Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Mesgarani, H.; Kermani, Mahya; Abbaszadeh, Mostafa

doi:10.1108/hff-07-2020-0459

Cited by 8 publications

(5 citation statements)

References 69 publications

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“…This part primarily examines the impact of host migration and spatial variation on disease dynamics. Mathematical software is used for numerical simulation based on the references (Zhenxiang and Linfei, 2022; Erdogan et al , 2019; Mesgarani et al , 2022; Li et al , 2022; Marom et al , 2023b).…”

Section: Numerical Simulationmentioning

confidence: 99%

Numerical analysis of the SIS infectious disease model with spatial heterogeneity

Zhang,

2024

HFF

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Purpose The susceptible-infectious-susceptible (SIS) infectious disease models without spatial heterogeneity have limited applications, and the numerical simulation without considering models’ global existence and uniqueness of classical solutions might converge to an impractical solution. This paper aims to develop a robust and reliable numerical approach to the SIS epidemic model with spatial heterogeneity, which characterizes the horizontal and vertical transmission of the disease. Design/methodology/approach This study used stability analysis methods from nonlinear dynamics to evaluate the stability of SIS epidemic models. Additionally, the authors applied numerical solution methods from diffusion equations and heat conduction equations in fluid mechanics to infectious disease transmission models with spatial heterogeneity, which can guarantee a robustly stable and highly reliable numerical process. The findings revealed that this interdisciplinary approach not only provides a more comprehensive understanding of the propagation patterns of infectious diseases across various spatial environments but also offers new application directions in the fields of fluid mechanics and heat flow. The results of this study are highly significant for developing effective control strategies against infectious diseases while offering new ideas and methods for related fields of research. Findings Through theoretical analysis and numerical simulation, the distribution of infected persons in heterogeneous environments is closely related to the location parameters. The finding is suitable for clinical use. Originality/value The theoretical analysis of the stability theorem and the threshold dynamics guarantee robust stability and fast convergence of the numerical solution. It opens up a new window for a robust and reliable numerical study.

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Section: Numerical Simulationmentioning

confidence: 99%

Numerical analysis of the SIS infectious disease model with spatial heterogeneity

Zhang,

2024

HFF

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“…where X 0 is the initial guess to be taken for (k + 1) th iteration and m indicates the m th component of X 0 . In (16), the following indexes have to be used: 2 ≤ i ≤ N x − 1 for all terms in the inequality; 2 ≤ j ≤ N y − 1 for the term in the left side and for the frst and second terms in the right side; 2 ≤ j ≤ N y − 2 for the third term; and 3 ≤ j ≤ N y − 1 for the fourth term in the right side. Te accuracy of the numerical scheme is calculated by the absolute maximum error formula given by […”

Section: International Journal Of Mathematics and Mathematical Sciencesmentioning

confidence: 99%

“…Various numerical techniques have been employed in the literature to compute numerical solutions for two-dimensional nonlinear ADR equations. For instance, Mesgarani et al [16] used radial basis functions to solve time-dependent nonlinear ADR equations with variable coefcients. Tey transformed the ADR equation into system ordinary diferential equations and used the fourth-order Runge-Kutta method to compute solution of the system.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

Tsega

2024

International Journal of Mathematics and Mathematical Sciences

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The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

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“…The RBF approximations can be constructed through differentiation (DRBF) or integration (IRBF). The governing equations of fluid dynamics have been successfully solved by the DRBF- and IRBF-based methods (Mai-Duy and Tanner, 2005, 2007; Dehghan and Shokri, 2008; Kosec and Šarler, 2008, 2013; Mohebbi et al , 2014; Ngo-Cong et al , 2017; Ebrahimijahan and Dehghan, 2021; Ebrahimijahan et al , 2020, 2022a, 2022b; Abbaszadeh et al , 2022; Mesgarani et al , 2022).…”

Section: Introductionmentioning

confidence: 99%

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

Mai‐Duy

Tien

Strunin

et al. 2023

HFF

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Purpose The purpose of this paper is to present a new discretisation scheme, based on equation-coupled approach and high-order five-point integrated radial basis function (IRBF) approximations, for solving the first biharmonic equation, and its applications in fluid dynamics. Design/methodology/approach The first biharmonic equation, which can be defined in a rectangular or non-rectangular domain, is replaced by two Poisson equations. The field variables are approximated on overlapping local regions of only five grid points, where the IRBF approximations are constructed to include nodal values of not only the field variables but also their second-order derivatives and higher-order ones along the grid lines. In computing the Dirichlet boundary condition for an intermediate variable, the integration constants are used to incorporate the boundary values of the first-order derivative into the boundary IRBF approximation. Findings These proposed IRBF approximations on the stencil and on the boundary enable the boundary values of the derivative to be exactly imposed, and the IRBF solution to be much more accurate and not influenced much by the RBF width. The error is reduced at a rate that is much greater than four. In fluid dynamics applications, the method is able to capture well the structure of steady highly non-linear fluid flows using relatively coarse grids. Originality/value The main contribution of this study lies in the development of an effective high-order five-point stencil based on IRBFs for solving the first biharmonic equation in a coupled set of two Poisson equations. A fast rate of convergence (up to 11) is achieved.

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Application of SPD-RBF method of lines for solving nonlinear advection–diffusion–reaction equation with variable coefficients

Cited by 8 publications

References 69 publications

Numerical analysis of the SIS infectious disease model with spatial heterogeneity

Numerical analysis of the SIS infectious disease model with spatial heterogeneity

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

An effective high-order five-point stencil, based on integrated-RBF approximations, for the first biharmonic equation and its applications in fluid dynamics

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