Of Helmholtz Equation Compact Difference Scheme And Algorithm

The new compact finite-difference method for solving the general Helmholtz equation in two and three dimensions, is developed and analyzed. The second-order schemes, fourth-order schemes and sixth-order schemes are based on the relations of the difference operator and the differential operator. In the paper [7], the sixth-order scheme and the sixth-order accurate approximation of the boundary in two dimensions have been conducted, the sixth-order scheme and the sixth-order approximation of the boundary by the same method have been studied in the three dimensions further.To the schemes of Helmholtz equation, we study the eigenvalues of the three-diagonal matrices of the schemes by the concept of matrix tensor product. And the accuracy orders of the schemes are also analyzed theoretically. In order to validate our schemes and examine their behavior, the orthogonal transformation method is used for the sixth-order and fourth-order in two dimensions. Due to we have two sixth-order schemes in three dimensions, the GCR method which is in Krylov iterate methods to solve the large sparse systems for different problems, and the relations of the behavior and wave-number are analyzed, too. Finally, we study the other Krylov iterate methods, such as GMRES, BI-CGSTAB, IDR(s), et.