Accuracy-preserving algorithm for difference schemes applied to non-uniform grids

To address the issue of accuracy degradation in finite difference schemes on non-uniform grids, this study investigated the mechanism behind geometry-induced errors in one-dimensional non-uniform grid configurations. A simplified grid model with only one-dimensional non-uniformity was constructed, and both theoretical analysis and numerical experiments were conducted to reveal the root causes of accuracy loss in traditional schemes. Theoretical findings indicated that existing difference schemes, such as first-order upwind, MUSCL and WENO schemes, failed to achieve their designed order of accuracy on non-uniform grids due to the uncontrolled influence of higher-order derivative terms introduced by coordinate transformations on the truncation error. Based on a re-examination of the fundamental goal of derivative approximation in finite difference methods, an innovative accuracy-preserving algorithm was proposed. This method reconstructed the derivative approximation model by combining directional difference quotients in a weighted manner, thereby correcting the discretization strategy of traditional schemes and effectively eliminating geometry-induced errors. Numerical validations using linear and quadratic flow fields as initial conditions demonstrated that the improved first-order scheme could strictly preserve linear flow fields (with error magnitude on the order of 10⁻¹⁷), and the second-order scheme could further preserve quadratic flow fields (with error magnitude on the order of 10⁻¹⁶). The findings of this study offer new insights for the development of error-reduction algorithms on non-uniform grids.