A High-Order-Accurate Unstructured Mesh Finite-Volume Scheme for the Advection–Diffusion Equation

High-order-accurate methods for viscous flow problems have the potential to reduce the computational effort required for a given level of solution accuracy. The state of the art in this area is more advanced for structured mesh methods and finite-element methods than for unstructured mesh finite-volume methods. In this paper, we present and analyze a new approach for high-order-accurate finite-volume discretization for diffusive fluxes that is based on the gradients computed during solution reconstruction. Our analysis results show that our schemes based on linear and cubic reconstruction can be reasonably expected to achieve second- and fourth-order accuracy in practice, respectively, while schemes based on quadratic reconstruction are expected to be only second-order accurate in practice. Numerical experiments show that in fact nominal accuracy is attained in all cases for two advection–diffusion problems, provided that curved boundaries are properly represented. To enforce boundary conditions on curved boundaries, we introduce a technique for constraining the least-squares reconstruction in boundary control volumes. Simply put, we require that the reconstructed solution satisfy the boundary condition exactly at all boundary flux integration points. Numerical experiments demonstrate the success of this approach, both in the reconstruction results and in simulation results.