A study on the dynamic spectral indices for SEP events on 2000 July 14 and 2005 January 20

We have studied the dynamic proton spectra for the two solar energetic particle (SEP) events on 2000 July 14 (hereafter GLE59) and 2005 January 20 (hereafter GLE69). The source locations of GLE59 and GLE69 are N22W07 and N12W58 respectively. Proton fluxes >30 MeV have been used to compute the dynamic spectral indices of the two SEP events. The results show that spectral indices of the two SEP events increased more swiftly at early times, suggesting that the proton fluxes >30 MeV might be accelerated particularly by the concurrent flares at early times for the two SEP events. For the GLE69 with source location at N12W58, both flare site and shock nose are well connected with the Earth at the earliest time. However, only the particles accelerated by the shock driven by eastern flank of the CME can propagate along the interplanetary magnetic field line to the Earth after the flare. For the GLE59 with source location at N22W07, only the particles accelerated by the shock driven by western flank of the associated CME can reach the Earth after the flare. Results show that there was slightly more than one hour during which the proton spectra for GLE69 are softer than that for GLE59 after the flares, suggesting that the shock driven by eastern flank of the CME associated with GLE69 is weaker than the shock driven by the western flank of the CME associated with GLE59. The results support that quasi-perpendicular shock has stronger potential in accelerating particles than the quasi-parallel shock. The results also suggest that only a small part of the shock driven by western flank of the CME associated with the GLE59 is quasi-perpendicular.Comment: Accepted by RA

A second-order accurate implicit difference scheme for time fractional reaction-diffusion equation with variable coefficients and time drift term

An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergence order in the $L_2$-norm is $\mathcal{O}(\tau^2 + h^2)$ with time step $\tau$ and mesh size $h$. Then, the same measure is exploited to solve the two-dimensional case of this problem and a rigorous theoretical analysis of the stability and convergence is carried out. Several numerical simulations are provided to show the efficiency and accuracy of our proposed schemes and in the last numerical experiment of this work, three preconditioned iterative methods are employed for solving the linear system of the two-dimensional case.Comment: 27 pages, 5 figures, 5 table

Can we estimate the intensities of great geomagnetic storms(Δ\DeltaSYM-H≤−\le -200 nT) by Burton equation or by O'Brien and McPherron equation?

We input solar wind parameters responsible for the main phases of 15 great geomagnetic storms (GGSs: $\Delta$SYM-H$\le-$200 nT) into the empirical formulae created by \cite{Burton1975}(hereafter Burton equation), and by \cite{OBrien2000}(hereafter OM equation) to evaluate whether \textbf{two equations} can correctly estimate the intensities of GGSs. The results show that the intensities of most GGSs estimated by OM equation are much smaller than the observed intensities. The RMS error between the intensities estimated by OM equation and the observed intensities is \textbf{203} nT, implying that the estimated storm intensity deviates significantly from the observed one. The RMS error between the intensities estimated by Burton equation and the observed intensities is 130.8 nT. The relative error caused by Burton equation for the storms with intensities $\Delta$SYM-H$<$-400 nT is larger than 27\%, implying that the absolute error will be large for the storms with $\Delta$SYM-H$<$-400 nT. The results indicate that the two equations cannot work effectively in the estimation of GGSs. On the contrary, the intensity of a GGS estimated by the empirical formula created by \cite{WangCB2003} can always be very close to the observed one if we select the right weight for solar wind dynamic pressure, proving that solar wind dynamic pressure is an important factor for GGS intensity, but it is overlooked in the ring current injection terms of Burton equation or OM equation. This is the reason why the two equations cannot work effectively in the estimation of GGSs

A limited-memory block bi-diagonal Toeplitz preconditioner for block lower triangular Toeplitz system from time-space fractional diffusion equation

A block lower triangular Toeplitz system arising from time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block bi-diagonal Toeplitz preconditioner is developed for the block lower triangular Toeplitz system, whose storage is of $\mathcal{O}(N)$ with $N$ being the spatial grid number; (ii) A new skew-circulant preconditioner is designed to fast calculate the inverse of the block bi-diagonal Toeplitz preconditioner multiplying a vector. Numerical experiments are given to demonstrate the efficiency of our preconditioners.Comment: 19 pages, 3 figures, 5 tabl

Fast second-order implicit difference schemes for time distributed-order and Riesz space fractional diffusion-wave equations

In this paper, fast numerical methods are established for solving a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by the weighted and shifted Gr$\ddot{\rm{u}}$nwald formula in time and the fractional centered difference formula in space. The unconditional stability and second-order convergence in time, space and distributed-order of the difference schemes are analyzed. In the one-dimensional case, the Gohberg-Semencul formula utilizing the preconditioned Krylov subspace method is developed to solve the symmetric positive definite Toeplitz linear systems derived from the proposed difference scheme. In the two-dimensional case, we also design a global preconditioned conjugate gradient method with a truncated preconditioner to solve the discretized Sylvester matrix equations. We prove that the spectrums of the preconditioned matrices in both cases are clustered around one, such that the proposed numerical methods with preconditioners converge very quickly. Some numerical experiments are carried out to demonstrate the effectiveness of the proposed difference schemes and show that the performances of the proposed fast solution algorithms are better than other numerical methods.Comment: 36 pages, 7 figures, 12 table

