipative boundary conditions. Upwind-downwind differencing techniques arise as the formal hyperbolic limit of the convective-diffusion equation. Finally, a finite difference “chain-rule” transforms the schemes from rectangular to quadrilateral subdomains.