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In this study, we examine the perturbed FitzHugh-Nagumo (FHN) neuron model, which incorporates a minor parameter. Owing to its pronounced nonlinearity, devising a numerical scheme for the perturbed FHN model that is both highly accurate and stable, as well as efficient, presents a significant challenge. To overcome this challenge, we have developed an effective numerical scheme by employing the Strang splitting technique, which leverages the Fourier spectral method combined with the second-order Strong Stability Preserving Runge-Kutta (SSP-RK2) method. Additionally, we will perform a thorough and rigorous convergence analysis of our proposed scheme. To conclude, we will conduct extensive numerical experiments to demonstrate the scheme's accuracy and efficiency. [ABSTRACT FROM AUTHOR]

An efficient and easy-to-implement second-order Strang splitting approach is mainly applied to study the scalar Allen-Cahn (AC) equation in this paper. Base on the idea of dimensional splitting, a new time dependent function (called exponential integrating factor) is introduced for the scalar AC equation. Then we propose the Strang splitting approach which is aim to decompose the original equation into linear part and nonlinear part. In particular, the explicit 2-stage strong stability preserving Runge-Kutta(SSP-RK2) method is employed for the nonlinear part. Furthermore, we rigorously demonstrate the maximum principle, energy stability and convergence of the proposed scheme. Various numerical simulations in 2D and 3D are presented to confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

In the present manuscript, the Benjamin-Bona-Mahony-Burgers (BBMB) equation will be handled numerically by Strang time-splitting technique. While applying this technique, collocation method based on quintic B-spline basis functions is applied. In line with our purpose, after splitting the BBM-Burgers equation given with appropriate initial boundary conditions into two subequations containing the derivative in terms of time, the quintic B-spline based collocation finite element method (FEM) for spatial discretization and the suitable finite difference approaches for time discretization is applied to each subequation and hereby two different systems of algebraic equations are obtained. Four test problems are utilized to test the efficiency and reliability of the presented method. The error norms L2 and L∞ with mass, energy, and momentum conservation constants I1,I2 and I3, respectively, are computed. To do a comparison with the other studies in the literature, the newly found approximate solutions are exhibited in both tabular and graphical formats. Also, stability analysis of numerical approach by the von Neumann method is researched. [ABSTRACT FROM AUTHOR]

In this paper an enhanced version of the semi-explicit Particle Finite Element Method for incompressible flow problems is presented. This goal is achieved by improving the solution of the advective sub-problem that results of applying the Strang operator splitting to the Navier–Stokes equations. An acceleration term is taken into account in the solution of the advective step and the Stokes problem. The solution of the advetive step is perfomed using a SPH kernel. Two test cases are solved for validating the methodology and estimating its accuracy. The numerical results demonstrate that the proposed scheme improves the accuracy of the semi-explicit PFEM scheme. [ABSTRACT FROM AUTHOR]

This work is devoted to the analysis of a numerical approximation to a general multi-dimensional kinetic Fokker–Planck (FP) equation with reaction and source terms and subject to specular reflection boundary conditions. This numerical approximation is based on splitting the kinetic FP model into a transport equation in space and a FP diffusive model in the velocity coordinates. The former is discretized by a Kurganov-Tadmor finite-volume scheme, while the latter is approximated by a generalized Chang & Cooper finite-volume method. Time integration is performed by a strong stability-preserving Runge-Kutta method where the reaction and source terms are accommodated with a Strang splitting technique and the use of a Magnus integrator. It is proved that the resulting numerical solution method is conservative and positive preserving, in the case where the continuous model has these properties, and it is second-order accurate in time and in phase space in the L 1 -norm, subject to a CFL condition. Results of numerical experiments are reported that validate these theoretical results. • A rigorous analysis of a class of numerical approximations to a generalized kinetic Fokker-Planck (FP) equation is presented. • The generalized FP equation includes reaction and source terms and is subject to specular reflection boundary equations. • A new numerical splitting scheme is developed that is proven to be positive, conservative, and second order accurate. • Numerical experiments with different scenarios are presented to validate the theoretical findings on the proposed method. [ABSTRACT FROM AUTHOR]

