The numerical analysis of stochastic time-fractional equations exhibits a significantly low-order convergence rate since the limited regularity of model caused by the nonlocal operator and the presence of noise. In this work, we consider stochastic time-fractional equations driven by integrated white noise, where CDt alpha psi(x, t), 0 G alpha G 2 and It gamma W-center dot (x, t), 0 G gamma G 1. We first establish the regularity of the mild solution. Then superlinear convergence rate (E||psi(& centerdot;,tn)-psi n||2)21=O(tau alpha+gamma-alpha d4-2 1-epsilon) with sufficiently small epsilon term in the exponent is established based on the modified twostep backward difference formula methods. Here d represents the spatial dimension, psi n denotes the approximate solution at the n-th time step, and E is the expectation operator. Numerical experiments are performed to verify the theoretical results. To the best of our knowledge, this is the first topic on the superlinear convergence analysis for the stochastic time-fractional equations with integrated white noise.
This paper investigates two spectral volume (SV) methods applied to 2D linear hyperbolic conservation laws on rectangular meshes. These methods utilize upwind fluxes and define control volumes using Gauss-Legendre (LSV) and right-Radau (RRSV) points within mesh elements. Within the framework of Petrov-Galerkin method, a unified proof is established to show that the proposed LSV and RRSV schemes are energy stable and have optimal error estimates in the L2 norm. Additionally, we demonstrate superconvergence properties of the SV method at specific points and analyze the error in cell averages under appropriate initial and boundary discretizations. As a result, we show that the RRSV method coincides with the standard upwind discontinuous Galerkin method for hyperbolic problems with constant coefficients. Numerical experiments are conducted to validate all theoretical findings.
The saturation assumption plays a central role in much of the analysis of a posteriori error estimates and refinement algorithms for adaptive finite element methods.In this work we provide an analysis of this assumption in the simple setting of interpolation.We have proved elsewhere[Bank and Yserentant,Numer.Math.,131:1(2015)]that interpolation error is both reliable and efficient as an a posteriori error estimate.Thus behavior of interpolation error is indicative of the behavior of the error in the exact finite element solution of a PDE as well as any practical a posteriori error estimate that is also reliable and efficient.
In this paper, a numerical method for solving nonlinear stochastic delay differential equations is proposed: two-step Milstein method. The mean square consistent and mean square convergence of the numerical method are studied. Through the relevant derivation, the conditions that the coefficients need to be satisfied when the numerical method is mean-square consistent and mean-square convergent are obtained, and it is proved that the mean-square convergence order of the numerical method is 1. Finally, the theoretical results are verified by numerical experiments.
In this research article,we present convenient analytical-approximate solutions for fluid flow models known as multi-dimensional Navier-Stokes equations containing time-fractional order by using a relatively new analytical method called modified generalized Mittag-Leffler function method.The Caputo fractional derivative is used to describe fractional mathematical formalism.The approximate solutions for five problems are implemented to demonstrate the validity and accuracy of the proposed method.It is also demonstrated that the solutions obtained from our method when α=1 coincide with the exact solutions,this is displayed by using some 2D and 3D plots for each problem.Moreover,the comparison between our outcomes with given exact solutions and results obtained by other methods in the literature besides absolute error is provided in some tables.Additionally,we offer some plots when α has different values to present the effect of fractional order on the solution of each suggested problem.The numerical simulation presented in this work indicates that the proposed method is efficient,reliable,accurate and easy which has less computational ability to give analytical-approximate solution form.So,this method can be extended to implement on different related problems arising in various areas of innovation and research.
Achieving linear complexity is crucial for demonstrating optimal convergence rates in adaptive refinement. It has been shown that the existing linear complexity local refinement algorithm for T-splines generally produces more degrees of freedom than the existing greedy refinement, which lacks linear complexity. This paper introduces a novel greedy local refinement algorithm for analysis-suitable T-splines, which achieves linear complexity and requires fewer control points than existing algorithms with linear complexity. Our approach is based on the observation that confining refinements around each T-junction to a preestablished feasible region ensures the algorithm's linear complexity. Building on this constraint, we propose a greedy optimization local refinement algorithm that upholds linear complexity while significantly reducing the degrees of freedom relative to previous linear complexity local refinement methods.
The rigorous error analysis of a class of serendipity virtual element methods applied to numerically solve semilinear parabolic integro-differential equations on curved domains is the focus of this study. Different from the standard virtual element method, the serendipity virtual element method eliminates all the internal-moment degrees of freedom only under certain conditions of the mesh and the degree of approximation. Consequently, if the interpolation operators are utilized to approximate the nonlinear terms, the implementation of Newton's iteration algorithm can be simplified. Nonhomogeneous Dirichlet boundary conditions are considered in this paper. The strategy of approximating curved domains with polygonal domains is taken into consideration, and to overcome the issue of suboptimal convergence caused by enforcing Dirichlet boundary conditions strongly, Nitsche-based projection method is employed to impose the boundary conditions weakly. For time discretization, Crank-Nicolson scheme incorporating trapezoidal quadrature rule is adopted. Based on the concrete formulation of Nitsche-based projection method, a Ritz-Volterra projection is introduced and its approximation properties are rigorously analyzed. Building upon these approximation properties, error estimates are derived for the fully discrete scheme. Additionally, the extension of the fully discrete scheme to 3D case is also included. Finally, we present two numerical experiments to corroborate the theoretical findings.
