First, a modified stencil approximation method is introduced, which improves the second-order polynomial approximation of the numerical flux on each candidate sub-stencil in the classical fifth-order WENO-JS scheme. The stencil approximation reaches the fourth-order accuracy by adding a cubic correction term, and it has ENO property by introducing an adjustable function.Then the modified stencil approximation method is applied to WENO-Z+and WENO-Z+M schemes, and the modified WENO-Z+schemes based on the modified stencil type (WENO-MS-Z+, WENO-MS-Z+M) are developed.A series of numerical examples are used to test the new schemes. The results show that the new schemes have a strong ability to capture shock waves and high resolution for small-scale wave structures, which is significantly improved compared with the original WENO-Z+and WENO-Z+M schemes.

Compared with those of hydrodynamics, the existence of magnetic field leads to extra characteristic waves for magnetohydrodynamics (MHD), and further leads to an inconsistency of jump condition across the contact discontinuity. Usually, for the magnetic field variables in an HLLC Riemann solver, a single HLL intermediate state is used to replace two HLLC intermediate states to achieve conservation and computational stability, at the cost of insufficient simulation accuracy for tangential discontinuity. In this paper, a previouly developed HLLC solver is specially constructed to deal with MHD tangential discontinuities accurately and satisfy the so-called Toro condition. With numerical tests, such as the time-dependent simulation of one-dimensional shock tube, the tangential discontinuities, and the global MHD simulation of Earth's magnetosphere, we compare numerical results of the modified HLLC solver with those of the standard HLLC and HLLD solvers. It indicates that the modified HLLC solver has better capture accuracy for tangential discontinuities than the previously developed HLLC solver, and has the accuracy of the more time-consuming HLLD solver in some situations.

Artificial-Neural Network (ANN) is a complex network system composed of many simple elements connected extensively each other and can be regarded as a simulation or abstraction of biological neural system. Multi-Layer Network (MLN) is one of the most important ANN. A essential algorithm to train MLN is BP (Back-Propagation) algorithm. Based on BP algorithm, a hgih-speed algorithm-train MLN layer-by-layer algorithm is proposed with main points:4) train MLN layer-by-layer instead of all together; b) giving instruction to hidden units; c) giving appropriate "energy function' according to specific Problems; d) preserving the merits of BP algorithm. The simulation results increase by orders of magnitude in training-speed. This algorithm applies also some research on running mechanism of MLN.

Propagation characters of optical soliton in single-mode fiber is studied by using Bäcklund transformation. The results show that optical soliton pulse width broaden as excp(2Γξ) and the faster soliton speed is, the smaller soliton intensity decays because of fiber loss.