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

Abstract

A major objective of the study described in the present paper is to achieve a high accuracy in the time integration of transient convection-diffusion problems and to eventually combine this with new methods for space discretization, such as meshless methods. In this way, a uniformly high-order accurate methodology could be made available for the numerical solution of convection-diffusion problems. Both Padé approximations of the exponential function and Runge-Kutta methods are considered for deriving multi-stage schemes involving first time derivatives only, thus easier to implement than standard Taylor-Galerkin schemes which incorporate second and third time derivatives. After a brief discussion of the stability and accuracy properties of the stability and accuracy properties of the multi-stage schemes and the presentation of illustrative examples, the paper closes with some considerations on the coupling of high-order accurate temporal schemes and meshless methods for the spatial representation.