Numerical simulation of time-periodic problems is a special area of research, since the time periodicity modifies the problem structure, and then it is desirable to use parallel methods to solve such problems. The classical parareal algorithm for time-periodic problems, which is parallel in time, solving an initial value coarse problem, called the periodic parareal algorithm with initial value coarse problem (PP-IC), usually converges very slowly, and even diverges for wave propagation problems. In this paper, we first present a new PP-IC algorithm based on a diagonalization technique proposed recently. In this new algorithm, we approximate the coarse propagator G in the classical PP-IC algorithm with a head-tail coupled condition such that G can be parallelized using diagonalization in time. We analyze the convergence factors of the diagonalization-based PP-IC algorithm for both the linear and nonlinear cases. Then, we further design and analyze a new parallel-intime algorithm for time-periodic problems by combining the Krylov subspace method with the diagonalization-based PP-IC algorithm to accelerate the convergence. Finally, we also determine an appropriate choice of the parameter alpha in the head-tail coupling condition, and illustrate our theoretical results with several numerical experiments, both for model problems and the realistic application of Maxwell's equations.
The weak convergence analysis plays an important role in error estimates for stochastic differential equations, which concerns with the approximation of the probability distribution of solutions. In this paper, we investigate the weak convergence order of a splitting-up method for stochastic differential equations. We first construct a splitting-up approximation, based on which we also set up a splitting-up numerical solution. We prove both of these two approximation methods are of first order of weak convergence with the help of Malliavin calculus. Finally, we present several numerical experiments to illustrate our theoretical analysis.
The h-version analysis technique developed in [Banjai et al., SIAM J. Numer. Anal., 55 (2017)] for Trefftz discontinuous Galerkin (DG) discretizations of the second order isotropic wave equation is extended to the time-dependent Maxwell equations in anisotropic media. While the discrete variational formulation and its stability and quasi-optimality are derived parallel to the acoustic wave case, the derivation of error estimates in a mesh-skeleton norm requires new transformation stabilities for the anisotropic case. The error estimates of the approximate solutions with respect to the condition number of the coefficient matrices are proved. Furthermore, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent nonhomogeneous Maxwell equations. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.
In this paper, we present a posteriori error estimates of the weak Galerkin finite element method for the steady-state Poisson-Nernst-Planck equations.The a posteriori error estimators for the electrostatic potential and ion concentrations are constructed.The reliability and efficiency of the estimators are verified by the upper and lower bounds of the energy norm of the error.The a posteriori error estimators are applied to the adaptive weak Galerkin algorithm for triangle, quadrilateral and polygonal meshes with hanging nodes.Finally, numerical results demonstrate the effectiveness of the adaptive algorithm guided by our constructed estimators.
The main purpose of this paper is to give stability analysis and error estimates of the ultra-weak local discontinuous Galerkin (UWLDG) method coupled with a spectral deferred correction (SDC) temporal discretization method up to fourth order, for solving the fourth-order equation. The UWLDG method introduces fewer auxiliary variables than the local discontinuous Galerkin method and no internal penalty terms are required for stability, which is efficient for high order partial differential equations (PDEs). The SDC method we adopt in this paper is based on second-order time integration methods and the order of accuracy is increased by two for each additional iteration. With the energy techniques, we rigorously prove the fully discrete schemes are unconditionally stable. By the aid of special projections and initial conditions, the optimal error estimates of the fully discrete schemes are obtained. Furthermore, we generalize the analysis to PDEs with higher even-order derivatives. Numerical experiments are displayed to verify the theoretical results.
This report presents a series of implicit-explicit (IMEX) variable stepsize algorithms for natural convection equations. The presented method requires a minimally intrusive modification to an existing program, does not add to the computational complexity, and is conceptually simple. Here, IMEX means the nonlinear term is treated fully explicitly, while the remaining terms are treated implicitly. Due to the increasing demand for low memory solvers, the addition of time adaptive can improve the accuracy and efficiency of the algorithms. For the first-order algorithm, we prove the stability of the variable stepsize backward Euler scheme combined with Adams-Bashforth 2 (VSS BE-AB2) and analyze convergence. Then, the stability of Constant Timestep Filtered-BE-AB2 (BE-AB2+F) is proved. Moreover, we construct adaptive algorithms by extending the approach to variable stepsize. Finally, numerical tests confirm the convergence rates of our method and validate the theoretical results.
