Essence and composition of truncation errors of discrete equations based on the finite volume method

Based on the principle of the finite volume method, the essence of truncation errors of discrete equations is revealed that represents the approximation degree to the conservative integral equation instead of the differential equation at the node or the average value over the control volume. This idea differs from that the truncation error expresses the approximation degree between the discrete expression and the differential equation in the finite difference method. The truncation error expression of discrete equations obtained by the finite volume method is derived, explaining that the truncation errors mainly involve three levels. The first level comes from the variables at the cell face, including the target variable and its first-order derivative, velocity, diffusion coefficient, etc. The second level originates from the flux at the cell face, consisting of the normal and tangential truncation errors: The former is caused by the approximations of the target variable and its first derivative, mass flow, and diffusion coefficient at the cell interface; the latter is induced by taking the value at one point as the average value of the cell face. The third level derives from discrete equations, and truncation errors include the convection and diffusion fluxes errors at the cell face and source term errors of the control volume. It can be found that there is an essential difference between the finite volume method and the finite difference method in truncation errors since there are no approximations of unknown parameters and tangential truncation errors at the cell face for the latter. Moreover, the truncation errors of the finite difference method are derived based on the grid node. Therefore, even for ideal two-dimensional or three-dimensional problems with constant physical properties, the truncation error expressions of these two methods are different. Because there are many sources of errors for the finite volume method, the overall accuracy of discrete equations cannot be improved by only raising the discrete accuracy of the target variable at the cell face. However, a high-accuracy scheme is easier to obtain to solve the constant property problems by the finite difference method without these errors. In addition, theoretical derivations and numerical tests indicate that the calculation accuracy of the finite volume method is related to the node position of the target variable in the grid. Based on the non-uniform orthogonal structured grid, it is deduced that the discrete accuracies of convection and diffusion terms are second-order and first-order in the normal and tangential directions, respectively, when the node positions are determined by the cell vertex method; the discrete accuracy of the convection term is second-order accuracy in both normal and tangential directions when the node positions are determined by the cell centered method, while that of the diffusion term is reduced to first-order in the normal direction. The simulation results of diffusion problems with and without a source term on the regular area discretized by the non-uniform orthogonal grid indicate that the cell centered method has higher accuracy than the cell vertex method, which is consistent with the theoretical analysis of truncation errors.