DEVELOPMENT OF A MONOTONIC MULTIDIMENSIONAL ADVECTION-DIFFUSION SCHEME

Abstract

The objective of this study is to present a finite-element advection-diffusion scheme for the steady scalar transport equation. The novelty is the use of two advection-diffusion schemes in combination in a way which ensures the satisfaction of the monotonicity property in their matrix equation. Common to these two fundamental finite-element models is that matrix equations are all classified to be irreducibly diagonal dominant. The resulting M-matrix finite-element method is the method of choice to resolve sharp profiles in the flow. The first finite-element method unconditionally provides monotonic solutions. The gain in the stability is due to the introduction of the upwind information along the local streamline. The second basic scheme is classified as conditionally monotonic and is well suited to predicting lower Peclet number flows. This Petrov-Galerkin finite-element model manifests itself by the use of Legendre polynomials to span finite-element spaces. An inherent feature of this formulation is the orthogonal property, which enables a considerable saving in the numerical evaluation of integral terms. Computational evidence reveals that the Legendre-polynomial finite-element model can provide more accurate solutions in low Peclet number conditions. As the Peclet number is increased to higher values that forbid a monotonic solution, the unconditionally monotonic finite-element model is used to complement the Legendre-polynomial finite-element model. This helps enhance the stability. A combined formulation renders a composite scheme that offers promise to optimize the scheme performance. In order to show that the present composite scheme is computationally efficient, the method needs to be rigorously tested against available analytic results. This composite scheme was found to provide monotonic solutions under high and low Peclet number conditions and provided accurate solutions at less computational cost. Use of this composite scheme promises a wider range of practical problems that can be modeled numerically.