A transformation-free HOC scheme and multigrid method for solving the 3D Poisson equation on nonuniform grids

A high-order compact (HOC) difference scheme is proposed to solve the three-dimensional (3D) Poisson equation on nonuniform orthogonal Cartesian grids involving no coordinate transformation from the physical space to the computational space. Theoretically, the proposed scheme has third to fourth-order accuracy; its fourth-order accuracy is achieved under uniform grid settings. Then, a multigrid method is developed to solve the linear system arising from this HOC difference scheme and the corresponding multigrid restriction and interpolation operators are constructed using the volume law. Numerical experiments are conducted to show the computed accuracy of the HOC scheme and the computational efficiency of the multigrid method.