A comparison of integration and interpolation Eulerian‐Lagrangian methods

Selected finite element Eulerian‐Lagrangian methods for the solution of the transport equation are compared systematically in the relatively simple context of 1D, constant coefficient, conservative problems. A combination of formal analysis and numerical experimentation is used to characterize the stability and accuracy that results from alternative treatments of the concentrations at the feet of the characteristic lines. Within the methods analyzed, those that approach such treatment with the perspective of ‘integration’ rather than ‘interpolation’ tend to have superior accuracy. Exact integration leads to unconditional stability and excellent accuracy. Quadrature integration leads only to conditional stability, but newly derived criteria show that stability restrictions are relatively mild and should not preclude the usefulness of quadrature integration methods in a range of practical applications. While conclusions cannot be extended directly to multiple dimensions and complex flows and geometries, results should provide useful insight to the development and behaviour of specific Eulerian‐Lagrangian transport models.