Applications of the Newton-Raphson method in a SDFEM for inviscid Burgers equation

In this paper we investigate in detail the applications of the classical Newton-Raphson method in connection with a spacetime finite element discretization scheme for the inviscid Burgers equation in one dimensional space. The underlying discretization method is the socalled streamline diffusion method, which combines good stability properties with high accuracy. The coupled nonlinear algebraic equations thus obtained in each spacetime slab are solved by the generalized Newton-Raphson method. Exploiting the bandstructured properties of the Jacobian matrix, two different algorithms based on the Newton-Raphson linearization are proposed. In a series of examples, we show that in each timestep a quadratic convergence order is attained when the Newton-Raphson procedure applied to the corresponding system of nonlinear equations.