Optimized compact finite difference schemes with high accuracy and maximum resolution

The pentadiagonal compact finite difference scheme and asymmetric boundary schemes are optimized with high accuracy and maximum resolution in this paper. Through Fourier analysis, the optimization is reduced to the problem of finding the minimum of a multivariable nonlinear function with multiple constraints. The advanced sequential quadratic programming method is employed to find the minimum. In order to extend the resolution characteristic of the schemes, the wavenumber domain for optimization is nearly identical to the well-resolved domain, and the maximum well-resolved wavenumber is obtained by means of equal step length searching. The optimized schemes are strictly stable as confirmed by an eigenvalue analysis. The increased performances of the schemes are demonstrated through their application to one- and two-dimensional examples and are compared with other schemes optimized before.