Dynamically Consistent Nonstandard Finite Difference Methods For Several Classes Of Partial Differential Equations

In the mid–18th century, Euler and D’Alembert proposed partial differential equa-tions to describe the string vibration process, respectively, since then the study on partialdifferential equations has developed. Generally, the solutions of partial differential equa-tions under specific initial and boundary conditions are di?cult to obtain, thus peopleconstruct various numerical methods to give the approximate solutions. As numericalmethods that can correctly reflect the dynamical properties of the continuous equationshave practical significance, the design of numerical methods that can preserve dynamicalproperties of partial differential equations is a topic worth studying.This paper constructs dynamically consistent nonstandard finite difference methodsfor several classes of partial differential equations(systems) with respect to the preserva-tion of the skew–symmetry of the partial differential equation, the positivity, boundedness,temporal and spatial monotonicity of solutions, and the existence and globally asymptot-ical stability of constant steady states.Firstly, a nonstandard finite difference method is proposed for a class of advection–diffusion–reaction equations. Two exact finite difference methods are proposed for thediffusion–free case, that is, the corresponding advection–reaction equation. Due to thecomplexity of exact finite difference methods, a nonstandard finite difference method isderived from one exact finite difference method, and it is proved that this method is capa-ble of preserving the monotonicity and boundedness of solutions, and the local stabilityof constant steady states of the advection–reaction equation. Based on the nonstandardfinite difference method for the advection–reaction equation, a nonstandard finite differ-ence method is proposed for the advection–diffusion–reaction equation. It is proved thatthis method is capable of preserving the positivity and boundedness of solutions, and theexistence of constant steady states of this class of differential equations unconditionally.Secondly, two nonstandard finite difference methods are developed for a class ofgeneralized Fisher–KPP equations. A nonstandard finite difference method is construct-ed for the spatial independent case, and it is proved that this method can preserve themonotonicity and boundedness of the solutions, and the local stability of equilibria of thecorresponding differential equation. Based on this nonstandard finite difference method,two nonstandard finite difference methods are constructed for the generalized Fisher–KPP equation. It is proved that these two methods are capable of preserving the skew–symmetry of this equation, the positivity, boundedness, temporal monotonicity and spa-tial monotonicity of solutions, and the existence of constant steady states. In addition, thesolvability and convergence of these two nonstandard finite difference methods are alsoanalyzed.Thirdly, nonstandard finite difference methods are constructed for a class of Fitz Hugh–Nagumo equations. Though a nonstandard finite difference method which is capable of p-reserving the dynamical properties of the spatial independent equation, nonstandard finitedifference methods are constructed for the Fitz Hugh–Nagumo equation with one spacevariable or two space variables, respectively. It is proved that the proposed methods arecapable of preserving the existence of constant steady states, and the positivity and bound-edness of solutions of the corresponding differential equations unconditionally. Besides,the solvability and convergence of these methods are discussed.Fourthly, a nonstandard finite difference method is proposed for a hepatitis B virusinfection model with spatial diffusion. The solvability of the proposed method is analyzed,and it is proved that this method is able to unconditionally preserve the positivity of thesolution and the existence of constant steady states of the continuous system. Throughconstructing Lyapunov functions, it is established that this method could unconditionallypreserve the globally asymptotical stability of the disease–free steady state and the chronicdisease steady state of the continuous system.Finally, a nonstandard finite difference method is designed for a SIR epidemic modelwith spatial diffusion. The proposed nonstandard finite difference method can be solveduniquely, and this method is capable of preserving the positivity and boundedness of solu-tions, and the existence of constant steady states of the continuous system unconditionally.In addition, through constructing Lyapunov functions, it is established that this method iscapable of preserving the globally asymptotical stability of the disease–free steady stateand the endemic steady state of the continuous system unconditionally.