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ABSTRACT
An improved finite difference method with a compact correction term is proposed to solve the Poisson’s equations. The compact correction term is developed by coupled high-order compact and low-order classical finite difference formulations. The numerical solutions obtained by the classical finite difference method are considered as fundamental solutions with lower accuracy, whereas a compact correction term is added into the source term of classical discrete formulation to improve the accuracy of numerical solutions. The proposed method can be extended from two-dimensional to multidimensional cases straightforwardly. Numerical experiments are carried out to verify the accuracy and efficiency of this method.
Nomenclature
| ae | = | coefficients of discretized Poisson’s equation |
| ai | = | coefficients of discretized equation for second derivative |
| an | = | coefficients of discretized Poisson’s equation |
| ap | = | coefficients of discretized Poisson’s equation |
| as | = | coefficients of discretized Poisson’s equation |
| bi | = | coefficients of discretized equation for second derivative |
| = | compact correction term | |
| ci | = | coefficients of discretized equation for second derivative |
| f | = | forcing function |
| H | = | numerical method with higher-order accuracy |
| L | = | numerical method with lower-order accuracy |
| Lx | = | horizontal length of computational domain |
| Ly | = | vertical length of computational domain |
| M | = | number of grid points in x-direction |
| N | = | number of grid points in y-direction |
| x | = | horizontal coordinate |
| xi | = | grid point in x-direction |
| y | = | vertical coordinate |
| yi | = | grid point in y-direction |
| αi | = | coefficient of discretized equations for second derivative |
| βi | = | coefficient of discretized equations for second derivative |
| γ | = | stretch ratio of neighbor grid |
| ξ | = | fundamental grid points with uniform intervals in the domain [−1, 1] |
| Δxi | = | grid spacing in x-direction |
| Δyi | = | grid spacing in y-direction |
| φ | = | variable value |
| φi,j | = | numerical solution |
| = | numerical solution with lower accuracy | |
| Ω | = | computational domain |
Nomenclature
| ae | = | coefficients of discretized Poisson’s equation |
| ai | = | coefficients of discretized equation for second derivative |
| an | = | coefficients of discretized Poisson’s equation |
| ap | = | coefficients of discretized Poisson’s equation |
| as | = | coefficients of discretized Poisson’s equation |
| bi | = | coefficients of discretized equation for second derivative |
| = | compact correction term | |
| ci | = | coefficients of discretized equation for second derivative |
| f | = | forcing function |
| H | = | numerical method with higher-order accuracy |
| L | = | numerical method with lower-order accuracy |
| Lx | = | horizontal length of computational domain |
| Ly | = | vertical length of computational domain |
| M | = | number of grid points in x-direction |
| N | = | number of grid points in y-direction |
| x | = | horizontal coordinate |
| xi | = | grid point in x-direction |
| y | = | vertical coordinate |
| yi | = | grid point in y-direction |
| αi | = | coefficient of discretized equations for second derivative |
| βi | = | coefficient of discretized equations for second derivative |
| γ | = | stretch ratio of neighbor grid |
| ξ | = | fundamental grid points with uniform intervals in the domain [−1, 1] |
| Δxi | = | grid spacing in x-direction |
| Δyi | = | grid spacing in y-direction |
| φ | = | variable value |
| φi,j | = | numerical solution |
| = | numerical solution with lower accuracy | |
| Ω | = | computational domain |