High-order accurate time-stepping schemes for convection-diffusion problems

Abstract

The paper discusses the formulation of high-order accurate time-stepping schemes for transient convection–diffusion problems to be combined with finite element methods of the least-squares type for a stable discretization of highly convective problems. Padé approximations of the exponential function are considered for deriving multi-stage time integration schemes involving first time derivatives only, thus easier to implement in conjunction with C0 finite elements than standard time-stepping schemes which incorporate higher-order time derivatives. After a brief discussion of the stability and accuracy properties of the multi-stage Padé schemes and having underlined the similarity between Padé and Runge–Kutta methods, the paper closes with the presentation of illustrative examples which indicate the effectiveness of the proposed methods.