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The Crank-Nicolson (CN) method is a well-known time integrator for evolutionary partial differential equations (PDEs) arising in many real-world applications. Since the solution at any time depends on the solution at previous time steps, the CN method is inherently difficult to parallelize. In this paper, we consider a parallel method for the solution of evolutionary PDEs with the CN scheme. Using an all-at-once approach, we can solve for all time steps simultaneously using a parallelizable over time preconditioner within a standard iterative method. Due to the diagonalization of the proposed preconditioner, we can prove that most eigenvalues of preconditioned matrices are equal to 1 and the others lie in the set: $\left\{z\in\mathbb{C}: 1/(1 + \alpha) < |z| < 1/(1 - \alpha)~{\rm and}~\Re{\rm e}(z) > 0\right\}$, where $0 < \alpha < 1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES method exhibits a mesh-independent convergence rate. Finally, we will verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs., Comment: 18 pages, 5 figures and 4 tables (update some contexts)

The Black-Scholes (B-S) equation has been recently extended as a kind of tempered time-fractional B-S equations, which becomes an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B-S model. To achieve high-order accuracy in space and overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods., Comment: 26 pages, 4 tables and 3 figures; correct and make some sentences clear. Ver3: deeply revise and reconstruct the context in order to make it more concise

Damped wave equations have been used in many real-world fields. In this paper, we study a low-rank solution of the strongly damped wave equation with the damping term, visco-elastic damping term and mass term. Firstly, a second-order finite difference method is employed for spatial discretization. Then, we receive a second-order matrix differential system. Next, we transform it into an equivalent first-order matrix differential system, and split the transformed system into three subproblems. Applying a Strang splitting to these subproblems and combining a dynamical low-rank approach, we obtain a low-rank algorithm. Numerical experiments are reported to demonstrate that the proposed low-rank algorithm is robust and accurate, and has second-order convergence rate in time., Comment: 14 pages, 3 figures, 2 tables

The extended Fisher--Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biology systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence and the maximum principle of the proposed scheme are proved rigorously. Based on the frame of the full-rank splitting scheme, a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation and the discrete maximum principle., Comment: 17pages, 4 figures, 4 tables

Time-space fractional Bloch-Torrey equations (TSFBTEs) are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order implicit difference scheme for TSFBTEs by employing the recently proposed L2-type formula [A.~A.~Alikhanov, C.~Huang, Appl.~Math.~Comput.~(2021) 126545]. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed to accelerate the convergence of Krylov subspace solvers according to the special structure of the coefficient matrix of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order $\alpha \in (0,0.3624)$. Numerical results are reported to show the efficiency of our method., Comment: 25 pages, 6 tables, 4 figures

Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well.The graded $L1$ scheme is often chosen to discretize such problems since it can handle the singularity of the solution near $t = 0$. In this paper, we propose a modification. We first split the time interval $[0, T]$ into $[0, T_0]$ and $[T_0, T]$, where $T_0$ ($0 < T_0 < T$) is reasonably small. Then, the graded $L1$ scheme is applied in $[0, T_0]$, while the uniform one is used in $[T_0, T]$. Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method., Comment: 3 figures; 3 tables. This preprint has been accepted by Journal of Scientific Computing

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 tables

The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the $p$-step BDF ($p\geq 2$) is not selfstarting, when they are exploited to solve time-dependent PDEs. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, but its resultant all-at-once discretized system is a block triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first give an estimation of the condition number of the all-at-once systems and then adapt the previous work to construct two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. The efficient implementation of these BC preconditioners is also presented especially for handling the computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings., Comment: 18 pages. 2 figures. 6 Table. Tech. Rep.: Institute of Mathematics, Southwestern University of Finance and Economics. Revised-1: refine/shorten the contexts and correct some typos; Revised-2: correct some references. Accepted by {\em Numer. Math. Theor. Meth. Appl.}, 14 Mar. 2021

In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded $L1$ formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the $M$-matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum-of-exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from $\mathcal{O}(MN^3 + M^2N)$ by a direct solver to $\mathcal{O}(MN(\log N + N_{exp}))$ per preconditioned Krylov subspace iteration and a memory requirement from $O(MN^2)$ to $O(NN_{exp})$, where $N$ and $(N_{exp} \ll)~M$ are the number of spatial and temporal grid nodes. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods., Comment: 31pages, 11 tables, 4 figures. Version4: accepted for publication by Mathematical Methods in Applied Sciences

The space fractional Cahn-Hilliard phase-field model is more adequate and accurate in the description of the formation and phase change mechanism than the classical Cahn-Hilliard model. In this article, we propose a temporal second-order energy stable scheme for the space fractional Cahn-Hilliard model. The scheme is based on the second-order backward differentiation formula in time and a finite difference method in space. Energy stability and convergence of the scheme are analyzed, and the optimal convergence orders in time and space are illustrated numerically. Note that the coefficient matrix of the scheme is a $2 \times 2$ block matrix with a Toeplitz-like structure in each block. Combining the advantages of this special structure with a Krylov subspace method, a preconditioning technique is designed to solve the system efficiently. Numerical examples are reported to illustrate the performance of the preconditioned iteration., Comment: 26 pages, 5 tables, 4 figures

Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is used. The convergence of our method is proved rigorously. Numerical examples are reported which show that the proposed method is robust and accurate., Comment: 17 pages; 3 figures; 4 tables

