Improving finite-volume diffusive fluxes through better reconstruction

The overarching goal of CFD is to compute solutions with low numerical error. For finite-volume schemes, this error originates as error in the flux integral. For diffusion problems on unstructured meshes, the diffusive flux (computed from reconstructed gradients) is one order less accurate than the reconstructed solution. Worse, the gradient errors are not smooth, and so no error cancellation accompanies the flux integration, reducing the flux integral to zero order for second-order schemes. Our aim is to compute the gradient and flux more accurately at the cell boundaries and hence obtain a better flux integral for a slight increase in computational cost. We propose a novel reconstruction method and flux discretization to improve diffusive flux accuracy on cell-centred, isotropic unstructured meshes. Our approach uses a modified least-squares system to reconstruct the solution to second-order accuracy in the H₁ norm instead of the prevalent L₂ norm, thus ensuring second-order accurate gradients. Either circumcentres or containment centres are chosen as the control-volume reference points based on a criteria to facilitate calculation of second-order gradients at flux quadrature points using a linear interpolation scheme along with a high-accuracy jump term to enhance stability of the system. Numerical results show a significant improvement in the order of accuracy of the computed diffusive flux as well as the flux integral. When applied to a channel flow advection-diffusion problem, the scheme resulted in an increased order of accuracy for the flux integral along with gains in solution accuracy by a factor of two. The characteristics of the new scheme were studied through stability, truncation error and cost analysis. The increase in computational costs were modest and affordable. The behaviour of the scheme was also tested by implementing a variation of it within the ANSYS Fluent discretization framework.Applied Science, Faculty ofMechanical Engineering, Department ofGraduat