An explicit higher order difference scheme on a compact stencil for elliptic equations on curvilinear geometries

A fourth order explicit finite difference scheme on a compact stencil is proposed for a general steady convection–diffusion equation on an arbitrary curvilinear coordinate system with computationally generated non-uniform grid. The method is dependent on the governing differential equation and attains higher order through consistent discretization and accurate evaluation of the transformation metrics. Extensive comparison of accuracy, convergence rate and computation time is made with standard explicit and implicit schemes to highlight the relatively high efficiency of using the proposed scheme. Applicability of the scheme to irregular flows and geometries is validated by solving for (i) potential flow past a cambered airfoil at an angle of attack and (ii) full incompressible Navier–Stokes calculations for flow past a cylinder performing rotary oscillations at low Reynolds number.