Numerical Solution of the Advection-Diffusion Equation Using the Radial Basis Function

The advection-diffusion equation is a form of partial differential equation. This equation is also known as the transport equation. The purpose of this research is to approximatio the solution of advection-diffusion equation  by numerical approach using radial basis functions network. The approximation is performed by using the multiquadrics basis function. The simulation of the numerical solution is run with the help of the Matlab program. The one-dimensional advection-diffusion equation used is  ∂u/∂t+C ∂u/∂x=D (∂^2 u)/(∂x^2 )  with given initial conditions, boundary conditions, and exact solution u(x,t). The numerical solution approximation using the radial basis function network with dt=0.004 and dx=0.02 produces the value at each discretization point is close to the exact solution. In this study, the smallest error between numerical solution and the exact solution is obtained 2.18339 ×〖10〗^(-10)

Solution of the advection equation using finite difference schemes and the method of characteristics

Numerical models are important engineering tools when considering the prediction of pollution transport in a body of water. Such a prediction is achieved by the solution of the advection-diffusion equation. At present, there exist many numerical techniques which can be used to solve the advection-diffusion equation. The major difficulty when considering undertaking such a simulation, is what method should be used to calculate the advection term. It is now accepted that the appropriate method to follow would involve, splitting up this water quality equation into two separate terms, advection and diffusion. By using this process, each term can be solved individually and the numerical difficulties associated with each term, treated separately. This work discusses the various numerical modelling techniques which can be used to solve the advection term. Two-dimensional finite difference schemes, including QUICKEST, are compared with multi-point method of characteristics techniques. These are analysed in terms of solving advection for various distributions of concentration. The adaptation of these schemes to allow for the use of Courant numbers exceeding unity is also explored. The ultimate aim is to develop a numerical scheme which can be implemented in an industrial model

Numerical solutions of time-space fractional advection-dispersion equations

Summary This paper establishes a difference approximation on time-space fractional advectiondispersion equations. Based on the difference approximation an ideal numerical example has been solved, and the result is compared with the one of the rigorous time fractional advection-dispersion equation and the rigorous space fractional advection-dispersion equation respectively. The results show: when time fractional order parameter γ=1 or space fractional order parameter α=2, the numerical calculation result of the time-space fractional advection-dispersion equations is in accordance with that of the rigorous time fractional advection-dispersion equation or the rigorous space fractional advection-dispersion equation. The variation law of the result with parameter is also similar to them, that is when γ is smaller, diffusion is slower; when α is smaller, diffusion is faster. The simulation calculation for a practical example indicates that time-space fractional advection-dispersion equations can simulate the skewness and the tail of anomalous diffusion. This paper provides a efficient tool for the research of fractional advection-dispersion equations

Mathematical modeling and numerical simulation of a bioreactor landfill using Feel++

In this paper, we propose a mathematical model to describe the functioning of a bioreactor landfill, that is a waste management facility in which biodegradable waste is used to generate methane. The simulation of a bioreactor landfill is a very complex multiphysics problem in which bacteria catalyze a chemical reaction that starting from organic carbon leads to the production of methane, carbon dioxide and water. The resulting model features a heat equation coupled with a non-linear reaction equation describing the chemical phenomena under analysis and several advection and advection-diffusion equations modeling multiphase flows inside a porous environment representing the biodegradable waste. A framework for the approximation of the model is implemented using Feel++, a C++ open-source library to solve Partial Differential Equations. Some heuristic considerations on the quantitative values of the parameters in the model are discussed and preliminary numerical simulations are presented

Spatial eigensolution analysis of discontinuous Galerkin schemes with practical insights for under-resolved computations and implicit LES

The study focusses on the dispersion and diffusion characteristics of discontinuous spectral element methods - specifically discontinuous Galerkin (DG) - via the spatial eigensolution analysis framework built around a one-dimensional linear problem, namely the linear advection equation. Dispersion and diffusion characteristics are of critical importance when dealing with under-resolved computations, as they affect both the numerical stability of the simulation and the solution accuracy. The spatial eigensolution analysis carried out in this paper complements previous analyses based on the temporal approach, which are more commonly found in the literature. While the latter assumes periodic boundary conditions, the spatial approach assumes inflow/outflow type boundary conditions and is therefore better suited for the investigation of open flows typical of aerodynamic problems, including transitional and fully turbulent flows and aeroacoustics. The influence of spurious/reflected eigenmodes is assessed with regard to the presence of upwind dissipation, naturally present in DG methods. This provides insights into the accuracy and robustness of these schemes for under-resolved computations, including under-resolved direct numerical simulation (uDNS) and implicit large-eddy simulation (iLES). The results estimated from the spatial eigensolution analysis are verified using the one-dimensional linear advection equation and successively by performing two-dimensional compressible Euler simulations that mimic (spatially developing) grid turbulence