Equivalence of finite element, volume and difference methods used to solve one-dimensional convectiondiffusion problems - Application to a solute transport process

Abstract

This work shows that, by adequately rewriting the one-dimensional convection-diffusion equation in a diffusional form, all the usual numerical procedures, that is, the finite element, finite volume and finite difference procedures, lead to the same and optimal steady-state stencil for numerical solutions. Additionally, a lumped capacitance (mass) matrix first-order time integration technique is presented. The resulting scheme, expressed with a transient, special lumped capacitance matrix (PMGV or prevailing main grid value method), was used to solve the problem of onedimensional transient saturated soil solute transport. A benchmark problem with analytical solution, representing a moving concentration front in a saturated soil was solved by both the diffusional method and by the Two-Step Taylor-Galerkin (TSTG) finite element method. The diffusional (PMGV) method proved itself to be an excellent option for solving advection dominated problems such as soil solute transport problems. The simulation results showed that the explicit PMGV scheme leads to the same accuracy as the TSTG scheme. The TSTG scheme shows numerical dispersion while the diffusional method showed false diffusion. However, the stability region for the PMGV is broader. The PMGV scheme is more effective than the TSTG in problems involving either low Pe numbers or high Pe numbers, if mesh refinement is made.