Unconditionally Positive, Explicit, Fourth Order Method for the Diffusion- and Nagumo-Type Diffusion–Reaction Equations

Kovács, Endre; Majár, János; Saleh, Mahmoud

doi:10.1007/s10915-023-02426-9

Cited by 5 publications

(13 citation statements)

References 59 publications

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“…9. The first two stages of the three-stage Constant-Linear-Quadratic-neighbor (CLQ) method [25] are the same as those of the LNe method, with the exception that the second LNe stage has to be made with not only a full but also a half time step size using (16). If we denote these results using u L i and u…”

Section: The Applied 12 Convex Combination Scheme For the Diffusion E...mentioning

confidence: 99%

“…This iteration can be further repeated in the very same way as in point 10 above. In this way, one can obtain the CLQ3 algorithm (five stages altogether) and the CLQ4 algorithm (six stages altogether) [25].…”

Section: The Applied 12 Convex Combination Scheme For the Diffusion E...mentioning

confidence: 99%

“…The proof of the theorem, as well as some more detailed descriptions of the discretization and the methods can be found in our earlier publications, e.g., [24,25], and the references therein.…”

Section: The Applied 12 Convex Combination Scheme For the Diffusion E...mentioning

confidence: 99%

“…This paper may be seen as a substantial extension of our earlier publication [24], where we performed tests of some positivity-preserving methods with a first or second order of convergence for the linear heat or diffusion equation. Since then, we have constructed third-and fourth-order numerical methods [25] that can solve some nonlinear equations as well, with the property of dynamical consistency. In this research paper, we involve these methods as well and focus on Fisher's equation.…”

Section: Introductionmentioning

confidence: 99%

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Systematic Investigation of the Explicit, Dynamically Consistent Methods for Fisher’s Equation

Khayrullaev,

Omle,

Kovács

2024

Computation

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We systematically investigate the performance of numerical methods to solve Fisher’s equation, which contains a linear diffusion term and a nonlinear logistic term. The usual explicit finite difference algorithms are only conditionally stable for this equation, and they can yield concentrations below zero or above one, even if they are stable. Here, we collect the stable and explicit algorithms, most of which we invented recently. All of them are unconditionally dynamically consistent for Fisher’s equation; thus, the concentration remains in the unit interval for arbitrary parameters. We perform tests in the cases of 1D and 2D systems to explore how the errors depend on the coefficient of the nonlinear term, the stiffness ratio, and the anisotropy of the system. We also measure running times and recommend which algorithms should be used in specific circumstances.

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Section: The Applied 12 Convex Combination Scheme For the Diffusion E...mentioning

confidence: 99%

Section: The Applied 12 Convex Combination Scheme For the Diffusion E...mentioning

confidence: 99%

Section: The Applied 12 Convex Combination Scheme For the Diffusion E...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Systematic Investigation of the Explicit, Dynamically Consistent Methods for Fisher’s Equation

Khayrullaev,

Omle,

Kovács

2024

Computation

Self Cite

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“…In this study, the solutions were found to be highly accurate in solving the concurrent transport equation for the one-dimensional advection-diffusion equation. Other examples can be found in [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The need for the development of HUPFD approaches arise from the limitations observed in numerical methods.…”

Section: Introductionmentioning

confidence: 99%

Developing Higher-Order Unconditionally Positive Finite Difference Methods for the Advection Diffusion Reaction Equations

Ndou,

Dlamini,

Jacobs

2024

Axioms

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This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space.

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Revealing Hidden Features of Chaotic Systems Using High-Performance Bifurcation Analysis Tools Based on CUDA Technology

Rybin,

Butusov,

Shirnin

et al. 2024

Int. J. Bifurcation Chaos

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Bifurcation analysis is an essential tool in nonlinear dynamics. Bifurcation diagrams help to discover subtle features of investigated dynamics such as chaotic and periodic regimes, hidden attractors and fixed points. However, plotting high-resolution bifurcation diagrams can be a computationally challenging task, especially in multiparametric evaluation. It should be noted that the bifurcation analysis is a task with natural parallelism and thus can be efficiently solved using hybrid central-graphics processing architectures. In this paper, we propose an advanced algorithm and special software for plotting bifurcation diagrams using calculations accelerated by graphics processing unit in combination with a highly efficient semi-implicit ordinary differential equation solver. Time series processing is based on the extraction of amplitude and phase features for the density-based spatial clustering of applications with noise to determine oscillation periodicity. We showcase the features of the application of proposed solutions on a set of test chaotic systems. The performance of the analysis algorithms is investigated in comparison with conventional solutions based on central processing unit and several approaches known from the literature. We explicitly show that the proposed algorithm outperforms known solutions in both calculation speed and precision.

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Unconditionally Positive, Explicit, Fourth Order Method for the Diffusion- and Nagumo-Type Diffusion–Reaction Equations

Cited by 5 publications

References 59 publications

Systematic Investigation of the Explicit, Dynamically Consistent Methods for Fisher’s Equation

Systematic Investigation of the Explicit, Dynamically Consistent Methods for Fisher’s Equation

Developing Higher-Order Unconditionally Positive Finite Difference Methods for the Advection Diffusion Reaction Equations

Revealing Hidden Features of Chaotic Systems Using High-Performance Bifurcation Analysis Tools Based on CUDA Technology

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