Numerical Modeling of Pollutant Transport: Results and Optimal Parameters

Abstract

In this work, we used three finite difference schemes to solve 1D and 2D convective diffusion equations. The three methods are the Kowalic–Murty scheme, Lax–Wendroff scheme, and nonstandard finite difference (NSFD) scheme. We considered a total of four numerical experiments; in all of these cases, the initial conditions consisted of symmetrical profiles. We looked at cases when the advection velocity was much greater than the diffusion of the coefficient and cases when the coefficient of diffusion was much greater than the advection velocity. The dispersion analysis of the three methods was studied for one of the cases and the optimal value of the time step size k, minimizing the dispersion error at a given value of the spatial step size. From our findings, we conclude that Lax–Wendroff is the most efficient scheme for all four cases. We also show that the optimal value of k computed by minimizing the dispersion error at a given value of a spacial step size gave the lowest l2 and l∞ errors.

Comparison of the Performance of New and Traditional Numerical Methods for Long-Term Simulations of Heat Transfer in Walls with Thermal Bridges

“…Energies 2023, 16, 4604 2 of 27 EN ISO 13789 [8] is a commonly used standard for calculating transmission heat transfer through a building envelope; which in turn mentions two standards for thermal bridges; EN ISO 10211 [9] provides a structure for thorough calculations of thermal bridges in building construction and EN ISO 14683 [10] provides a simplified approach with default linear thermal transmittance values.…”

“…Innovative analytical approaches [15] exist for spatially homogeneous systems, and mathematicians create and evaluate most numerical techniques for homogeneous cases. However, some analytical solutions for steady-state and transient situations are also available for the onedimensional inhomogeneous system, for example, multilayer systems [16]. Most of the time, these systems are used to determine how much heat is gained or lost through the outside envelope and how much heat is stored inside buildings [3].…”

“…Energies 2023, 16, 4604 3 of 27 Heat transfer and other similar problems are routinely solved by lots of numerical methods. Most of them fall into one of two categories: explicit method or implicit method.…”

“…The widely used explicit algorithms, such as the classical FTCS (forward time central space), execute time steps in a small amount of time and are quite straightforward to parallelize. The problem is that the so-called Courant-Friedrichs-Lewy (CFL) limit constrains their stability domain and causes them to become unstable if the time step size is above this limit [16]. Moreover, since an insulated wall consists of very different materials, the coefficients can change sharply from mesh point to mesh point; thus, this limit is rather tiny and the stiffness ratio is large.…”

Prediction and Optimization of Thermal Loads in Buildings with Different Shapes by Neural Networks and Recent Finite Difference Methods

“…There are rather few exact solutions for even one-dimensional inhomogeneous systems in steady-state or transient scenarios. For instance, multi-layer systems can be solved analytically, and these systems are useful for studying the thermal behavior of buildings, such as how they lose or gain heat [15]. However, the properties of the materials in a structure, such as density, thermal conductivity, and specific heat, can vary widely [16].…”