Improved Finite Analytic Methods for Solving Advection-dominated Transport Equation in Highly Variable Velocity Field
Solute transport studies frequently rely on numerical solutions of the classical advection-diffusion equation. Unfortunately, solutions obtained with traditional finite difference and finite element techniques typically exhibit excessive numerical diffusion or spurious oscillation when advection dominates, especially when velocity field is highly variable. One recently developed technique, the finite analytic method, offers an attractive alternative. Finite analytic methods utilize local
... c solutions in discrete elements to obtain the algebraic representations of the governing partial differential equations, thus eliminating the truncation error in the finite difference and the use of approximating functions in the finite element method. The finite analytic solutions have been shown to be stable and numerically robust for advection-dominated transport in heterogeneous velocity fields. However, the existing finite analytic methods for solute transport in multiple dimensions have the following disadvantages. First, the method is computationally inefficient when applied to heterogeneous media due to the complexity of the formulation. Second, the evaluation of finite analytic coefficients is when the Peclet number is large. Third, the method introduces significant numerical diffusion due to inadequate temporal approximation when applied to transient problems. This thesis develops improved finite analytic methods for two-dimensional steady as well as unsteady solute transports in steady velocity fields. For steady transport, the new method exploits the advantages of the existing finite analytic and finite difference methods. The analytically difficult diffusion terms are approximated by finite difference and numerically difficult advection and reaction terms are treated analytically in a local element in deriving the numerical schemes. The new finite analytic method is extended to unsteady transport through application of Laplace transformation. Laplace transformation converts the transient equation to a steady-state expression that can be solved with the steady version of the improved finite analytic method. Numerical inversion of the transformed variables is used to recover solute concentration in the physical space-time domain. The effectiveness and accuracy of the new finite analytic method is demonstrated through stringent test examples of two dimensional steady-state transport in highly variable velocity fields. The results clearly demonstrated that the improved finite analytic methods are efficient, robust and accurate.