Novel numerical analysis for nonlinear advection–reaction–diffusion systems

Abstract

<jats:title>Abstract</jats:title><jats:p>In this article, a numerical model for a Brusselator advection–reaction–diffusion (BARD) system by using an elegant numerical scheme is developed. The consistency and stability of the proposed scheme is demonstrated. Positivity preserving property of the proposed scheme is also verified. The designed scheme is compared with the two well-known existing classical schemes to validate the certain physical properties of the continuous system. A test problem is also furnished for simulations to support our claim. Prior to computations, the existence and uniqueness of solutions for more generic problems is investigated. In the underlying system, the nonlinearities depend not only on the desired solution but also on the advection term that reflects the pivotal importance of the study.</jats:p>

A Numerical Study on One-Dimensional Reaction-Diffusion Equation and Fisher’s Equation

“…When r=0, equation 7becomes the diffusion equation. Then for this diffusion equation, using (8), the Explicit Central Difference Scheme (ECDS) becomes, It is equilibrium at = 0, 1. We write,…”

“…Here we use the scheme (8) to find the comparison between the analytical and the numerical solution of Fisher's equation. Hence we find how much error is contained in the scheme.…”

“…Thus we can say the unit interval as an invariant region for this model [17]. We discuss about the diffusion coefficient D and the reaction coefficient/rate of growth r. According to (8), the discretized scheme for (12) will be, Now by numerical experiment we try to understand the effect of the parameter r on the stability condition. For this we present here graphical representation of different situations of the solution for different values of D and r, when ∆ = 0.01 ∆ = 0.1 .…”

“…Therefore, it is reliable to use the numerical solution methods for the precise solutions of the RDE where the initial and boundary data can be implemented in a much more efficient way. So we study finite difference method for the numerical solution of the RDE [6][7][8][9][10][11].…”

A discretization-free deep neural network-based approach for advection-dispersion-reaction mechanisms

“…Even if there are several analytical methods for accomplishing this, they are restricted to particular unique types to reduce errors [16][17][18][19]. Additionally, several numerical techniques such as the boundary element method (BEM), the finite volume method (FVM), the finite difference method (FDM), and the finite element method (FEM) are designed to address constraints associated with the model structure [20][21][22]. These methods often need the domain to be discretized into finite numbers, which is an issue that has an implementation on complex domains and considerably affects the efficiency.…”