有限差分法计算热传导方程的几何诱导误差

Firstly, as an example, the unstructured finite difference method (UFDM) based on discrete equivalent equations was proposed, to enhance the geometric adaptability of the finite difference method. During the solution of the heat conduction equations with the finite difference method in the curvilinear coordinate system, the coordinate transformation will lead to geometric-induced errors. This phenomenon was illustrated by means of the central difference scheme to solve the temperature field equation. Based on the precision definition of the truncation errors of the difference scheme, the geometry-induced error was theoretically demonstrated to inevitably leads to a reduction in order. Secondly, a linear preservation assessment model was built to verify the 1st-order accuracy, and the difference scheme was proved to be difficult to guarantee the 1st-order accuracy of the assessment model on non-uniform grids. On this basis, a linear preservation algorithm based on gradient reconstruction was proposed. Numerical calculations show that, for structured grids with any shape to calculate linearly distributed temperature fields, numerical solutions with errors at the machine precision level of 0 can be obtained. This study provides a theoretical and practical foundation for developing fully automatic temperature field calculation software.