| Abstract: | ![]()
Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two- and three-dimensions aredeveloped and analyzed. Different from a few sixth-order compact finite differenceschemes in the literature, the finite difference and weight coefficients of the newmethods have analytic simple expressions. One of the new ideas is to use a weightedcombination of the source term at staggered grid points which is important for gridpoints near the boundary and avoids partial derivatives of the source term. Furthermore, the new compact schemes are exact for 2D and 3D Poisson equations if thesolution is a polynomial less than or equal to 6. The coefficient matrices of the newschemes are $M$-matrices for Helmholtz equations with wave number $K≤0,$ whichguarantee the discrete maximum principle and lead to the convergence of the newsixth-order compact schemes. Numerical examples in both 2D and 3D are presentedto verify the effectiveness of the proposed schemes. |
