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A fundamental task in particle-in-cell (PIC) simulations of plasma physics is solving for charged particle motion in electromagnetic fields. This problem is especially challenging when the plasma is strongly magnetized due to numerical stiffness arising from the wide separation in time scales between highly oscillatory gyromotion and overall macroscopic behavior of the system. In contrast to conventional finite difference schemes, we investigated exponential integration techniques to numerically simulate strongly magnetized charged particle motion. Numerical experiments with a uniform magnetic field show that exponential integrators yield superior performance for linear problems (i.e. configurations with an electric field given by a quadratic electric scalar potential) and are competitive with conventional methods for nonlinear problems with cubic and quartic electric scalar potentials. [ABSTRACT FROM AUTHOR]

Recently, an exponential integrator Fourier pseudo-spectral (EIFP) scheme for the Klein–Gordon–Dirac (KGD) equation in the nonrelativistic limit regime has been proposed (Yi et al., 2019). The scheme is fully explicit and numerical experiments show that it is very efficient due to the fast Fourier transform (FFT). However, the authors did not give a strict convergence analysis and error estimate for the scheme. In addition, the scheme did not satisfy time symmetry which is an important characteristic of the exact solution. In this paper, by setting two-level format for Klein–Gordon part and three-level format for Dirac part, respectively, we proposed a new EIFP scheme for the KGD equation with periodic boundary conditions. The new scheme is time symmetric and fully explicit. By using the standard energy method and the mathematical induction, we make a rigorously convergence analysis and establish error estimates without any CFL condition restrictions on the grid ratio. The convergence rate of the proposed scheme is proved to be at the second-order in time and spectral-order in space, respectively, in a generic H m -norm. The numerical experiments are carried out to confirm our theoretical analysis. Because that our error estimates are given under the general H m -norm, the conclusion can easily be extended to two- and three-dimensional problems without the stability (or CFL) condition under sufficient regular conditions. [ABSTRACT FROM AUTHOR]

Highlights • The weights of the Gaussian RBF-FD for a non-equidistant grid with a concentration on the hot zone are brought forward. • To reduce the burdensome, a Krylov method is taken into account. • The proposed computational technique is capable to handle even 6D time-dependent financial PDEs. • Systems of ODEs up to 7.5 million ODEs are handled in a short piece of time using the contributed GRBF-FD approach. Abstract The objective of this work is to present a novel procedure for tackling European multi-asset option problems, which are modeled mathematically in terms of time-dependent parabolic partial differential equations with variable coefficients. To use as low as possible of number computational grid points, a non-uniform grid is generated while a radial basis function-finite difference scheme with the Gaussian function is applied on such a grid to discretize the model as efficiently as possible. To reduce the burdensome for tackling the resulting set of ordinary differential equations, a Krylov method, which is due to the application of exponential matrix function on a vector, is taken into account. The combination of these techniques reduces the computational effort and the elapsed time. Several experiments are brought froward to illustrate the superiority of the new improved approach. In fact, the contributed procedure is capable to tackle even 6D PDEs on a normally-equipped computer quickly and efficiently. [ABSTRACT FROM AUTHOR]

Understanding the connection between spatial patterns in population densities and ecological heterogeneity is significant to the understanding of population dynamics and the governance of species in a given domain. In this paper, the spatiotemporal complexity of prey–predator dynamics with fractional Laplacian derivative and migration is investigated. To provide good guidelines on the choice of parameters, the linear stability analysis of the models was investigated. The fractional Laplacian operator was defined in terms of the left- and right-handed Riemann–Liouville derivative which in turn were approximated by using the fourth-order compact difference scheme, and the resulting system of ODEs was advanced in time using the fourth-order exponential time-differencing Runge–Kutta method. In the simulation experiments, different Turing dynamics such as spots, stripes, and other chaotic patterns are observed. Overall, pattern formation in predator–prey models is useful for understanding the dynamics of ecological systems, predicting the long-term behavior of the system, and studying the impact of environmental factors on the dynamics of the system. • Modification of predator–prey models with different functional responses. • The adoption of viable and reliable numerical techniques. • Local stability analysis. • Pattern formation scenarios in superdiffusive processes. • Numerical experiments in high-dimensional spaces. [ABSTRACT FROM AUTHOR]

A class of generic spatially extended fractional reaction-diffusion systems that modelled predator-prey interactions is considered. The first order time derivative is replaced with the Caputo fractional derivative of order γ  ∈ (0, 1). The local analysis where the equilibrium points and their stability behaviours are determined is based on the adoption of qualitative theory for dynamical systems ordinary differential equations. We derived conditions for Hopf bifurcation analytically. Most significantly, existence conditions for a unique stable limit cycle in the phase plane are determined analytically. Our analytical findings are in agreement with the numerical results presented in one and two dimensions. The system of fractional nonlinear reaction-diffusion equations has demonstrated the usefulness of understanding the dynamics of nonlinear phenomena. [ABSTRACT FROM AUTHOR]

In this paper, some nonlinear space-fractional order reaction-diffusion equations (SFORDE) on a finite but large spatial domain x ∈ [0, L ], x = x ( x , y , z ) and t ∈ [0, T ] are considered. Also in this work, the standard reaction-diffusion system with boundary conditions is generalized by replacing the second-order spatial derivatives with Riemann-Liouville space-fractional derivatives of order α , for 0 < α < 2. Fourier spectral method is introduced as a better alternative to existing low order schemes for the integration of fractional in space reaction-diffusion problems in conjunction with an adaptive exponential time differencing method, and solve a range of one-, two- and three-components SFORDE numerically to obtain patterns in one- and two-dimensions with a straight forward extension to three spatial dimensions in a sub-diffusive (0 < α < 1) and super-diffusive (1 < α < 2) scenarios. It is observed that computer simulations of SFORDE give enough evidence that pattern formation in fractional medium at certain parameter value is practically the same as in the standard reaction-diffusion case. With application to models in biology and physics, different spatiotemporal dynamics are observed and displayed. [ABSTRACT FROM AUTHOR]