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This research devises a rapid and effective numerical approach tailored to a three-dimensional integro-partial differential equation that encompasses three weakly singular kernels. The second-order convolution quadrature rule is employed to approximate the Riemann-Liouvile integral term in time, and the compact difference scheme is utilized in space. Moreover, the Crank-Nicolson alternating direction implicit method is adopted. The discrete energy method is then used to rigorously prove the solvability, convergence, and stability of the constructed scheme. The convergence order is proved with order O(τ2+hx4+hy4+hz4), where τ represents the time step, and hx, hy, hz are the step sizes in the x, y, z directions, respectively. The ADI algorithm is capable of converting the three-dimensional problem under discussion in this paper into the continuous solution of three one-dimensional problems, thereby circumventing the necessity of solving linear equations with large sparse matrices serving as coefficient matrices. This approach remarkably curtails the computational time. Finally, some numerical examples are provided to verify the accuracy of the theoretical analysis.

A large family of paradigmatic models arising in the area of image/signal processing, machine learning and statistics regression can be boiled down to consensus optimization. This paper is devoted to a class of consensus optimization by reformulating it as monotone plus skew-symmetric inclusion. We propose a distributed optimization method by deploying the algorithmic framework of generalized alternating direction implicit method. Under some mild conditions, the proposed method converges globally. Furthermore, the preconditioner is exploited to expedite the efficiency of the proposed method. Numerical simulations on sparse logistic regression are implemented by variant distributed fashions. Compared to some state-of-the-art methods, the proposed method exhibits appealing numerical performances, especially when the relaxation factor approaches to zero.

This work presents a second-order in time variable step BDF2 ADI orthogonal spline collocation (OSC) scheme for the three-dimensional nonlinear fractional evolution equation. Non-uniform meshes can overcome the difficulties arising from the singularity of the exact solution at t = 0 . The time-stepped second-order backward differentiation formula (BDF2) is utilised in the temporal direction, and the spatial direction combines the alternating direction implicit (ADI) method and the arbitrary order OSC method. Then we prove the stability of the numerical scheme and give error estimates. The theoretical analysis shows that the proposed numerical scheme based on variable time step can solve the singularity problem effectively. Finally, numerical examples are given to verify our theoretical analysis.

A new method is formulated and analyzed for the approximate solution of a two-dimensional time-fractional diffusion-wave equation. In this method, orthogonal spline collocation is used for the spatial discretization and, for the time-stepping, a novel alternating direction implicit method based on the Crank–Nicolson method combined with the L 1 -approximation of the time Caputo derivative of order α ∈ ( 1 , 2 ) . It is proved that this scheme is stable, and of optimal accuracy in various norms. Numerical experiments demonstrate the predicted global convergence rates and also superconvergence.

In this paper, a second-order difference scheme is developed to solve two-dimensional two-sided space distributed-order fractional diffusion equation with variable coefficients. In the spatial direction, a second-order difference scheme is proposed, the distribution-order integral is discretized by the Gauss–Legendre quadrature formula and the space fractional derivative is approximated by the weighted and shifted Grünwald–Letnikov operators. In addition, the time direction is discretized into a second-order difference scheme by the Crank–Nicolson method. Therefore, the main numerical scheme is developed. Furthermore, a small perturbation is added to the main difference scheme to construct an alternating-direction implicit scheme. Also, the stability and convergence of the numerical scheme are proved. Finally, some numerical results are provided to show the accuracy and efficiency of the proposed method.

A nonlinear mathematical model is developed for unsteady groundwater flow in a three-layer heterogeneous aquifer system, comprising a confined aquifer, a covering layer, and a weakly permeable barrier. The model incorporates infiltration and evaporation governed by the M.M. Krylov–S.F. Averyanov law, where evaporation intensity depends on a critical groundwater level. Governing equations are nondimensionalized and solved using the alternating direction implicit (ADI) method with quasi-linearization to treat nonlinearities. Periodic variations in precipitation and evaporation are considered, alongside variable boundary permeabilities. The approach enables realistic simulation of multi-layer aquifer dynamics under diverse climatic and hydrogeological conditions, offering a robust tool for sustainable groundwater management, drought risk assessment, and aquifer protection strategies.

In this paper, we develop an efficient spectral Galerkin method for the three-dimensional (3D) multi-term time-space fractional diffusion equation. Based on the L 2-1 σ formula for time stepping and the Legendre-Galerkin spectral method for space discretization, a fully discrete numerical scheme is constructed and the stability and convergence analyses are rigorously established. The results show that the fully discrete scheme is unconditionally stable and has second-order accuracy in time and optimal error estimation in space. In addition, we give the detailed implementation and apply the alternating-direction implicit (ADI) method to reduce the computational complexity. Furthermore, numerical experiments are presented to confirm the theoretical claims. As an application of the proposed method, the fractional Bloch-Torrey model is also solved.

In this paper, we establish the pricing framework of European options in the presence of the looping contagion risk. First, the looping contagion risk model is transformed into a martingale form and the semi-explicit risk-neutral pricing formula is established for the pricing of European options. Then the pricing partial differential equations (PDEs) are derived and solved by the stable alternating direction implicit (ADI) methods. Moreover, the ADI methods in this paper are proved to be unconditionally stable through Fourier analysis framework. Finally, the Monte Carlo simulations are performed and combined with the numerical solutions of PDEs to compute the desired option prices via the semi-explicit pricing formula. Numerical examples are given to confirm the convergence of the numerical methods and the economic analysis is provided to illustrate the economic effect of the looping contagion risk on option prices.

In this study, the system of 2-D Burgers equations is numerically solved by using alternating direction implicit method. Two model problems are studied to demonstrate the efficiency and accuracy of the alternating direction implicit method. Numerical results obtained by present method are compared with the exact solutions and numerical solutions given by other researchers. It is displayed that the method is unconditionally stable by using the von-Neumann (Fourier) stability analysis method. It is shown that all results are in good agreement with the results given by existing numerical methods in the literature. nema

We discuss the numerical solution of large-scale sparse projected periodic discrete-time Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Low-rank versions of these methods are also presented, which can be used to compute low-rank approximations to the solutions of projected periodic Lyapunov equations in lifted form with low-rank right-hand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discrete-time descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.