ECOLOGICAL DISASTERS AS A QUASI-ONE-DIMENSIONAL DIFFUSION PROBLEM

Abstract

Accidents during maritime transportation and unplanned emissions of toxic substances have traditionally been the focus of ecologists and merchant ship owners. Oil spill disasters cause significant damage to the environment and have a long-term negative impact on the development of biota. It is also known that the spent toxic substance is buried in highstrength containers on the seabed. However, their long stay in the marine environment (50–70 years) leads to thinning and destruction of the shell, causing cracks through which the substance seeps into the water. In this paper, we have investigated a computer simulation of the substance emissions propagation in seawater, by considering the corresponding models as quasi-one-dimensional diffusion problems. Using central point symmetry, the substance concentration function was reduced to dependence on a single spatial variable, which allowed us to reduce the problem under consideration the solution of one-dimensional partial differential equation with the corresponding Laplace operator on the right-hand side. The basic method for solving problems is the method of finite differences by the second order accuracy, and the calculation tool is open computer algebra system CAS Maxima. The first problem considers the release of a toxic substance from a flooded container at the bottom of the sea, which is modeled by Robin boundary value problem (boundary problem of 3d kind). The presence of a constant source of diffusing impurity at the beginning of numerical integration interval made it possible to use a direct two-step calculation method; the result of the calculations is the time distribution of the concentration on the water surface above the container and in its vicinity for 24 hours. In the second problem considers a liquid substance spill near a shallow bank with a semi-circle form, which is modeled by the Neumann boundary value problem (boundary problem of 2nd kind). Zero impurity flux at both edges of the integration interval led to pick as solution the modified indirect Crank-Nicholson scheme; the result of the calculations is the spatial distribution of the maximum concentration of impurity along the coastline during 24 hours.