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The aim of this paper is to investigate the high-order approximation of a class of SPDEs with cubic nonlinearity driven by multiplicative noise with the help of the amplitude equations. The highlight of our work is the provision of approximate solutions with enhanced accuracy. Precisely, previous researches primarily concentrated on deriving approximate solutions via the first-order amplitude equations. However, this paper constructs approximate solutions by utilizing both first-order and second-order amplitude equations. And, we rigorously prove that such approximate solutions enjoy improved convergence property. To further illustrate our demonstration intuitively, we apply our main theorem to stochastic Allen–Cahn equation and present a numerical analysis. • The high-order amplitude equations of SPDEs with multiplicative noise is obtained. • The provision of approximate solutions with enhanced accuracy is given. • The approximate solutions enjoy improved convergence property is rigorously proved. • Applications and numerical analysis to stochastic Allen–Cahn equation are presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish some sufficient conditions for the existence of random limit cycle generated by stochastic Liénard equation. Our technique involve Lyapunov functions and truncation arguments. Furthermore, using polar coordinate transformation and rigid rotation, we further established existence (non-existence) of a possible minimal period of random periodic solution of stochastic van der Pol oscillator. • In section 1 and 2, we introduced the work and reviewed the notion of random periodicity. • In section 3, we proved the existence and uniqueness of stable random periodic solutions for stochastic Liénard equations. • In section 4, we proved the existence of possible minimal period for stochastic van der Pol oscillator. • In section 5, we concluded the paper and pointed out a follow-up of this paper in our future publications. [ABSTRACT FROM AUTHOR]

This paper introduces, analyses, and implements efficient time parallel methods for solving the Cahn–Hilliard (CH) equation. Efficient numerical methods for the CH equation are crucial due to its wide range of applications. In particular, simulating the CH equation often requires long computational times to obtain the solution during the phase coarsening stage. Therefore, there is a need to accelerate the computations using parallel methods in the time dimension. We propose linear and nonlinear Parareal methods for the CH equation, depending on the choice of the fine approximation. The effectiveness of our approach is demonstrated through numerical experiments. • Formulation of linear and non-linear Parallel-in-time methods for the Cahn–Hilliard equation. • Convergence analysis of proposed time parallel methods. • Numerical illustration of convergence behaviour of proposed time parallel methods. • Possibility of parallel computing. [ABSTRACT FROM AUTHOR]

This paper presents a Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. The spatial discretization adopts the Galerkin finite element method (FEM), and the temporal discretization employs the CNLF method for linear terms and the semi-implicit method for nonlinear terms. The first step uses Stokes style's scheme, the second step employs the Crank–Nicolson extrapolation scheme, and others apply the CNLF scheme. We establish that the fully discrete scheme is stable and convergent when the time step is less than or equal to a positive constant. Firstly, we show the stability of the scheme by means of the mathematical induction method. Next, we focus on analyzing error estimates of the CNLF method, where the convergence order of the velocity and magnetic field reach second-order accuracy, and the pressure is the first-order convergence accuracy. Finally, the numerical examples demonstrate the optimal error estimates of the proposed algorithm. • Crank–Nicolson leap-frog (CNLF) scheme for the unsteady incompressible magnetohydrodynamics (MHD) equations. • The CNLF method for linear terms and the semi-implicit method for nonlinear terms. • Fully discrete scheme is stable and convergent. • The convergence order of the velocity and magnetic field reach second-order accuracy. [ABSTRACT FROM AUTHOR]

The present paper introduces the distributed-order (DO) Bessel fractional derivative for study of the Euler–Poisson–Darboux (EPD) equation including the spatial Riesz fractional derivative (RFD). For this purpose, we discretize the integral term of the DO fractional derivative and approximate the RFD derivative. We thereafter apply an implicit difference method (IDM) for numerical analysis and solvability of the reduced system from the IDM is discussed. The stability and convergent theorems are stated and the numerical tests are given to show efficiency of the proposed technique and to verify good agreement with the theoretical concepts. • We introduced the concept of the distributed order fractional derivative for the Bessel derivative. • We showed numerical analysis of the DO fractional Bessel derivative and the RFD for the EPD equation on bounded domains. • We applied an implicit difference method for numerical analysis. [ABSTRACT FROM AUTHOR]

In this paper we present a novel way to mathematically frame the concept of ecological memory of plant water stress in the context of root water uptake in unsaturated flow equations. Inspired by recent eco-hydrological papers, we model the water absorption by roots with a non-local sink term, accounting also for a memory effect. In order to model such a memory effect, an integral equation is defined; the main purpose of this work is to provide sufficient conditions on the functions at play for ensuring existence and uniqueness of its solution. Finally, tailored numerical methods are implemented, and numerical simulations are also provided. • We mathematically frame the ecological memory of water stress by plant roots. • We model water absorption by roots by a non-local sink term. • We define an integral equation to model such a memory effect in Richards' equation. • We provide assumptions ensuring existence and uniqueness of its solution. • Tailored numerical methods are implemented, and numerical simulations are provided. [ABSTRACT FROM AUTHOR]