In the present paper, two effective numerical schemes depending on a second-order Strang splitting technique are presented to obtain approximate solutions of the one-dimensional Burgers' equation utilizing the collocation technique and approximating directly the solution by multiquadric-radial basis function (MQ-RBF) method. To show the performance of both schemes, we have considered two examples of Burgers' equation. The obtained numerical results are compared with the available exact values and also those of other published methods. Moreover, the computed L 2 and L ∞ error norms have been given. It is found that the presented schemes produce better results as compared to those obtained almost all the schemes present in the literature. [ABSTRACT FROM AUTHOR]

Our aim is to analyze operator splitting for the fractional Korteweg‐de Vries (KdV) equation, ut=uux+Dαux, α∈[1,2], where Dα=−(−Δ)α/2 is a non‐local operator with α∈[1,2). Under the appropriate regularity of the initial data, we demonstrate the convergence of approximate solutions obtained by the Godunov and Strang splitting. Obtaining the Lie commutator bound, we show that for the Godunov splitting, first order convergence in L2 is obtained for the initial data in H1+α and in case of the Strang splitting, second order convergence in L2 is obtained by estimating the Lie double commutator for initial data in H1+2α. The obtained rates are expected in comparison with the KdV (α=2) case. [ABSTRACT FROM AUTHOR]

In this article, we offer a comparison in terms of computational efficiency between two techniques to avoid order reduction when using Strang method to integrate nonlinear initial boundary value problems with time‐dependent boundary conditions. We see that it is important to consider an exponential method for the integration of the linear nonhomogeneous and stiff part in the technique by Einkemmer et al. so that the latter is comparable in efficiency with that suggested by Alonso et al. Some other advantages of the technique suggested by Alonso et al. are stated in the conclusions. [ABSTRACT FROM AUTHOR]

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results. [ABSTRACT FROM AUTHOR]

The Strang splitting method, formally of order two, can suffer from order reduction when applied to semilinear parabolic problems with inhomogeneous boundary conditions. The recent work [L. Einkemmer and A. Ostermann, SIAM J. Sci. Comput., 37, 2015; SIAM J. Sci. Comput., 38, 2016] introduces a modification of the method to avoid the reduction of order based on the nonlinearity. In this paper we introduce a new correction constructed directly from the flow of the nonlinearity and which requires no evaluation of the source term or its derivatives. The goal is twofold. One, this new modification requires only one evaluation of the diffusion flow and one evaluation of the source term flow at each step of the algorithm and it reduces the computational effort to construct the correction. Second, numerical experiments suggest it is well suited in the case where the nonlinearity is stiff. We provide a convergence analysis of the method for a smooth nonlinearity and perform numerical experiments to illustrate the performances of the new approach. [ABSTRACT FROM AUTHOR]

We approximate the solution for the time dependent Schrödinger equation in two steps. We first use a pseudospectral collocation method that uses samples of the functions on rank-1 or rank-r lattice points. We then get a system of ordinary differential equations in time, which we solve approximately by stepping in time using the Strang splitting method. We prove that the numerical scheme proposed converges quadratically with respect to the time step size, given that the potential is in a Korobov space with the smoothness parameter greater than 9/2. Particularly, we prove that the required degree of smoothness is independent of the dimension of the problem. We demonstrate our new method by comparing with results using sparse grids from [V. Gradinaru, SIAM J. Numer. Anal., 46 (2007), pp. 103--123], with several numerical examples showing the large advantage for our new method and pushing the examples to higher dimensionality. The proposed method has two distinctive features from a numerical perspective: (i) numerical results show the error convergence of time discretization is consistent even for higher-dimensional problems; (ii) by using the rank-1 lattice points, the solution can be efficiently computed (and further time stepped) using only one-dimensional fast Fourier transforms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the convergence property of the Strang splitting for the Gardner equation. We assume that the Gardner equation is locally well-posed and the solution is bounded. We first obtain the regularity properties of the nonlinear divided equation. With these regularity properties, the Strang splitting is proved to converge at the expected rate in L 2. Numerical experiments demonstrate the theoretical result and serve to compare the accuracy and efficiency of different time stepping methods. Finally, the proposed method is applied to simulate the multi solitons collisions for the Gardner equation. [ABSTRACT FROM AUTHOR]

Bu çalışmada, daha önce çözdüğümüz, iki-boyutlu konvektif sınır koşullu erime probleminde, türevlerin bir kısmında açık yöntem kullanırken bir kısmında da kapalı yöntem kullanarak sonlu farklar oluşturulmuştur ve bu denklemlerin çözümü için bir iteratif yöntem geliştirilmiştir. Metod (x, y) koordinatlarında ikinci dereceden doğruluğa sahiptir. Bu metodla elde edilen sonuçlar, önceki araştırmacılar tarafından verilen sonuçlarla tamamen uyumludur.

