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A non-standard finite difference method is proposed for an epidemic model which describes the hepatitis B virus infection with spatial dependence. Using the theory of M-matrix, it is shown that the proposed method is unconditionally positive. Moreover, this method preserves all constant steady-state solutions of the corresponding continuous system. Through constructing discrete Lyapunov functions, the globally asymptotical stabilities of the steady-state solutions are fully determined by the basic reproduction number R 0 independent of the time and space step sizes, which coincides with the continuous system. Finally, numerical experiments are provided to illustrate the theoretical results.

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In this note, a non-standard finite difference (NSFD) scheme is proposed for an advection-diffusion-reaction equation with nonlinear reaction term. We first study the diffusion-free case of this equation, that is, an advection-reaction equation. Two exact finite difference schemes are constructed for the advection-reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection-reaction equation is combined with a finite difference space-approximation of the diffusion term to provide a NSFD scheme for the advection-diffusion-reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. f inite difference scheme; non-standard f inite difference scheme; positivity; boundedness As the exact solution of Equation (2) difficult to obtain, numerical scheme that provides a discretization approximation of the exact solution is widely used [1]. For a good numerical scheme, the numerical solution is qualitatively the same as the exact solution, such as the number and stability of the fixed points, the positivity, the boundedness, and the monotonicity of the exact solution. However, traditional numerical schemes are often unable to achieve this, while non-standard finite difference (NSFD) scheme by Mickens et

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In this work, a version of continuous stage stochastic Runge–Kutta (CSSRK) methods is developed for stochastic differential equations (SDEs). First, a general order theory of these methods is established by the theory of stochastic B-series and multicolored rooted tree. Then the proposed CSSRK methods are applied to three special kinds of SDEs and the corresponding order conditions are derived. In particular, for the single integrand SDEs and SDEs with additive noise, we construct some specific CSSRK methods of high order. Moreover, it is proved that with the help of different numerical quadrature formulas, CSSRK methods can generate corresponding stochastic Runge–Kutta (SRK) methods which have the same order. Thus, some efficient SRK methods are induced. Finally, some numerical experiments are presented to demonstrate those theoretical results.