An Improved Theta-method for Systems of Ordinary Differential Equations

Lubuma, Jean M.‐S.; Roux, Alet

doi:10.1080/1023619031000146904

Cited by 23 publications

(30 citation statements)

References 5 publications

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“…As far as the accuracy of the NSFD scheme is concerned, its local truncation error can be obtained from the Taylor expansion and the symmetric nature of the scheme when θ =θ = 1/2, as in the case of the classical theta-method in [40] or [10]. This yields the following result:…”

Section: Theorem 14mentioning

confidence: 92%

“…Since this approximation is essentially different from the weighted average of values at t k and t k+1 , which is applied to all other terms, we use a parameterθ, which need not equal θ. This is the main difference with [40] and [10].…”

Section: Application To the Mseir Modelmentioning

confidence: 99%

“…where Λ is a transition matrix, we can circumvent this difficulty, following the approach in [4,5] or [40]. Setting ε = x − x * , the linearization of the system (1) at x * reads as…”

Section: Theoremmentioning

confidence: 99%

“…The disease-free equilibrium x * = (y * , 0), viewed as a fixed point of the implicit NSFD scheme (40) or (41) on Ω, is GAS. (D 2 ) When z = 0, we obtain the reduced system…”

Section: Theorem 11mentioning

confidence: 99%

“…However, they will be investigated as particular cases of a new family of schemes where the underlying guidelines stated in Section 3 on the nonlocal approximation of nonlinear terms and on complex denominator functions for the discrete derivatives are reinforced. The new family, an extension of the nonstandard theta-method developed in [40], is as follows: given θ,θ ∈ [0, 1], we consider the scheme…”

Section: Application To the Mseir Modelmentioning

confidence: 99%

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Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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This work is the numerical analysis and computational companion of the paper by Kamgang and Sallet (Math. Biosc. 213 (2008), pp. 1-12) where threshold conditions for epidemiological models and the global stability of the disease-free equilibrium (DFE) are studied. We establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability (GAS) of the DFE for general epidemiological models. Then, we design nonstandard finite difference (NSFD) schemes in which the Metzler matrix structure of the continuous model is carefully incorporated and both Mickens' rules (World Scientific, Singapore, 1994) on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way. As a result of these strategies, our NSFD schemes are proved to be dynamically consistent with the continuous model, i.e., they replicate their basic features, including the GAS of the DFE, the linear stability of the endemic equilibrium (EE), the positivity of the solutions, the dissipativity of the system, and its inherent conservation law. The general analysis is made detailed for the MSEIR model for which the NSFD theta method is implemented, with emphasis on the computational aspects such as its convergence, or local truncation error. Numerical simulations that illustrate the theory are provided.

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Section: Theorem 14mentioning

confidence: 92%

Section: Application To the Mseir Modelmentioning

confidence: 99%

“…where Λ is a transition matrix, we can circumvent this difficulty, following the approach in [4,5] or [40]. Setting ε = x − x * , the linearization of the system (1) at x * reads as…”

Section: Theoremmentioning

confidence: 99%

“…The disease-free equilibrium x * = (y * , 0), viewed as a fixed point of the implicit NSFD scheme (40) or (41) on Ω, is GAS. (D 2 ) When z = 0, we obtain the reduced system…”

Section: Theorem 11mentioning

confidence: 99%

Section: Application To the Mseir Modelmentioning

confidence: 99%

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Dynamically consistent nonstandard finite difference schemes for epidemiological models

Anguelov

Dumont

Lubuma

et al. 2014

Journal of Computational and Applied Mathematics

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Non‐standard finite difference schemes for multi‐dimensional second‐order systems in non‐smooth mechanics

Dumont

Lubuma

2006

Math Methods in App Sciences

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SUMMARYThis work is an extension of the paper (Proc. R. Soc. London 2005; 461A:1927-1950) to impact oscillators with more than one degree of freedom. Given the complex and even chaotic behaviour of these nonsmooth mechanical systems, it is essential to incorporate their qualitative physical properties, such as the impact law and the frequencies of the systems, into the envisaged numerical methods if the latter is to be reliable. Based on this strategy, we design several non-standard finite difference schemes. Apart from their excellent error bounds and unconditional stability, the schemes are analysed for their efficiency to preserve some important physical properties of the systems including, among others, the conservation of energy between consecutive impact times, the periodicity of the motion and the boundedness of the solutions. Numerical simulations that support the theory are provided.

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Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation

Garba

Gumel

Hassan

et al. 2015

Applied Mathematics and Computation

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One-dimensional models are important for developing, demonstrating and testing new methods and approaches, which can be extended to more complex systems. We design for a linear delay differential equation a reliable numerical method, which consists of two time splits as follows: (a) It is an exact scheme at the early time evolution −τ ≤ t ≤ τ, where τ is the discrete value of the delay; (b) Thereafter, it is a nonstandard finite difference (NSFD) scheme obtained by suitable discretizations at the backtrack points. It is shown theoretically and computationally that the NSFD scheme is dynamically consistent with respect to the asymptotic stability of the trivial equilibrium solution of the continuous model. Extension of the NSFD to nonlinear epidemiological models and its good performance are tested on a numerical example.

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An Improved Theta-method for Systems of Ordinary Differential Equations

Cited by 23 publications

References 5 publications

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Dynamically consistent nonstandard finite difference schemes for epidemiological models

Non‐standard finite difference schemes for multi‐dimensional second‐order systems in non‐smooth mechanics

Switching from exact scheme to nonstandard finite difference scheme for linear delay differential equation

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