Improving finite-difference schemes in the context of Nero-hydrodynamic calculations: A conservative approach to solving transfer equations

The purpose of this study was to address critical limitations in existing finite-difference schemes for solving convective transfer equations in aero-hydrodynamic calculations, particularly the non-conservative behavior of the widely used Courant-Isaacson-Rees scheme under specific velocity distributions. The methodology involved a comprehensive analysis of finite-difference schemes using the mass conservation equation for transported substances in compressible media. Test calculations were performed on one-dimensional and two-dimensional problems with varying velocity fields, including cases with velocity sign changes and zero-velocity zones. The proposed scheme uses max and min functions to blend positive and negative velocity components. This approach maintains conservation. The findings demonstrate that the traditional Courant scheme loses conservation when velocity signs change, particularly at stagnation points, leading to mass loss or artificial mass generation. In contrast, the new conservative finite-difference scheme maintains exact mass conservation, stability, and symmetry. It performs well with all tested velocity distributions, even in challenging cases where traditional schemes struggle. In conclusion, the developed scheme eliminates non-conservative behavior that affected existing methods, ensuring accurate representation of physical processes in hydrodynamic calculations. The practical implications include the ability to use larger time steps in numerical calculations while maintaining accuracy and stability, making it particularly valuable for complex aero-hydrodynamic simulations involving flow separation, stagnation points, and variable velocity fields.