Kejia Pan

Wentao Cai, Dongdong He, Kejia Pan

In this paper, we propose a linearized finite element method (FEM) for solving the cubic nonlinear Schrödinger equation with wave operator. In this method, a modified leap-frog scheme is applied for time discretization and a Galerkin finite element method is applied for spatial discretization. We prove that the proposed method keeps the energy conservation in the given discrete norm. Comparing with non-conservative schemes, our algorithm keeps higher stability. Meanwhile, an optimal error estimate for the proposed scheme is given by an error splitting technique. That is, we split the error into two parts, one from temporal discretization and the other from spatial discretization. First, by introducing a time-discrete system, we prove the uniform boundedness for the solution of this time-discrete system in some strong norms and obtain error estimates in temporal direction. With the help of the preliminary temporal estimates, we then prove the pointwise uniform boundedness of the finite element solution, and obtain the optimal $L^2$-norm error estimates in the sense that the time step size is not related to spatial mesh size. Finally, numerical examples are provided to validate the convergence-order, unconditional stability and energy conservation.

Xiaoqiang Yue, Shi Shu, Xiaowen Xu et al.

The paper investigates a non-intrusive parallel time integration with multigrid for space-fractional diffusion equations in two spatial dimensions. We firstly obtain a fully discrete scheme via using the linear finite element method to discretize spatial and temporal derivatives to propagate solutions. Next, we present a non-intrusive time-parallelization and its two-level convergence analysis, where we algorithmically and theoretically generalize the MGRIT to time-dependent fine time-grid propagators. Finally, numerical illustrations show that the obtained numerical scheme possesses the saturation error order, theoretical results of the two-level variant deliver good predictions, and significant speedups can be achieved when compared to parareal and the sequential time-stepping approach.

Kejia Pan, Xiaoxin Wu, Xiaoqiang Yue et al.

In this paper, an efficient and high-order accuracy finite difference method is proposed for solving multidimensional nonlinear Burgers' equation. The third-order three stage Runge-Kutta total variation diminishing (TVD) scheme is employed for the time discretization, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. Our method is third-order accurate in time and sixth-order accurate in space. The CCD-TVD method treats the nonlinear term explicitly thus it is very efficient and easy to implement. In addition, we prove the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments including both two-dimensional and three-dimensional problems have been conducted to demonstrate the high efficiency and accuracy of the proposed method.

Dongdong He, Kejia Pan, Hongling Hu

In this paper, by using Strang's second-order splitting method, the numerical procedure for the three-dimensional (3D) space fractional Allen-Cahn equation can be divided into three steps. The first and third steps involve an ordinary differential equation, which can be solved analytically. The intermediate step involves a 3D linear fractional diffusion equation, which is solved by the Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique can convert the multidimensional problem into a series of one-dimensional problems, which greatly reduces the computational cost. A fourth-order difference scheme is adopted for discretization of the space fractional derivatives. Finally, Richardson extrapolation is exploited to increase the temporal accuracy. The proposed method is shown to be unconditionally stable by Fourier analysis. Another contribution of this paper is to show that the numerical solutions satisfy the discrete maximum principle under reasonable time step constraint. For fabricated smooth solutions, numerical results show that the proposed method is unconditionally stable and fourth-order accurate in both time and space variables. In addition, the discrete maximum principle is also numerically verified.

Dongdong He, Kejia Pan

In this paper, the coupled fractional Ginzburg-Landau equations are first time investigated numerically. A linearized implicit finite difference scheme is proposed. The scheme involves three time levels, is unconditionally stable and second-order accurate in both time and space variables. The unique solvability, the unconditional stability and optimal pointwise error estimates are obtained by using the energy method and mathematical induction. Moreover, the proposed second-order method can be easily extended into the fourth-order method by using an average finite difference operator for spatial fractional derivatives and Richardson extrapolation for time variable. Finally, numerical results are presented to confirm the theoretical results.

Xin Zhong, Weiwei Ling, Kejia Pan et al.

