Different Methods analysing Convection-Diffusion

Many countries in this world have lack of drinking water. Austria has advantage of drinking water coming from the mountains. This article contains a study focusing on mathematical modelling using different methods for the analysis of groundwater pollution. The distribution of pollution follows the convection-diffusion equation. Therefore different methods ranging from analytical and numerical to alternative approaches dealing with random walk are compared. The analysis of the approaches is mostly done for one and two dimensional case. Introduction In order to analysis the po llution distribution in water of sim ilar circumstances the m athematical equation describing t his be haviour i s a c onvection-diffusion equation. This equation can not only be used to analysis the be haviour of pollution. Also in biology, chem istry and ot her fiel ds of study t his e quation i s im portant. Regarding biology the equation can be use d to predict the development of fur pattern for cats. In chemistry the mixture of different substances follows this equation. In the field of physical modelling and simulation this equation is often c alled heat equation bec ause it describe s the distribution of heat em anating from a source. De spite disci plines in nat ural sciences also the fi nance market uses this equation t o foresee the behaviour of buyers of st ocks. In general the convectiondiffusion equation looks as follows: (1) Equation (1) is a partial differential equation of second or der a nd c ontains tw o dif ferent variables which can be ti me-dependent, position-dependent or simply constant. In t he following we assume that all the variables are constant. The first term of this equation describes a re gular distribution in every di rection. It is similar to spreading of waves after throwi ng in a little stone into water. The variable in th e secon d term of (1) symbolises the velocity field of oriente d movement. Assuming for example a river with a ce rtain flux the n the distribution would be influence by the velocity of the flux. This information will be transformed into t he equation using the variable . To sum it up, t he convection-diffusion equation contains one part de scribing the chaotic movement in all dir ections and an orie nted distribution depending on the circumstances. In the following a fl ux only in xdirection is assumed. This pr oblem description wi ll be a nalysed usi ng three different a pproaches applied in one and two dimensions. 1 Analytical Solution In this case, due to the used initial and boundary conditions, an anal ytical solutio n can be given. The initial condition describes a pollutio n sources which releases all the pollution at ti me without inj ecting any further pollution. B oth so lutions, oneand twodimensional, are used to validate the different methods. One-dimensional. Using the regarded equation is given as follows