关键词: 有限差分法, 几何守恒律, 离散等价方程, 几何诱导误差, 坐标变换

Abstract: Coordinate transformation is required when the finite difference method is applied to the mesh with complex geometries, and the errors induced by the coordinate transformation are often introduced in this process. These errors are proved to be inevitable in the uniform flow field calculation with uniform grids in cylindrical coordinate systems, even if the transformation function of the physical coordinates to the calculated coordinates is continuously derivable, or the coordinate transformation coefficients in the calculation process are calculated by the accurate analytical formula, or the grid is completely orthogonal and fully smooth. Theoretical analysis shows the mechanism of the coordinate transformation induced errors:when the conservative Euler equation is transformed from the Cartesian coordinate system to the body fitted coordinate system, a source term is added. Currently, scholars usually use the geometric conservation law to construct a method based on coordinate transformation coefficients, which are matched with the format of the finite difference, to eliminate the source term. In this work, we introduce a new algorithm that processes the direct discretization from the discrete equivalent functions including the source term. Based on the above new algorithm, error analysis is carried out for the reconstruction process of MUSCL format under non-equidistant grid conditions. Theoretical derivation shows that the influence coefficient of the non-equidistant interpolation formula needs to be considered in reconstruction, only when the variables are transformed into the computational space for MUSCL reconstruction can the interpolation accuracy be guaranteed under uniform grid. Our theoretical analysis and numerical experiments have proven that this algorithm will not introduce coordinate transformation errors to the uniform flow field calculations.

Key words: finite difference method, geometric conservation law, discrete equivalence equation, geometrically induced errors, coordinate transformation