Qualitatively stable finite difference schemes for advection–reaction equations

A systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection–reaction equations. In the case of hyperbolic fixed-points, a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions. Non-hyperbolic fixed-points are described with the help of two new monotonic properties the construction of schemes, which preserve these properties, being done by nonlocal approximation of nonlinear terms in the reaction terms.     

Conservative finite difference schemes for the generalized Zakharov–Kuznetsov equations

This paper is concerned with the construction of conservative finite difference schemes by means of discrete variational method for the generalized Zakharov–Kuznetsov equations and the numerical solvability of the two-dimensional nonlinear wave equations. A finite difference scheme is proposed such that mass and energy conservation laws associated with the generalized Zakharov–Kuznetsov equations hold. Our arguments are based on the procedure that D. Furihata has recently developed for real-valued nonlinear partial differential equations. Numerical results are given to confirm the accuracy as well as validity of the numerical solutions and then exhibit remarkable nonlinear phenomena of the interaction and behavior of pulse wave solutions.     

Runge–Kutta Lawson schemes for stochastic differential equations

In this paper, we present a framework to construct general stochastic Runge–Kutta Lawson schemes. We prove that the schemes inherit the consistency and convergence properties of the underlying Runge–Kutta scheme, and confirm this in some numerical experiments. We also investigate the stability properties of the methods and show for some examples, that the new schemes have improved stability properties compared to the underlying schemes.

     

A note on a delay Lotka–Volterra competitive system with random perturbations

This note is concerned with a stochastic competitive system with time delays. Under a simple assumption, almost sufficient and necessary conditions for stability in time average and extinction of each population are established. Some numerical simulations are introduced to illustrate the main results.     

Conservative finite difference schemes for the Degasperis–Procesi equation

We consider the numerical integration of the Degasperis–Procesi equation, which was recently introduced as a completely integrable shallow water equation. For the equation, we propose nonlinear and linear finite difference schemes that preserve two invariants associated with the bi-Hamiltonian form of the equation at the same time. We also prove the unique solvability of the schemes, and show some numerical examples.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

Two energy-conserving and compact finite difference schemes for two-dimensional Schrödinger-Boussinesq equations

In this paper, we study two compact finite difference schemes for the Schrödinger-Boussinesq (SBq) equations in two dimensions. The proposed schemes are proved to preserve the total mass and energy in the discrete sense. In our numerical analysis, besides the standard energy method, a “cut-off” function technique and a “lifting” technique are introduced to establish the optimal H¹ error estimates without any restriction on the grid ratios. The convergence rate is proved to be of O(τ² + h⁴) with the time step τ and mesh size h. In addition, a fast finite difference solver is designed to speed up the numerical computation of the proposed schemes. The numerical results are reported to verify the error estimates and conservation laws.

     

Implicit–explicit predictor–corrector schemes for nonlinear parabolic differential equations

In the present paper, a family of predictor–corrector (PC) schemes are developed for the numerical solution of nonlinear parabolic differential equations. Iterative processes are avoided by use of the implicit–explicit (IMEX) methods. Moreover, compared to the predictor schemes, the proposed methods usually have superior accuracy and stability properties. Some confirmation of these are illustrated by using the schemes on the well-known Fisher’s equation.     

Dynamical properties of discrete Lotka–Volterra equations

A discrete version of the Lotka–Volterra differential equations for competing population species is analyzed in detail in much the same way as the discrete form of the logistic equation has been investigated as a source of bifurcation phenomena and chaotic dynamics. It is found that in addition to the logistic dynamics – ranging from very simple to manifestly chaotic regimes in terms of governing parameters – the discrete Lotka–Volterra equations exhibit their own brands of bifurcation and chaos that are essentially two-dimensional in nature. In particular, it is shown that the system exhibits “twisted horseshoe” dynamics associated with a strange invariant set for certain parameter ranges.     

On exact convergence rates for solutions of linear systems of Volterra difference equations†

The asymptotic behaviour of the solution of general linear Volterra non-convolution difference equations on a finite dimensional space, is investigated. It is proved under appropriate assumptions that the solution converges to a limit, which is in general non-trivial. These results are then used to obtain the exact rate of decay of solutions of a class of convolution Volterra difference equations, which have no characteristic roots. In particular, we obtain the exact rate of convergence of the solution of equations whose kernel does not converge exponentially. A useful formula for the weighted limit of a discrete convolution is also obtained.     

A reduced-order finite difference extrapolation algorithm based on POD technique for the non-stationary Navier–Stokes equations

In this paper, a proper orthogonal decomposition (POD) technique is used to establish a reduced-order finite difference (FD) extrapolation algorithm with lower dimensions and sufficiently high accuracy for the non-stationary Navier–Stokes equations, and the error estimates between the reduced-order FD solutions and the classical FD solutions and the implementation for solving the reduced-order FD extrapolation algorithm are provided. Two numerical examples illustrate the fact that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is shown that the reduced-order FD extrapolation algorithm based on POD method is feasible and efficient for solving the non-stationary Navier–Stokes equations.     

Rational Jacobi elliptic solutions for nonlinear differential–difference lattice equations

In this work, we present a direct new method for constructing the rational Jacobi elliptic solutions for nonlinear differential–difference equations, which may be called the rational Jacobi elliptic function method. We use the rational Jacobi elliptic function method to construct many new exact solutions for some nonlinear differential–difference equations in mathematical physics via the lattice equation. The proposed method is more effective and powerful for obtaining the exact solutions for nonlinear differential–difference equations.     

Necessary conditions for ɛ-uniform convergence of finite difference schemes for parabolic equations with moving boundary layers

A grid approximation of the boundary value problem for a singularly perturbed parabolic reaction-diffusion equation is considered in a domain with the boundaries moving along the axis x in the positive direction. For small values of the parameter ? (this is the coefficient of the higher order derivatives of the equation, ? ∈ (0, 1]), a moving boundary layer appears in a neighborhood of the left lateral boundary S ₁ ^L . In the case of stationary boundary layers, the classical finite difference schemes on piece-wise uniform grids condensing in the layers converge ?-uniformly at a rate of O(N ^?1lnN + N ₀), where N and N ₀ define the number of mesh points in x and t. For the problem examined in this paper, the classical finite difference schemes based on uniform grids converge only under the condition N ^?1 + N ₀ ^?1 ? ?. It turns out that, in the class of difference schemes on rectangular grids that are condensed in a neighborhood of S ₁ ^L with respect to x and t, the convergence under the condition N ^?1 + N ₀ ^?1 ≤ ?^1/2 cannot be achieved. Examination of widths that are similar to Kolmogorov’s widths makes it possible to establish necessary and sufficient conditions for the ?-uniform convergence of approximations of the solution to the boundary value problem. These conditions are used to design a scheme that converges ?-uniformly at a rate of O(N ^?1lnN + N ₀).