Fourth-order finite difference scheme for a system of quasilinear elliptic equations

The system of two quasilinear elliptic equations is approximated by the method of lines, which has the truncation error O(h²) at points neighboring the boundary and O(h⁴) at the most interior points. It is proved that the global error of the method is O(h⁴) at all mesh points. The two-point boundary value problem for the system of ordinary differential equations that arises from the method of lines is solved by the O(h⁴) convergent finite difference scheme, suitable to the equations of the form u_xx = f(x, u) without the first derivative u_x. The system of algebraic equations obtained by the full discretization is solved by Gauss elimination method for three diagonal matrices combined with the method of iterations. A numerical example is presented.     

Dispersion of linear and nonlinear optical susceptibilities and the hyperpolarizability of 3-methyl-4-phenyl-5-(2-pyridyl)-1,2,4-triazole

As a starting point for our calculation of 3-methyl-4-phenyl-5-(2-pyridyl)-1,2,4-triazole we used the XRD data obtained by C. Liu, Z. Wang, H. Xiao, Y. Lan, X. Li, S. Wang, Jie Tang, Z. Chen, J. Chem. Crystallogr., 2009 39 881. The structure was optimized by minimization of the forces acting on the atoms keeping the lattice parameters fixed with the experimental values. Using the relaxed geometry we have performed a comprehensive theoretical investigation of dispersion of the linear and nonlinear optical susceptibilities of 3-methyl-4-phenyl-5-(2-pyridyl)-1,2,4-triazole using the full potential linear augmented plane wave method. The local density approximation by Ceperley-Alder (CA) exchange-correlation potential was applied. The full potential calculations show that this material possesses a direct energy gap of 3.4 eV for the original experimental structure and 3.2 eV for the optimized structure. We have calculated the complex's dielectric susceptibility ε(ω) dispersion, its zero-frequency limit ε(1)(0) and the birefringence. We find that a 3-methyl-4-phenyl-5-(2-pyridyl)-1,2,4-triazole crystal possesses a negative birefringence at the low-frequency limit Δn(0) which is equal to about -0.182 (-0.192) and at λ = 1064 nm is -0.193 (-0.21) for the non-optimized structure (optimized one), respectively. We also report calculations of the complex second-order optical susceptibility dispersions for the principal tensor components: χ(ω), χ(ω) and χ(ω). The intra- and inter-band contributions to these susceptibilities are evaluated. The calculated total second order susceptibility tensor components at the low-frequency limit |χ(0)| and |χ(ω)| at λ = 1064 nm for all the three tensor components are evaluated. We established that the calculated microscopic second order hyperpolarizability, β(ijk), the vector component along the dipole moment direction, at the low-frequency limit and at λ = 1064 nm, for the dominant component |χ(ω)| is 4.99 × 10(-30) esu (3.4 × 10(-30) esu) and 7.72 × 10(-30) esu (5.1 × 10(-30) esu), respectively for the non-optimized structure (optimized structure).     

On the maximum principle and its application to diffusion equations

In this article, an analog of the maximum principle has been established for an ordinary differential operator associated with a semi‐discrete approximation of parabolic equations. In applications, the maximum principle is used to prove O(h²) and O(h⁴) uniform convergence of the method of lines for the diffusion Equation (1). The system of ordinary differential equations obtained by the method of lines is solved by an implicit predictor corrector method. The method is tested by examples with the use of the enclosed Mathematica module solveDiffusion. The module solveDiffusion gives the solution by O(h²) uniformly convergent discrete scheme or by O(h⁴) uniformly convergent discrete scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007