A High-Order Piecewise Polynomial Reconstruction for Finite-Volume Methods Solving Convection and Diffusion Equations

In finite-volume methods for fluid flows, the average of field variables over local mesh cells are the unknowns that are integrated in time based on the integral conservation laws. In order to compute the cell-face fluxes, nodal variables and derivative values at cell faces are needed, which ultimately determine the accuracy of the finite-volume method. In this work, a piecewise fourth-order polynomial reconstruction model based on volume averages is developed for smoothly varying flow fields over finite-volume cells. The obtained polynomial is used to extrapolate the cell-face variables and derivative values with high-order accuracy. The extrapolated cell-face quantities are used directly to compute the convection or diffusion integrals to construct high-order finite-volume methods. Some one-dimensional examples are shown to demonstrate the fifth-order accuracy of the proposed approach when solving the linear convection equation, and the fourth-order solution accuracy when solving the linear diffusion equation.