Time Discretizations for Scalar Nonlinear Convection–Diffusion Problems

This paper deals with the numerical solution of a scalar nonlinear convection–diffusion equation. The space semi–discretization is carried out by the discontinuous Galerkin finite element method. Several possibilities of the time discretizations are discussed. To obtain a stable and efficient schemes we use implicit scheme for linear terms in combination with suitable explicit scheme for nonlinear terms, which leads to the necessity to solve only linear problem at each time level. Introduction The investigation of convection-diffusion problems is a very topical subject. These problems play an important role in fluid dynamics, hydrology, heat and mass transfer, environmental protection and other physical-related problems at the one side and financial mathematics, image processing at the other side. Our aim is to develop sufficiently robust, efficient and accurate numerical scheme for the solution of nonlinear convection–diffusion problems. To obtain such a scheme we employ the discontinuous Galerkin finite element method for the space semi–discretization and then a suitable time discretization. Discontinuous Galerkin finite element method appears to be very suitable for problems with solutions containing discontinuities or steep gradients. The DGFEM is based on piecewise polynomial discontinuous approximations. There is a number of works devoted to theory and applications of DGFEM. Let us mention, e.g. [3], [5], [13]. The time discretization can be carried out by many ways, see e.g. [8], [9], [14]. Explicit Runge-Kutta methods, which are very popular for solving ordinary differential equations, have high order of accuracy and are simple for implementation. But the resulting scheme is conditionally stable and the time step is drastically limited. In order to avoid this disadvantage, it seems suitable to apply an implicit method, which allows us to use a much longer time step. However, fully implicit DGFEM leads to a large, strongly nonlinear algebraic system, whose solution is rather complicated. This is the reason that we propose here a semi-implicit scheme, which appears quite efficient and robust. The linear diffusion and stabilization terms are treated implicitly, whereas the nonlinear convective terms explicitly in such a way that we do not loose the order of accuracy in time. This approach for multistep methods and self–adjoint operators was analyzed in [1], [2]. The content of this paper is the following. In section Continuous problem, we formulate the initial-boundary value problem for scalar nonlinear convection-diffusion equation in the classical way. Then we reformulate this problem in the weak sence and formulate adequate regularity conditions for proving the error estimates. In section Space discretization, we carry out the space discretization of the problem by DGFEM and achieve the semi-discretized problem. In section Time discretization we discuss the Euler method, the Backward difference formulae, the Runge–Kutta methods and the Time discontinuous Galerkin method. In section Conclusion we introduce some concluding remarks. Continuous problem Let Ω ⊂ IR (d = 2 or 3) be a bounded polyhedral domain and T > 0. (For d = 2 under the concept of a polyhedral domain we mean a polygonal domain.) We set QT = Ω× (0, T ). By WDS'07 Proceedings of Contributed Papers, Part I, 227–232, 2007. ISBN 978-80-7378-023-4 © MATFYZPRESS