Resonant tunneling diodes (RTDs) exhibit a distinctive characteristic known as negative resistance. Accurately calculating the tunneling bias energy is indispensable for the design of quantum devices. This paper conducts a thorough investigation into the currentvoltage (I-V) characteristics of RTDs utilizing various numerical methods. Through a series of numerical experiments, we verified that the transfer matrix method ensures robust convergence in I-V curves and proficiently determines the tunneling bias for energy potential functions with discontinuities. Our numerical analysis underscores the significant impact of variations in effective mass on I-V curves, emphasizing the need to consider this effect. Furthermore, we observe that increasing the doping concentration results in a reduction in tunneling bias and an enhancement in peak current. Leveraging the unique features of the I-V curve, we employ shallow neural networks to accurately fit the I-V curves, yielding satisfactory results with limited data.
This paper deals with the numerical solutions of two-dimensional (2D) semi-linear tion term. A high-order compact difference method called I-type basic scheme is developed for solving the equations and it is proved under the suitable conditions that this method has the computational accuracy O(tau 2 + h4 x+ h4y), where tau, hx and hy are the calculation stepsizes of the method in t-, x- and y-direction, respectively. With the above method and Newton linearized technique, a II-type basic scheme is also suggested. Based on the both basic schemes, the corresponding I- and II-type alternating direction implicit (ADI) schemes are derived. Finally, with a series of numerical experiments, the computational accuracy and efficiency of the four numerical schemes are further illustrated.
For complex-valued or quaternionic neural networks, scholars and researchers usually decompose them into real-valued systems. The decomposed real-valued systems are equivalent to original systems. Then, the dynamical behaviors of real-valued systems obtained are investigated, including stability, synchronization, and chaos etc. In this paper, a class of quaternionic neural networks with time-varying delays is investigated. First, by designing a suitable PI controller, synchronization of the considered chaotic system is realized. By using a non-decomposition method and structuring a novel Lyapunov functional, sufficient conditions are derived to guarantee synchronization between the drive-response systems. It is worth mentioning that, unlike other methods, our approach does not require breaking down the quaternionic neural networks into four separate real-valued systems. Furthermore, we demonstrate the practical application of these chaotic quaternionic neural networks with time-varying delays in image encryption and decryption. Based on one sequence of chaotic signal from state trajectory of single quaternion-valued neuron and a new encryption algorithm, the application of chaotic system proposed, that is, image encryption, is researched. The process of image decryption is simply the reverse of the encryption process. Finally, numerical simulation examples are provided to validate the effectiveness of the designed PI controller and performance of image encryption and decryption.
The k-th (k = 3, 4, 5) order backward differential formula (BDFk) is applied to develop the high order energy stable schemes for the molecular beam epitaxial model with slope selection. The numerical schemes are established by combining the convex splitting technique with the k-th order accurate Douglas-Dupont stabilization term in the form of STk-1 Delta h(phi n - phi n-1). With the help of the new constructed discrete gradient structure of the k-th order explicit extrapolation formula, the stabilized BDFk scheme is proved to preserve energy dissipation law at the discrete levels and unconditionally stable in the energy norm. By using the discrete orthogonal convolution kernels and the associated convolution embedding inequalities, the L2 norm error estimate is established under a weak constraint of time-step size. Numerical simulations are presented to demonstrate the accuracy and efficiency of the proposed numerical schemes.
In this paper, we develop the stabilization-free virtual element method for the Helmholtz transmission eigenvalue problem on anisotropic media. The eigenvalue problem is a variable-coefficient, non-elliptic, non-selfadjoint and nonlinear model. Separating the cases of the index of refraction n =6 1 and n equivalent to 1, the stabilization-free virtual element schemes are proposed, respectively. Furthermore, we prove the spectral approximation property and error estimates in a unified theoretical framework. Finally, a series of numerical examples are provided to verify the theoretical results, show the benefits of the stabilization-free virtual element method applied to eigenvalue problems, and implement the extensions to high-order and high-dimensional cases.
In this paper, we propose and analyze two second-order accurate finite difference schemes for the one-dimensional heat equation with concentrated capacity on a computational domain Omega = [a, b]. We first transform the target equation into the standard heat equation on the domain excluding the singular point equipped with an inner interface matching (IIM) condition on the singular point x = xi is an element of (a, b), then adopt Taylor's expansion to approximate the IIM condition at the singular point and apply second-order finite difference method to approximate the standard heat equation at the nonsingular points. This discrete procedure allows us to choose different grid sizes to partition the two sub-domains [a, xi] and [xi, b], which ensures that x = xi is a grid point, and hence the proposed schemes can be generalized to the heat equation with more than one concentrated capacities. We prove that the two proposed schemes are uniquely solvable. And through in-depth analysis of the local truncation errors, we rigorously prove that the two schemes are second-order accurate both in temporal and spatial directions in the maximum norm without any constraint on the grid ratio. Numerical experiments are carried out to verify our theoretical conclusions.
In this paper, we propose a pressure-robust weak Galerkin (WG) finite element scheme to solve the Stokes-Darcy problem. To construct the pressure-robust numerical scheme, we use the divergence-free velocity reconstruction operator to modify the test function on the right side of the numerical scheme. This numerical scheme is easy to implement because it only need to modify the right side. We prove the error between the velocity function and its numerical solution is independent of the pressure function and viscosity coefficient. Moreover, the errors of the velocity function reach the optimal convergence orders under the energy norm, as validated by both theoretical analysis and numerical results.
