Streamline Diffusion Method for a Nonlinear Equation Governing the Spreading of Oil Spills

The paper describes a finite element method that uses a version of the space-time streamline diffusion technique and includes the control of total mass applied to a nonlinear convection--diffusion equation that governs the spreading of oil spills on moving water surfaces. The use of such equation for numerical predictions of the evolution of such spills, although highly desirable to help to lessen their consequences, brings several difficulties. In fact, from the theoretical point of view, the equation presents either parabolic or hyperbolic (in the sense of transport equations) character depending on the solution itself. This is due to the nonlinearity of the diffusion term that can pass from strictly positive to zero and vice-versa depending on the value of the solution. In such {\it a priori} unknown regions, fast transitions may occur, bringing spurious oscillations that may deteriorate the numerical solutions obtained with ordinary algorithms. The performance of the proposed method is compared in controled situations with the corresponding performances of more traditional methods. The results shows clear advantages in its use.