The numerical experiments of three upwind schemes in contact discontinuity were carried out on the numerical dissipation under different flow field parameters, and the mechanism of numerical dissipation was analyzed. The numerical calculation results and theoretical analysis show that when the flux vector splitting scheme is used for contact discontinuity calculation, if the flow field is static or there exists a subsonic region in the flow field, the generation of density dissipation will induce numerical perturbation errors moving with characteristic velocity. These errors have no effect on the magnitude of numerical dissipation, but it will affect the distribution of velocity and pressure parameters in the flow field, thus changing the structure of the flow field. In two-dimensional flow fields, the mutual interference of the induced errors will produce numerous complex small-scale structures, which bring difficulties to the flow field structure identification. Meanwhile, in the flow field with linear distribution of density parameters, if the object reconstructed by the spatial discrete scheme is convective flux, using the flux vector splitting scheme to calculate the flow field will generate numerical errors, making it difficult to reach the second order of computational accuracy.

The process of grid generation and flow field simulation in CFD calculation can be automated to enhance the efficiency of CFD simulation, which has a great potential for applications. An Auto-CFD technology framework based on the tie-dye algorithm which can generate grids automatically and initiate inviscid supersonic flow field calculation quickly without geometry clean-up after inputting solid models. A 2D numerical example verifies that the calculation accuracy of this technology is comparable to that of the unstructured finite volume method, but with a significant improvement in calculation efficiency. An Auto-CFD software that can start the calculation in real time based on hand-drawn models was developed in theory, which can automatically simulate any two-dimensional irregular shape. The Auto-CFD technology was also extended from 2D to 3D space, and the irregular point cloud obtained by laser scanning the car was used as the solid model to demonstrate the adaptability of this technology to complex shape problems. The Auto-CFD technology framework developed based on the tie-dye algorithm is compatible with mainstream difference schemes and has good mesh adaptability, which is expected to solve the technical problems existing in existing Auto-CFD software.

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.

When the shock capturing method is used to solve inviscid compressible flow with a mathematical discontinuity as the initial condition, the discontinuity will gradually evolve into a numerical transition region across several grid points, and non-physical fluctuations in two directions parallel to the discontinuity can be induced in this process. The initial shock, which is calculated as the initial condition, satisfies the Euler equation, while the flow field parameters should satisfy the modified equations, such conflict might be the reason for the initial shock-induced non-physical fluctuations. As the discontinuities such as shock waves are defined by the characteristic lines of the Euler equation, an upwind flux scheme based on characteristics (UFSC) is proposed in the present study. Using several conventional conservation flux schemes as a reference, the UFSC can eliminate the non-physical fluctuations induced by the Steger-Warming and Van Leer schemes at the initial contact discontinuities, and can reduce the amplitude of the initial shock-induced disturbances. For the flow field in smooth regions, the computational results from the UFSC and conserved flux scheme are similar, but larger pressure peaks appear near the strong shock. To overcome this defect, a hybrid scheme UFSC+S is constructed, which uses the Steger-Warming for calculation in the shock region and the UFSC in other regions.

The gradient reconstruction process determines the spatial discretization accuracy and robustness of the finite volume method. A novel gradient reconstruction method is proposed for the cell-centered finite volume method. Based on the weighted least squares principle, this method calculates the face-centered variables and the gradient of these variables and then solves the cell-centered gradient using either central difference or arithmetic averaging methods for different grid types. Finally, a new boundary constraint algorithm adapted to the new gradient reconstruction method is developed by combining boundary conditions with the gradient reconstruction process. A grid convergence study using an exact test function shows that the new method can achieve the linear reconstruction of the full-field gradient under smooth solution conditions. A series of inviscid and viscous flow cases show that, compared with the previous method, this method can effectively reduce the numerical dissipation in the near boundary region and improve the computational accuracy with better robustness under the large aspect ratio triangular grid.