Time-accurate solution of stabilized convection-difusion-reaction equations: I. Time and space discretization

Abstract

The paper addresses the development of time‐accurate methods for solving transient convection–diffusion–reaction problems using finite elements. Multi‐stage time‐stepping schemes of high accuracy are used. They are first combined with a Galerkin formulation to briefly recall the time–space discretization. Then spatial stabilization techniques are combined with high‐order time‐stepping schemes. Moreover, a least‐squares formulation is also developed for these high‐order time schemes combined with $C^{0}$ finite elements (in spite of the diffusion operator and without reducing the strong form into a system of first‐order differential equations). The weak forms induced by the SUPG, GLS, SGS and least‐squares formulations are presented and compared. In a companion paper (Part II of this work), the phase and damping properties of the developed schemes are analysed and numerical examples are included to confirm the effectiveness of the proposed methodology for solving time‐dependent convection–diffusion–reaction problems.

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