The aim of this study is to obtain numerical behavior of a one‐dimensional modified Burgers' equation using cubic B‐spline collocation finite element method after splitting the equation with Strang splitting technique. Moreover, the Ext4 and Ext6 methods based on Strang splitting and derived from extrapolation have also been applied to the equation. To observe how good and effective this technique is, we have used the well‐known the error norms L2 and L∞ in the literature and compared them with previous studies. In addition, the von Neumann (Fourier series) method has been applied after the nonlinear term has been linearized to investigate the stability of the method. [ABSTRACT FROM AUTHOR]

This article proposes some higher order splitting-up techniques based on the cubic B-spline Galerkin finite element method in analyzing the Burgers equation model. The strong form of both conservation and diffusion parts of the time-split Burgers equation have been considered in building the Galerkin approach. To integrate the corresponding ODE system, the Crank-Nicolson time discretization scheme is used. The proposed schemes are shown to be unconditionally stable. Three challenging examples have been considered that have changing values of the kinematic viscosity constant of the medium. Moreover, cases of shock waves of severe gradient are solved and compared with the exact solution and the literature. The qualitative and quantitative results demonstrate that our numerical approach has far higher accuracy than rival methods. [ABSTRACT FROM AUTHOR]

Strang splitting is a well established tool for the numerical integration of evolution equations. It allows the application of tailored integrators for different parts of the vector field. However, it is also prone to order reduction in the case of non-trivial boundary conditions. This order reduction can be remedied by correcting the boundary values of the intermediate splitting step. In this paper, three different approaches for constructing such a correction in the case of inhomogeneous Dirichlet, Neumann, and mixed boundary conditions are presented. Numerical examples that illustrate the effectiveness and benefits of these corrections are included. [ABSTRACT FROM AUTHOR]

In this paper, the convergence analysis of operator splitting methods for the Camassa–Holm equation is provided. The analysis is built upon the regularity of the Camassa–Holm equation and the divided equations. It is proved that the solution of the Camassa–Holm equation satisfies the locally Lipschitz condition in H 1 and H 2 norm, which ensures the regularity of the numerical solution. Through the calculus of Lie derivatives, we show that the Lie–Trotter and Strang splitting converge with the expected rate under suitable assumptions. Numerical experiments are presented to illustrate the theoretical result. [ABSTRACT FROM AUTHOR]

We propose and analyze a Strang splitting method for a cubic semilinear Schrödinger equation with forcing and damping terms and subject to periodic boundary conditions. The nonlinear part is solved analytically, whereas the linear part – space derivatives, damping and forcing – is approximated by the exponential trapezoidal rule. The necessary operator exponentials and ϕ -functions can be computed efficiently by fast Fourier transforms if space is discretized by spectral collocation. Under natural regularity assumptions, we first show global existence of the problem in H 4 ( T ) and establish global bounds depending on properties of the forcing. The main result of our error analysis is first-order convergence in H 1 ( T ) and second-order convergence in L 2 ( T ) on bounded time-intervals. [ABSTRACT FROM AUTHOR]

We present the first microscopic electron cooling simulations from first principles with accurate prediction of cooling time. These simulations were performed using our previously developed numerical method, PHAD, which is the first efficient large-scale collisional numerical method in beam physics. The simulation results are benchmarked with the experimental data of the low energy bunched electron cooling of ion beams at the storage ring CSRm at the IMP facility in China. We have accurately considered the nonlinear dynamics in the whole accelerator system in addition to the electron cooling section. As a result, our simulations correctly reproduced cooling times of the experiments from first principles and without any tuning or fitting parameters in the code. [ABSTRACT FROM AUTHOR]

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ordinary differential equation. However, Strang splitting suffers from order reduction which limits its efficiency. This reduction is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost significantly. Numerical examples including a coupled mechanical system illustrate the proved convergence results. [ABSTRACT FROM AUTHOR]

In this work, we extended our earlier study on the solution of two-dimensional heat equation problem by considering a class of time-split finite difference methods. Operator splitting is used as a procedure for computing, some derivatives are computed explicitly and some of them computed implicitly during this procedure. The procedure is second order accurate in time and in (x, y) coordinates. The results of computing by present procedure are in totally compatible with the results obtained previously by other researches. [ABSTRACT FROM AUTHOR]

