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This paper presents the implementation of a parameter-free third-order recon- struction method for cell-centered finite volume solvers on unstructured grids. The reconstruction is based on nodal gradients obtained using the least squares approach from solutions at adjacent cell centers. The cell and face gradients are computed by simple arithmetic averaging of vertex gradients, while the face values are obtained through quadratic interpolation. Importantly, the current reconstruction method does not require explicit second derivatives, and its stencil remains as compact as that used in traditional linear reconstruction methods. The third-order accuracy of the left and right states at the face values, along with the second-order accuracy of the face gradients, is numerically verified on various unstructured grids. This verified third-order accuracy is a crucial condition for ensuring the overall accuracy of the finite volume solver. 

This paper presents a robust mesh moving solver developed to address moving boundary problems. Crucially, the resulting deformed mesh retains the same topology as the original mesh without being overly distorted. The mesh is treated as an elastic material, and the deformation of the computational domain resulting from moving boundaries is determined by solving the equilibrium linear elasticity equations. The linear elasticity equations are discretized by the classic Galerkin finite element method and solved by the block conjugate gradient iterative method. To maintain the quality of the mesh after motion, the Young's modulus of each element is weighted by the reciprocal of the distance between the element center and the moving boundaries. The effectiveness of this approach is demonstrated through a set of 2D and 3D test cases featuring prescribed translational and/or rotational motion of the embedded object. The method is now ready for integration into our existing in-house CFD solvers.