A Taylor–Galerkin method for convective transport problems

TLDR

The paper proposes a method to derive finite element schemes for the scalar convection equation in one or more spatial dimensions. The method uses forward‑time Taylor series expansions up to third order to construct a generalized time‑discretized equation, which is then spatially discretized with the Bubnov–Galerkin finite element method and extended to variable‑coefficient and multi‑dimensional problems. The Taylor–Galerkin schemes achieve high phase accuracy and minimal numerical damping compared to standard Galerkin and Petrov–Galerkin methods when using linear elements and various time‑stepping schemes.

Abstract

A method is described to derive finite element schemes for the scalar convection equation in one or more space dimensions. To produce accurate temporal differencing, the method employs forward-time Taylor series expansions including time derivatives of second- and third-order which are evaluated from the governing partial differential equation. This yields a generalized time-discretized equation which is successively discretized in space by means of the standard Bubnov–Galerkin finite element method. The technique is illustrated first in one space dimension. With linear elements and Euler, leap-frog and Crank–Nicolson time stepping, several interesting relations with standard Galerkin and recently developed Petrov–Galerkin methods emerge and the new Taylor–Galerkin schemes are found to exhibit particularly high phase-accuracy with minimal numerical damping. The method is successively extended to deal with variable coefficient problems and multi-dimensional situations.