A comparison of Eulerian-Lagrangian methods for the solution of transport equation

Abstract

We present an extensive formal and experimental comparison of the accuracy and stability of selected finite-element Eulerian-Lagrangian methods (FE-ELMs) for the solution of the 1D transport equation.

The comparison shows that recent FE-ELMs that use the perspective of integration (rather than the more conventional perspective of interpolation) to treat initial conditions at the feet of the characteristic lines have excelent accuracy properties. Although some "integration" FE-ELMs are only conditionally stable, the associated constraints are expected to be minor for most applications of interest.

Motivation for the comparison was motivated by the on-going development of a new generation of multi-dimensional Eulerian-Lagrangian water quality models, oriented towards surface water applications, and addressing tracers with increasingly complex chemical and biological pathways.

While this motivation may appear remote for a theoretical study of numerical methods, the need for the information now generated is strong, and the practical contributions are significant. Indeed, the need stems from the fear (partly rooted in accummulated experience) that complex water quality models may magnify remaining shortcomings in fundamental aspects of the solution of the transport equation, such as stability, mass conservation, and ability to accurately transport sharp gradients of concentration.

Pratical contributions include: * the recommendation to initiate the systematic replacement of "interpolation" FE-ELMs by "integration" FE-ELMs, as the reference strategy to solve for hydraulic transport in the family of ELA water quality models (ELA, ELAcol, ELAsed, ELAmet, etc.); * the development of basic data and understanding necessary for the informed use of "integration" FE-ELMs; an example is the development of basic data necessary to drive the semi-automatic generation of finite element grids taylor-made for the simulation of transport processes by "integration" FE-ELMs.

Key Words: Transport equation. Eulerian-Lagrangian methods. Stability analysis. Accuracy analysis. Numerical experimentation.