Traditional three-dimensional magnetotelluric (MT) numerical forward modeling methods, such as the finite element method (FEM) and finite volume method (FVM), suffer from high computational costs and low efficiency due to limitations in mesh refinement and computational resources. We propose a novel neural network architecture named MTAGU-Net, which integrates an attention gating mechanism for 3D MT forward modeling. Specifically, a dual-path attention gating module is designed based on forward response data images and embedded in the skip connections between the encoder and decoder. This module enables the fusion of critical anomaly information from shallow feature maps during the decoding of deep feature maps, significantly enhancing the network's capability to extract features from anomalous regions. Furthermore, we introduce a synthetic model generation method utilizing 3D Gaussian random field (GRF), which accurately replicates the electrical structures of real-world geological scenarios with high fidelity. Numerical experiments demonstrate that MTAGU-Net outperforms conventional 3D U-Net in terms of convergence stability and prediction accuracy, with the structural similarity index (SSIM) of the forward response data consistently exceeding 0.98. Moreover, the network can accurately predict forward response data on previously unseen datasets models, demonstrating its strong generalization ability and validating the feasibility and effectiveness of this method in practical applications.

Kejia Pan, Dongdong He, Hongling Hu

In this paper, we develop an EXCMG method to solve the three-dimensional Poisson equation on rectangular domains by using the compact finite difference (FD) method with unequal meshsizes in different coordinate directions. The resulting linear system from compact FD discretization is solved by the conjugate gradient (CG) method with a relative residual stopping criterion. By combining the Richardson extrapolation and tri-quartic Lagrange interpolation for the numerical solutions from two-level of grids (current and previous grids), we are able to produce an extremely accurate approximation of the actual numerical solution on the next finer grid, which can greatly reduce the number of relaxation sweeps needed. Additionally, a simple method based on the midpoint extrapolation formula is used for the fourth-order FD solutions on two-level of grids to achieve sixth-order accuracy on the entire fine grid cheaply and directly. The gradient of the numerical solution can also be easily obtained through solving a series of tridiagonal linear systems resulting from the fourth-order compact FD discretizations. Numerical results show that our EXCMG method is much more efficient than the classical V-cycle and W-cycle multigrid methods. Moreover, only few CG iterations are required on the finest grid to achieve full fourth-order accuracy in both the $L^2$-norm and $L^{\infty}$-norm for the solution and its gradient when the exact solution belongs to $C^6$. Finally, numerical result shows that our EXCMG method is still effective when the exact solution has a lower regularity, which widens the scope of applicability of our EXCMG method.

Dongdong He, Kejia Pan

This paper concerns the numerical study for the generalized Rosenau-Kawahara-RLW equation obtained by coupling the generalized Rosenau-RLW equation and the generalized Rosenau-Kawahara equation. We first derive the energy conservation law of the equation, and then develop a three-level linearly implicit difference scheme for solving the equation. We prove that the proposed scheme is energy-conserved, unconditionally stable and second-order accurate both in time and space variables. Finally, numerical experiments are carried out to confirm the energy conservation, the convergence rates of the scheme and effectiveness for long-time simulation.

Kejia Pan, Dongdong He, Hongling Hu

In this paper, we develop a new extrapolation cascadic multigrid (ECMG$_{jcg}$) method, which makes it possible to solve 3D elliptic boundary value problems on rectangular domains of over 100 million unknowns on a desktop computer in minutes. First, by combining Richardson extrapolation and tri-quadratic Serendipity interpolation techniques, we introduce a new extrapolation formula to provide a good initial guess for the iterative solution on the next finer grid, which is a third order approximation to the finite element (FE) solution. And the resulting large sparse linear system from the FE discretization is then solved by the Jacobi-preconditioned Conjugate Gradient (JCG) method. Additionally, instead of performing a fixed number of iterations as cascadic multigrid (CMG) methods, a relative residual stopping criterion is used in iterative solvers, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple Richardson extrapolation is used to cheaply get a fourth order approximate solution on the entire fine grid. Test results are reported to show that ECMG$_{jcg}$ has much better efficiency compared to the classical MG methods. Since the initial guess for the iterative solution is a quite good approximation to the FE solution, numerical results show that only few number of iterations are required on the finest grid for ECMG$_{jcg}$ with an appropriate tolerance of the relative residual to achieve full second order accuracy, which is particularly important when solving large systems of equations and can greatly reduce the computational cost. It should be pointed out that when the tolerance becomes smaller, ECMG$_{jcg}$ still needs only few iterations to obtain fourth order extrapolated solution on each grid, except on the finest grid. Finally, we present the reason why our ECMG algorithms are so highly efficient for solving such problems.

Dongdong He, Kejia Pan

In this paper, we construct a semi-implicit finite difference method for the time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck system is a nonlinear system, the numerical method presented in this paper only needs to solve a linear system at each time step, which can be done very efficiently. The rigorous proof for the mass conservation and electric potential energy decay are shown. Moreover, mesh refinement analysis shows that the method is second order convergent in space and first order convergent in time. Finally we point out that our method can be easily extended to the case of multi-ions.