Eulerian‐Lagrangian solution of the vertically averaged groundwater transport equation

Abstract

Eulerian‐Lagrangian methods (ELMs) have been developed to reduce smearing and oscillations in numerically generated advection‐dominated transport solutions. Improvements in front propagation characteristics are gained, but new sources of errors are introduced in the tracking step of ELMs. The Eulerian‐Lagrangian least squares collocation method (ELLESCO) is applied to the vertically averaged transport equation, and the ELLESCO method is used to demonstrate how different types of discretization errors affect ELM solutions. The accuracy of ELMs is partially dependent on the accuracy of a tracking step. The tracking integration can be made as accurate as desired by dynamically adjusting the number of second‐order Runge‐Kutta integretion steps per finite difference transport step. However, the accuracy of the final position is dependent on accurate velocities. Inaccurate velocities can lead to poorly located concentration fronts and mass balance errors. Although ELLESCO can extend the spatial and temporal discretization limits of classical methods, concentration fronts must be spread over one and a half elements to avoid oscillations in the solution. Time truncation errors associated with concentration dependent source terms lead to mass balance errors, and source terms should be computed with the Crank‐Nicolson approximation to avoid excessive truncation error. The size of the finite difference transport time step and the magnitude of the local velocity determine the distance between the head and foot of the particle back tracks. This distance, referred to as the Lagrangian spatial resolution, must be less than the characteristic length of areal sources, or distortions will occur in the concentration front. The limit on the Lagrangian spatial resolution is an effective limit on the transport finite difference time step size.