An alternative procedure to avoid excessive numerical diffusion with the ELLAM

In this work, the Eulerian-Lagrangian Localized Adjoint Method (ELLAM) is improved to better reduce the numerical diffusion when solving Advection Diffusion Equations (ADE). The ELLAM preserves the performance of characteristic methods and treats general boundary conditions naturally in their formulations. ELLAM is accurate for large time steps. However, when using several time steps, it is known that they suffer from numerical diffusion [2]. Because the location of the foot of the characteristic does not coincide with a grid point, interpolation is necessary at each time step. When many interpolations are necessary (many time steps), numerical diffusion becomes significant. With higher order interpolation, negative weights will be necessary to avoid numerical diffusion, creating potential spurious oscillations [1]. This phenomenon is reduced for the one-dimensional problem in [4] by combining ELLAM with a moving grid procedure. However, this approach cannot be extended in the same way to 2 or 3 dimensions. Moreover, mass-lumping is often used to avoid oscillations with numerical methods and is known to add excessive numerical diffusion with Eulerian-Lagrangian methods [3]. To reduce this problem, one dimensional ELLAM scheme with a selective lumping has been developed in [3]. In this work, an alternative procedure is used to avoid excessive numerical diffusion with Eulerain Lagrangian methods when performing several time steps. Compared to the selective lumping approach, it adds less numerical diffusion and can be more easily extended to multidimensional problems and unstructured meshes. References [1] Ruan, F. and D. McLaughlin, An investigation of Eulerian-Lagrangian methods for solving heterogeneous advection-dominated transport problems, Water Resources Research (1999), Vol 35, No 8, pp 2359-2373. [2] Russell TF, Numerical dispersion in Eulerian-Lagrangian methods. Computational Methods in Water Resources, Vol. 2, S. M. Hassanizadeh et al., ed., Elsevier, Amsterdam (2002), pp. 963-970. [3] Russell TF, and P. Binning, Oh No, not the Wiggles Again! A Revisit of an Old Problem and a New Approach, Computational Methods in Water Resources, Vol. 1, C. T. Miller et al., ed., Elsevier, Amsterdam, (2004), pp. 483-494. [4] Younes A, An accurate moving grid Eulerian Lagrangian Localized Adjoint Method for solving the one-dimensional variable-coefficient ADE, Int. J. Numer. Meth. Fluids, 45, (2004), 157-178.