[en] It is well-known that reaction–diffusion systems are used to describe the pattern formation models. In this paper, we will investigate the pattern formation generated by the fractional reaction–diffusion systems. We first explore the mathematical mechanism of the pattern by applying the linear stability analysis for the fractional Gierer–Meinhardt system. Then, an efficient high-precision numerical scheme is used in the numerical simulation. The proposed method is based on an exponential time differencing Runge–Kutta method in temporal direction and a Fourier spectral method in spatial direction. This method has the advantages of high precision, better stability, and less storage. Numerical simulations show that the system control parameters and fractional order exponent have decisive influence on the generation of patterns. Our numerical results verify our theoretical results. (paper)

[en] General numerical methods for ordinary differential equations (ODE) initial-value problems are surveyed, with emphasis on second-order ODE's. Issues include truncation and roundoff error, stability, and starting/stopping. For nonstiff systems, predictor-corrector Adams methods, with variable step and order, are best overall. (author)

[en] We present a regularization method for solving nonlinear ill-posed problems by applying the family of Runge–Kutta methods to an initial value problem, in particular, to the asymptotical regularization method. We prove that the developed iterative regularization method converges to a solution under certain conditions and with a general stopping rule. Some particular iterative regularization methods are numerically implemented. Numerical results of the examples show that the developed Runge–Kutta-type regularization methods yield stable solutions and that particular implicit methods are very efficient in saving iteration steps

[en] In this Letter a new explicit fourth-order seven-stage Runge-Kutta method with a combination of minimal dispersion and dissipation error and maximal accuracy and stability limit along the imaginary axes, is developed. This method was produced by a general function that was constructed to satisfy all the above requirements and, from which, all the existing fourth-order six-stage RK methods can be produced. The new method is more efficient than the other optimized methods, for acoustic computations

[en] Switching the magnetization of a magnetic bit through flipping of soliton offers the possibility of developing a new innovative approach for data storage technologies. The spin dynamics of a site-dependent ferromagnet with antisymmetric Dzyaloshinskii-Moriya interaction is governed by a generalized inhomogeneous higher order nonlinear Schroedinger equation. We demonstrate the magnetization reversal through flipping of soliton in the ferromagnetic medium by solving the two coupled evolution equations for the velocity and amplitude of the soliton using the fourth order Runge-Kutta method numerically. We propose a new approach to induce the flipping behaviour of soliton in the presence of inhomogeneity by tuning the parameter associated with Dzyaloshinskii-Moriya interaction which causes the soliton to move with constant velocity and amplitude along the spin lattice. (author)

[en] A large number of numerical schemes have been developed for the integration of the hyperbolic system of partial differential equations (PDEs) arising in the magnetohydrodynamic (MHD) simulation of plasmas. These schemes can be based on either the combined space and time discretization such as the Lax-Wendroff type schemes, or one may perform first a separate space discretization leading to a semidiscretized set of ordinary differential equations (ODEs), which are then separately integrated in time. In this work, a comparative study of two schemes based on simultaneous discretization of space and time (Richtmyer two-step Lax-Wendroff scheme and MacCormack scheme) and one scheme based on centered-space semidiscretization followed by time integration by the fourth-order Runge-Kutta method, is presented. Particular attention is paid to the applicability of the linear stability criteria to the numerical integration of nonlinear MHD equations with geometry and field components of a linear θ-pinch. (author)

[en] In this paper, the numerical-analytical solution for the hyperchaotic Chen system is obtained via the multistage homotopy analysis method (MSHAM). An analytical form of the solution within each time interval is given, which is not possible using standard numerical methods. The numerical results obtained by the MSHAM and the classical fourth-order Runge-Kutta (RK4) method are in complete agreement. Moreover, the residual error for the MSHAM solution is given for each time interval.

[en] A multistep advective predictor has been developed within the context of projection schemes for incompressible flows. The key idea is to integrate with schemes of different order the different regions of the domain. In regions where advection dominates, multistepping yields a considerable benefit. In those regions where viscosity dominates, the scheme reverts naturally to the original one-step scheme. Several examples show savings of the order of 1:3-1:10 as compared with standard projection schemes, even for transient problems. Given that these benefits can be achieved with a very modest change in existing codes, the proposed multistage advective predictor should be widely applicable

[en] For the measurement of the time profile and the contrast information of the ultrashort laser pulse, based on the third-order intensity correlation principle, using optical pulse replication, a measurement method is proposed. Theoretical analysis is made about the measurement method. The simulation was done with split-step Fourier and Runge-Kutta methods. By measuring the pulse with pieces of windows and piecing the windows together, the measuring range can be enlarged. Thus a high resolution and large window measurement is achieved. The pre-pulse and main pulse are separated into different windows to avoid the use of gradient attenuator, and provides high-contrast measurement capability. (authors)

[en] Highlights: • Numerical solutions of Schrödinger equation. • Modified cubic B-splines. • Differential Quadrature method. - Abstract: In this study, an effective differential quadrature method (DQM) which is based on modified cubic B-spline (MCB) has been implemented to obtain the numerical solutions for the nonlinear Schrödinger (NLS) equation. After separating the Schrödinger equation into coupled real value differential equations,we have discretized using DQM and then obtained ordinary differential equation systems. For time integration, low storage strong stability-preserving Runge–Kutta method has been used. Numerical solutions of five different test problems have been obtained. The efficiency and accuracy of the method have been measured by calculating error norms L2 and Linfinity and two lowest invariants I1 and I2. Also relative changes of invariants are given. The newly obtained numerical results have been compared with the published numerical results and a comparison has shown that the MCB-DQM is an effective numerical scheme to solve the nonlinear Schrödinger equation.