| Magnetic reconnection plays a very important role in solar flares, corona mass ejections (CME), and other solar activities. Many authors focus their attention on the solar small-scale activities such as microflares, Ellerman bombs, chromospheric jets, and so on. The reason is that these small-scale activities have simpler structures than the big solar activities such as major flare and CME. It is relatively easy to study. This may provide a new understanding of the basic physical mechanisms and give us a clue to study the more complicated eruption events.Microflares, or subflares, or bright points, are small-scale and short-lifetime solar activities. The size of microflares ranges from several arcsec to about20arcsec, the duration and the total energy can be10-30minutes and1026-1029ergs, respectively. Microflares have been observed in many wavelengths including Ha, EUV, soft X-ray, hard X-ray, and microwave. However, not all microflares have emissions at all wave-lengths. Observational characteristics of microflares, such as the heating, the relation with magnetic field, the duration and the coincidence between different wavelengths, etc, imply that microflares are produced by magnetic reconnection, similar to big flares.In part of our works, we use the magnetohydrodynamics (MHD) simulation to study the formation and evolution of microflares. There are two codes designed by ourselves for calculation. The one is based on the CIP-MOCCT scheme, in which the hydrodynamics part is calculated by CIP scheme and magnetic field is handled by MOCCT method to keep the divergence free condition. The other code is MAP code, which is written in FORTRAN language for MHD calculation with the adaptive mesh refinement (AMR) and Message Passing Interface (MPI) parallelization. There are sev-eral optional numerical schemes in MAP code for computing the MHD part, namely, modified Mac Cormack (MMC) scheme, Lax-Friedrichs (LF) scheme and weighted essentially non-oscillatory (WENO) scheme. All of them are second order, two-step, component-wise schemes for hyperbolic conservative equations. The total variation di-minishing (TVD) limiters and approximate Riemann solvers are also equipped. A high resolution can be achieved by the hierarchical block-structured AMR mesh in this code. Besides, MAP code includes the extended generalized Lagrange multiplier (EGLM) MHD equations to reduce the non-divergence free error produced by the scheme in the magnetic induction equation. The numerical algorithms for the non-ideal terms, e.g., the resistivity and the thermal conduction, are also equipped in the MAP code. The details of the AMR and MPI algorithms are described in this thesis.With gravity, ionization, and radiation being considered, we perform2.5D com-pressible resistive MHD simulations of chromospheric magnetic reconnection using the CIP-MOCCT scheme to study the evolution of microflares. The temperature dis-tribution of the quiet-Sun atmospheric model VALC and the helium abundance (10%) are adopted. Our2.5D MHD simulation reproduces qualitatively the temperature en-hancement observed in chromospheric microflares. The temperature enhancement AT is demonstrated to be sensitive to the background magnetic field, whereas the total evo-lution time At is sensitive to the magnitude of the anomalous resistivity. Moveover, we found a scaling law, which is described as ΔT/Δt～nH-1.5B2.1η0.88. Our results also indicate that the velocity of the upward jet is much greater than that of the downward jet and the X-point may move up or down.However, the simulation discusses in the above paragraph uses a very simple mag-netic configuration, since that simulation do not care about how the anti-parallel field formed but the evolution after the reconnection. Thus, in order to explain the microflare in a self-consistent manner, we perform2.5D compressible resistive MHD simulations of microflares which are triggered by magnetic reconnection between emerging flux and a pre-existing background magnetic field. The background magnetic field is a canopy-type configuration which is rooted at the boundary of the solar supergranule. By chang-ing the bottom boundary conditions in the simulation, new magnetic flux emerges up at the center of the supergranule and reconnects with the canopy-type magnetic field. We successfully simulate the coronal and chromospheric microflares, whose current sheets are located at the corona and the chromosphere, respectively. The microflare of coronal origin has a larger size and a higher temperature enhancement than that of chromospheric origin. In the microflares of coronal origin, we also found a hot jet (-1.8×106K), which is probably related to the observational EUV/SXR jets, and a cold jet (-104K), which is similar to the observational Ha/Ca surges, whereas there is only an Ha/Ca bright point in the microflares of chromospheric origin. The study of parameter dependence shows that the size and strength of the emerging magnetic flux are the key parameters which determine the height of the reconnection location, and further determine the different observational features of the microflares.Recently, ubiquitous jets (for example, chromospheric anemone jets, penumbral microjets, umbral light bridge jets) have been observed by Solar Optical Telescope on board the satellite Hinode. These tiny and frequently occurring jets are considered to be a possible evidence of small-scale ubiquitous reconnection in the solar atmosphere. However, the details of three dimensional magnetic configuration are still not very clear. Here we propose a new model based on three dimensional simulations of mag-netic reconnection using a typical current sheet magnetic configuration with a strong guide field. The most interesting feature is that the jets produced by the reconnection eventually move along the guide field lines. This model provides a fresh understanding of newly discovered ubiquitous jets and moreover a new observational basis for the theory of astrophysical magnetic reconnection.
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