Sun-Earth connection Event of Super Geomagnetic Storm on March 31, 2001: the Importance of Solar Wind Density

An X1.7 flare at 10:15 UT and a halo CME with a projected speed of 942 km/s erupted from NOAA solar active region 9393 located at N20W19, were observed on 2001 March 29. When the CME reached the Earth, it triggered a super geomagnetic storm (hereafter super storm). We find that the CME always moved towards the Earth according to the intensity-time profiles of protons with different energies. The solar wind parameters responsible for the main phase of the super storm occurred on March 31, 2001 is analyzed taking into account the delayed geomagnetic effect of solar wind at the L1 point and using the SYM-H index. According to the variation properties of SYM-H index during the main phase of the super storm, the main phase of the super storm is divided into two parts. A comparative study of solar wind parameters responsible for the two parts shows the evidence that the solar wind density plays a significant role in transferring solar wind energy into the magnetosphere, besides the southward magnetic field and solar wind speed

An implicit integration factor method for a kind of spatial fractional diffusion equations

A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete ordinary differential system by using the implicit integration factor method, which is a class of efficient semi-implicit temporal scheme. Numerical results show that the proposed scheme is accurate even for the discontinuous coefficients.Comment: 7 pages, 1 figure and 4 tables. This paper is accepted by the Second International Conference on Physics, Mathematics and Statistics. It will be published in Journal of Physics: Conference Serie

Dependence of Intensities of Major Geomagnetic Storms (Dst ≤\le -100 nT) on Associated Solar Wind Parameters

A geomagnetic storm is the result of sustained interaction between solar wind with a southward magnetic field and the magnetosphere. To investigate the influence of various solar wind parameters on the intensity of major geomagnetic storm, 67 major geomagnetic storms that occurred between 1998 and 2006 were used to calculate the correlation coefficients (CCs) between the intensities of major geomagnetic storms and the time integrals of southward interplanetary magnetic field $B_s$, solar wind electric field ($E_y$) and injection function (Q) during the main phase of the associated geomagnetic storms. SYM-H$_{min}$ was used to indicate the intensity of the associated major geomagnetic storm, while I($B_z$), I($E_y$) and I(Q) were used to indicate the time integrals of $B_z$, $E_y$ and Q during the main phase of associated major geomagnetic storm respectively. The derived CC between I($B_z$) and SYM-H$_{min}$ is 0.33, while the CC between I($E_y$) and SYM-H$_{min}$ is 0.57 and the CC between I(Q) and SYM-H$_{min}$ is 0.86. The results provide statistical evidence that solar wind dynamic pressure or solar wind density plays a significant role in transferring solar wind energy into the magnetosphere, in addition to the southward magnetic field and solar wind speed. Solar wind that has a strong geoeffectiveness requires solar wind dynamic pressure $>$3 nPa or solar wind density $>3$ nPa$/V_{sw}^2$. Large and long duration $B_s$ alone cannot ensure a major geomagnetic storm, especially if the solar wind dynamic pressure is very low, as large and long duration Bs is not a full condition, only a necessary condition to trigger a major geomagnetic storm

A preconditioning technique for all-at-once system from the nonlinear tempered fractional diffusion equation

An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are proved under several suitable assumptions, and numerical examples show that the convergence orders of these two schemes are $1$ in both time and space. Secondly, a nonlinear all-at-once system is derived based on the nonlinear implicit scheme, which may suitable for parallel computations. Newton's method, whose initial value is obtained by interpolating the solution of the linearized implicit scheme on the coarse space, is chosen to solve such the nonlinear all-at-once system. To accelerate the speed of solving the Jacobian equations appeared in Newton's method, a robust preconditioner is developed and analyzed. Numerical examples are reported to demonstrate the effectiveness of our proposed preconditioner. Meanwhile, they also imply that such the initial guess for Newton's method is more suitable.Comment: 10 tables, 2 figure

Strang-type preconditioners for solving fractional diffusion equations by boundary value methods

The finite difference scheme with the shifted Gr\"{u}nwarld formula is employed to semi-discrete the fractional diffusion equations. This spatial discretization can reduce to the large system of ordinary differential equations (ODEs) with initial values. Recently, boundary value method (BVM) was developed as a popular algorithm for solving large systems of ODEs. This method requires the solutions of one or more nonsymmetric, large and sparse linear systems. In this paper, the GMRES method with the block circulant preconditioner is proposed for solving these linear systems. One of the main results is that if an $A_{\nu_1,\nu_2}$-stable boundary value method is used for an m-by-m system of ODEs, then the preconditioner is invertible and the preconditioned matrix can be decomposed as I+L, where I is the identity matrix and the rank of L is at most $2m(\nu_1+\nu_2)$. It means that when the GMRES method is applied to solve the preconditioned linear systems, the method will converge in at most $2m(\nu_1+\nu_2)+1$ iterations.Finally, extensive numerical experiments are reported to illustrate the effectiveness of our methods for solving the fractional diffusion equations.Comment: 19 pages,4 figure