High-fidelity simulations of ignition, extinction and oscillatory combustion processes are of practical interest in a broad range of combustion applications. Splitting schemes, widely employed in reactive flow simulations, could fail for stiff reaction–diffusion systems exhibiting near-limit flame phenomena. The present work first employs a model perfectly stirred reactor (PSR) problem with an Arrhenius reaction term and a linear mixing term to study the effects of splitting errors on the near-limit combustion phenomena. Analysis shows that the errors induced by decoupling of the fractional steps may result in unphysical extinction or ignition. The analysis is then extended to the prediction of ignition, extinction and oscillatory combustion in unsteady PSRs of various fuel/air mixtures with a 9-species detailed mechanism for hydrogen oxidation and an 88-species skeletal mechanism for n -heptane oxidation, together with a Jacobian-based analysis for the time scales. The tested schemes include the Strang splitting, the balanced splitting, and a newly developed semi-implicit midpoint method. Results show that the semi-implicit midpoint method can accurately reproduce the dynamics of the near-limit flame phenomena and it is second-order accurate over a wide range of time step size. For the extinction and ignition processes, both the balanced splitting and midpoint method can yield accurate predictions, whereas the Strang splitting can lead to significant shifts on the ignition/extinction processes or even unphysical results. With an enriched H radical source in the inflow stream, a delay of the ignition process and the deviation on the equilibrium temperature are observed for the Strang splitting. On the contrary, the midpoint method that solves reaction and diffusion together matches the fully implicit accurate solution. The balanced splitting predicts the temperature rise correctly but with an over-predicted peak. For the sustainable and decaying oscillatory combustion from cool flames, both the Strang splitting and the midpoint method can successfully capture the dynamic behavior, whereas the balanced splitting scheme results in significant errors. [ABSTRACT FROM AUTHOR]

Two formally second-order accurate, semi-implicit, iterative methods for the solution of scalar transport–reaction equations are developed for Direct Numerical Simulation (DNS) of low Mach number turbulent reacting flows. The first is a monolithic scheme based on a linearly implicit midpoint method utilizing an approximately factorized exact Jacobian of the transport and reaction operators. The second is an operator splitting scheme based on the Strang splitting approach. The accuracy properties of these schemes, as well as their stability, cost, and the effect of chemical mechanism size on relative performance, are assessed in two one-dimensional test configurations comprising an unsteady premixed flame and an unsteady nonpremixed ignition, which have substantially different Damköhler numbers and relative stiffness of transport to chemistry. All schemes demonstrate their formal order of accuracy in the fully-coupled convergence tests. Compared to a (non-)factorized scheme with a diagonal approximation to the chemical Jacobian, the monolithic, factorized scheme using the exact chemical Jacobian is shown to be both more stable and more economical. This is due to an improved convergence rate of the iterative procedure, and the difference between the two schemes in convergence rate grows as the time step increases. The stability properties of the Strang splitting scheme are demonstrated to outpace those of Lie splitting and monolithic schemes in simulations at high Damköhler number; however, in this regime, the monolithic scheme using the approximately factorized exact Jacobian is found to be the most economical at practical CFL numbers. The performance of the schemes is further evaluated in a simulation of a three-dimensional, spatially evolving, turbulent nonpremixed planar jet flame. [ABSTRACT FROM AUTHOR]

Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric properties. In the presence of nontrivial boundary conditions, however, splitting methods usually suffer from order reduction and some additional loss of accuracy. For diffusion-reaction equations with inhomogeneous oblique boundary conditions, a modification of the classic second order Strang splitting is proposed that successfully solves the problem of order reduction. The same correction also improves the accuracy of the classic first order Lie splitting. The proposed modification only depends on the available boundary data and, in the case of non-Dirichlet boundary conditions, on the computed numerical solution. Consequently, this modification can be implemented in an efficient way, which makes the modified splitting schemes superior to their classic versions. The framework employed in our error analysis also allows us to explain the fractional orders of convergence that are often encountered for classic Strang splitting. Numerical experiments that illustrate the theory are provided. [ABSTRACT FROM AUTHOR]

A fractional time derivative is introduced into Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differ- ential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coeficients is crucial to ensure both the well-posedness of the system and the computational eficiency of the diffusive ap- proximation. For this purpose, optimization with constraint is shown to be a very eficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme and the diffusive part exactly. Numerical experiments are proposed to assess the eficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation. [ABSTRACT FROM AUTHOR]