New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two-and three-dimensions are developed and analyzed.Different from a few sixth-order compact finite difference schemes in the literature,the finite difference and weight coefficients of the new methods have analytic simple expressions.One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points 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 the solution is a polynomial less than or equal to 6.The coefficient matrices of the new schemes are M-matrices for Helmholtz equations with wave number K≤0,which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes.Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

ON EXTRAPOLATION CASCADIC MULTIGRID METHOD

Based on an asymptotic expansion of （bi）linear finite elements, a new extrapolation formula and extrapolation cascadic multigrid method （EXCMG） are proposed. The key ingredients of the proposed methods are some new extrapolations and quadratic interpolations, which are used to provide better initial values on the refined grid. In the case of triple grids, the errors of the new initial values are analyzed in detail. The numerical experiments show that EXCMG has higher accuracy and efficiency.

TIME-EXTRAPOLATION ALGORITHM （TEA） FOR LINEAR PARABOLIC PROBLEMS

The fast solutions of Crank-Nicolson scheme on quasi-uniform mesh for parabolic prob- lems are discussed. First, to decrease regularity requirements of solutions, some new error estimates are proved. Second, we analyze the two characteristics of parabolic discrete scheme, and find that the efficiency of Multigrid Method （MG） is greatly reduced. Nu- merical experiments compare the efficiency of Direct Conjugate Gradient Method （DCG） and Extrapolation Cascadic Multigrid Method （EXCMG）. Last, we propose a Time- Extrapolation Algorithm （TEA）, which takes a linear combination of previous several level solutions as good initial values to accelerate the rate of convergence. Some typical extrapolation formulas are compared numerically. And we find that under certain accuracy requirement, the CG iteration count for the 3-order and 7-level extrapolation formula is about 1/3 of that of DCG＇s. Since the TEA algorithm is independent of the space dimension, it is still valid for quasi-uniform meshes. As only the finest grid is needed, the proposed method is regarded very effective for nonlinear parabolic problems.

Secretogranln-II plays a critical role in zebrafis neurovascular modeling

Secretoneurin （SN） is a neuropeptide derived from specific proteolytic processing of the precursor secretogranin II （Sgll）. In zebrafish and other teleosts, there are two paralogs named sglla and sgllb. Our results showed that neurons expressing sgllb were aligned with central arteries in the hindbrain, demonstrating a close neurovascular association. Both sgllb-/- and sgllo-/-/sgllb-/- mutant embryos were defective in hindbrain central artery development due to impairment of migration and proliferation of central artery cells. Further study revealed that sgllb is non-ceU autonomous and required for central artery development. Hindbrain arterial and venous network identities were not affected in sgllb-/- mutant embryos, and the mRNA levels of Notch and VEGF pathway-related genes were not altered. However, the activation of MAPK and PI3K/AKT pathways was inhibited in sgllb-/- mutant embryos. Reactivation of MAPK or PI3K/AKT in endothelial cells could partially rescue the central artery developmental defects in the sgllb mutants. This studV provides the first in vivo evidence that sgllb ptavs a critical rote in neurovascutar modeling of the hindbrain. Targeting the Sgll system may, therefore, represent a new avenue for the treatment of vascular defects in the central nervous system.

Effect of Fertilization on Anatomical and Physical-mechanical Properties of Neosinocalamus Affinis Bamboo

In order to maintain soil fertility of Neosinocalamus affinis plantations,fertilizers of N,P,and K were applied.The anatomical and physical-mechanical properties of N.affinis bamboo wood from different fertilization treatments were measured.The aim of this study was to elucidate the effect of fertilization practice on the properties of N.affinis bamboo wood.The results revealed that the fertilization of P and K resulted in a slight reduction in fiber length.The application of P,K,and low level(0.3 kg/clump)of N fertilizers had no significant effect on the fiber morphology,while high level(0.9 kg/clump)of N fertilizer contributed to short fibers.The specific gravity was significantly decreased by fertilization,while the volume shrinkage was increased.Since the effect of various fertilization treatments had different influence patterns on the properties of N.affinis,specific evaluations on the quality of the fertilized bamboo wood should be performed prior to its utilization.