In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shifted Gr\"{u}nwald difference (WSGD) formula in spatial discretization, respectively. Theoretical results and numerical tests are conducted to verify the $(2 - \gamma)$-order and 2-order of temporal and spatial convergence with $\gamma\in(0,1)$ the order of Caputo fractional derivative, respectively. The fast sum-of-exponential approximation of the generalized Caputo fractional derivative and Toeplitz-like coefficient matrices are also developed to accelerate the proposed implicit difference scheme. Numerical experiments show the effectiveness of the proposed numerical scheme and its good potential for large-scale simulation of GTSFDEs., Comment: 23 pages, 10 tables, 1 figure. Make several corrections again and have been submitted to a journal at Sept. 20, 2019. Version 2: Make some necessary corrections and symbols, 13 Jan. 2020. Revised manuscript has been resubmitted to journal

INTRODUCTION:Gastric cancer represents a great challenge for healthcare providers and requires a multidisciplinary treatment approach in which surgery plays a major role. Minimally invasive surgery has been progressively developed, first with the advent of laparoscopy and recently with the spread of robotic surgery, but a number of issues are currently being debated, including the limitations in performing an effective extended lymph node dissection, the real advantages of robotic systems, the role of laparoscopy for Advanced Gastric Cancer, the reproducibility of a total intracorporeal technique and the oncological results achievable during long-term follow-up. METHODS AND ANALYSIS:A multi-institutional international database will be established to evaluate the role of robotic, laparoscopic and open approaches in gastric cancer, comprising of information regarding surgical, clinical and oncological features. A chart review will be conducted to enter data of participants with gastric cancer, previously treated at the participating institutions. The database is the first of its kind, through an international electronic submission system and a HIPPA protected real time data repository from high volume gastric cancer centres. ETHICS AND DISSEMINATION:This study is conducted in compliance with ethical principles originating from the Helsinki Declaration, within the guidelines of Good Clinical Practice and relevant laws/regulations. A multicentre study with a large number of patients will permit further investigation of the safety and efficacy as well as the long-term outcomes of robotic, laparoscopic and open approaches for the management of gastric cancer. TRIAL REGISTRATION NUMBER:NCT02325453; Pre-results.

INTRODUCTION:Gastric cancer represents a great challenge for healthcare providers and requires a multidisciplinary treatment approach in which surgery plays a major role. Minimally invasive surgery has been progressively developed, first with the advent of laparoscopy and recently with the spread of robotic surgery, but a number of issues are currently being debated, including the limitations in performing an effective extended lymph node dissection, the real advantages of robotic systems, the role of laparoscopy for Advanced Gastric Cancer, the reproducibility of a total intracorporeal technique and the oncological results achievable during long-term follow-up. METHODS AND ANALYSIS:A multi-institutional international database will be established to evaluate the role of robotic, laparoscopic and open approaches in gastric cancer, comprising of information regarding surgical, clinical and oncological features. A chart review will be conducted to enter data of participants with gastric cancer, previously treated at the participating institutions. The database is the first of its kind, through an international electronic submission system and a HIPPA protected real time data repository from high volume gastric cancer centres. ETHICS AND DISSEMINATION:This study is conducted in compliance with ethical principles originating from the Helsinki Declaration, within the guidelines of Good Clinical Practice and relevant laws/regulations. A multicentre study with a large number of patients will permit further investigation of the safety and efficacy as well as the long-term outcomes of robotic, laparoscopic and open approaches for the management of gastric cancer. TRIAL REGISTRATION NUMBER:NCT02325453; Pre-results.

Bifurcations or related topics in two quantum systems are studied. The first system is a hydrogen ion in parallel electric and magnetic fields. I study cross section of electron detachment by a laser pulse. Effects of bifurcations of electronic periodic orbits on the cross section are demonstrated. We obtained a semiclassical and an approximate quantum-mechanical formula for the cross section. The results given by these theoretical formulas agree very well with numerical results of the exact quantum theory. The second system I study is a hydrogen atom, also in parallel electric and magnetic fields. I study a numerical method for pure quantum-mechanical calculation of photoabsorption spectrum of the atom to a state near the ionization limit. In solving the Schrodinger equation directly, the Hamiltonian operator is represented by a matrix in terms of the so-called Sturmiam basis functions. The matrix is symmetric and sparse. The inverted and shifted method for solving eigenvalue problem of a matrix, and the Lonczos method for tridiagonalization of a matrix are employed.

Bifurcations or related topics in two quantum systems are studied. The first system is a hydrogen ion in parallel electric and magnetic fields. I study cross section of electron detachment by a laser pulse. Effects of bifurcations of electronic periodic orbits on the cross section are demonstrated. We obtained a semiclassical and an approximate quantum-mechanical formula for the cross section. The results given by these theoretical formulas agree very well with numerical results of the exact quantum theory. The second system I study is a hydrogen atom, also in parallel electric and magnetic fields. I study a numerical method for pure quantum-mechanical calculation of photoabsorption spectrum of the atom to a state near the ionization limit. In solving the Schrodinger equation directly, the Hamiltonian operator is represented by a matrix in terms of the so-called Sturmiam basis functions. The matrix is symmetric and sparse. The inverted and shifted method for solving eigenvalue problem of a matrix, and the Lonczos method for tridiagonalization of a matrix are employed.