This paper aims to study an H 1 -Galerkin mixed finite element method (FEM) for Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity. The bilinear element and zero-order Raviart–Thomas element ( Q 11 / Q 10 × Q 01 ) are taken as approximation spaces for the original variables p , ϕ and the auxiliary variables u = ∇ p and ψ = ∇ ϕ , respectively. The supercloseness and superconvergence estimates of order O (h 2 + τ 2) are presented for p and ϕ in the H 1 norm and u and ψ in the L 2 norm, which improve the known results that the original variable ϕ can only reach optimal error estimate shown in numerical experiments without theoretical analysis. Here h is the subdivision parameter and τ is the time step. Finally, some numerical results are provided to support the theoretical analysis. • An H 1 -Galerkin mixed FEM for nonlinear KGZ equations is studied. • The superconvergence estimates are presented, which improve the known results. • The results are also valid to the nonconforming FE pair E Q 1 r o t / Q 10 × Q 01 . [ABSTRACT FROM AUTHOR]

In this paper, a two-dimensional variable-coefficients semilinear Sobolev equation under Neumann boundary condition is considered, and a novel Crank–Nicolson type linearized fourth-order block-centered finite difference scheme which preserves mass is developed and analyzed. Under local Lipschitz continuous assumption on the nonlinear source term, second-order temporal and fourth-order spatial accuracy are rigorously proved for the primal scalar variable p , its gradient u and its flux q simultaneously. Then, stability under a rough time stepsize condition is proved following the convergence results. Numerical experiments are presented to confirm the theoretical conclusions and well performance of the proposed scheme. • A linearlized conservative BCFD method is proposed for semilinear Sobolev equations. • Optimal fourth-order error estimates are rigorously proved for related variables. • Stability is given under a rough time stepsize condition. • Numerical examples are tested to show the accuracy and conservation of the scheme. [ABSTRACT FROM AUTHOR]

In this paper, we consider numerical approximations for solving the inductionless magnetohydrodynamic (MHD) equations. By utilizing the scalar auxiliary variable (SAV) approach for dealing with the convective and coupling terms, we propose some first- and second-order schemes for this system. These schemes are linear, decoupled, unconditionally energy stable, and only require solving a sequence of differential equations with constant coefficients at each time step. We further derive a rigorous error analysis for the first-order scheme, establishing optimal convergence rates for the velocity, pressure, current density and electric potential in the two-dimensional case. Numerical examples are presented to verify the theoretical findings and show the performances of the schemes. • Some SAV schemes for the inductionless MHD equations are proposed. • Unconditional and optimal error analysis are given. • Numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Bundle Adjustment (BA) plays a crucial role in 3D reconstruction methods and Structure from Motion (SFM) algorithms, as it involves nonlinear minimization of reprojection error. The efficiency and accuracy of BA solutions are of paramount importance for achieving high-quality 3D reconstructions. The solution of the normal equations derived from nonlinear minimization is typically the most time-consuming step in BA and directly impacts the reconstruction accuracy. In this paper, we propose a Preconditioned Landweber Iteration solver for the normal equations. This solver utilizes a preconditioner and a corresponding relaxation strategy to improve the conditioning of the normal equations, leading to improved convergence speed and accuracy of the iteration. Compared to traditional direct and iterative algorithms, our approach significantly enhances the accuracy and solution time of BA. Experimental results demonstrate that our proposed solver outperforms the Schur complement solver in terms of both speed and accuracy when solving the normal equations. This improvement contributes to further acceleration and accuracy in large-scale 3D reconstruction, particularly for challenging large-scale intensive scenarios. • Proposed Preconditioned Landweber Iteration for efficient Bundle Adjustment (BA). • Enhanced conditioning of normal equations for quicker convergence and accuracy. • Improved BA solution time and accuracy compared to traditional solvers. • Particularly beneficial for large-scale 3D reconstruction and Structure from Motion. • Outperformed Schur complement solver, especially in challenging large-scale problem. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in Hu et al. (2016). We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme. • Error analysis of a fully discrete finite element method for the MHD equations is given. • The proposed scheme is linear, unconditional stability and structure-preserving. • Numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical solution of a class of Riesz space fractional telegraph equation. In the spatial direction, the equations are discretized using the fractional central difference scheme, and an equivalent semi-linear form is obtained. Then, a fourth-order exponential Runge–Kutta method is chosen in the temporal direction. Moreover, an efficient method for calculating the matrix exponent and matrix φ -function is proposed by performing a series of matrix transformations on the coefficient matrix in the semi-linear form, improving the efficiency of the matrix functions calculation. Several numerical experiments show that the convergence order of the scheme is O (h 2 + τ 4) , where h is the space step and τ is the time step. The effectiveness of the scheme is also verified. • New numerical methods are proposed for Riesz space-fractional telegraph equations. • The methods convergent of order 2 in space and order 4 in time. • The matrix functions are calculated efficiently by a series of matrix transformations. [ABSTRACT FROM AUTHOR]