Combinations of nonstandard finite difference schemes and composition methods with complex time steps for population models

2016 ◽  
Vol 09 (04) ◽  
pp. 1650049
Author(s):  
Cuicui Liao ◽  
Xiaohua Ding ◽  
Jiuzhen Liang
Keyword(s):  
Finite Difference ◽  
Numerical Method ◽  
Numerical Solutions ◽  
Whooping Cough ◽  
Population Models ◽  
Finite Difference Schemes ◽  
Complex Time ◽  
Composition Methods ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

We propose an efficient numerical method for two population models, based on the nonstandard finite difference (NSFD) schemes and composition methods with complex time steps. The NSFD scheme is able to give positive numerical solutions that satisfy the conservation law, which is a key property for biological population models. The accuracy is improved by using the composition methods with complex time steps. Numerical tests on the plankton nutrient model and whooping cough model are presented to show the efficiency and advantage of the proposed numerical method.

scholarly journals Nonstandard finite-difference schemes for the two-level Bloch model

2018 ◽  
Vol 09 (04) ◽  
pp. 1850033 ◽  
Author(s):  
Marc E. Songolo ◽  
Brigitte Bidégaray-Fesquet
Keyword(s):  
Finite Difference ◽  
Physical Properties ◽  
Exact Solutions ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Bloch Equation ◽  
Finite Difference Schemes ◽  
Splitting Schemes ◽  
Matrix Exponentials ◽  
Nonstandard Finite Difference

In this paper, we present splitting schemes for the two-level Bloch model. After proposing two ways to split the Bloch equation, we show that it is possible in each case to generate exact numerical solutions of the obtained sub-equations. These exact solutions involve matrix exponentials which can be expensive to compute. Here, for [Formula: see text] matrices we develop equivalent formulations which reduce the computational cost. These splitting schemes are nonstandard ones and conserve all the physical properties (Hermicity, positiveness and trace) of Bloch equations. In addition, they are explicit, making effective their implementation when coupled with the Maxwell’s equations.

scholarly journals An unconditionally positive and global stability preserving NSFD scheme for an epidemic model with vaccination

2014 ◽  
Vol 24 (3) ◽  
pp. 635-646 ◽  
Author(s):  
Deqiong Ding ◽  
Qiang Ma ◽  
Xiaohua Ding
Keyword(s):  
Numerical Simulation ◽  
Global Stability ◽  
Difference Equations ◽  
Epidemic Model ◽  
Lyapunov Functions ◽  
Numerical Solutions ◽  
Time Step ◽  
The Stability ◽  
Nsfd Scheme ◽  
Nonstandard Finite Difference

Abstract In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivity of the solutions and the stability of the equilibria, without being restricted by the time step. Our proof of global stability utilizes the method of Lyapunov functions. Numerical simulation illustrates the effectiveness of our results

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