A high-order finite volume scheme for unsteady convection-dominated convection-diffusion equations

A finite volume scheme with high-order accuracy is proposed for solving the unsteady convection-dominated transport problems. In this scheme, some weighting parameters are introduced in discretizing both the convection term and time integral. These parameters are determined analytically by making the truncation error of the proposed scheme as small as possible. Since the discretization equations of the proposed scheme share the same band structure as that of the traditional finite volume method based on the central differencing scheme, the proposed scheme does not increase computing cost. Numerical results show that the proposed scheme not only can achieve sixth-order accuracy but also avoids any unphysical oscillations in the steep gradient region of the solutions of the linear and nonlinear convection-dominated convection-diffusion problems.