Exponential Spline Interpolation in Characteristic Based Scheme for Solving the Advective-Diffusion Equation

Abstract

This paper demonstrates the use of shape-preserving exponential spline interpolation in a characteristic based numerical scheme for the solution of the linear advective-diffusion equation. The results from this scheme are compared with results from a number of numerical schemes in current use using test problems in one and two dimensions. These test cases are used to assess the merits of using shape-preserving interpolation in a characteristic based scheme. The evaluation of the schemes is based on accuracy, efficiency, and complexity. The use of the shape-preserving interpolation in a characteristic based scheme is accurate, captures discontinuities, does not introduce spurious oscillations, and preserves the monotonicity and positivity properties of the exact solution. However, fitting exponential spline interpolants to the nodal concentrations is computationally expensive. Exponential spline interpolants were also fitted to the integral of the concentration profile. The integral of the concentration profile is a smoother function than the concentration profile. It requires less computational effort to fit an exponential spline interpolant to the integral than the nodal concentrations. By differentiating the interpolant, the nodal concentrations are obtained. This results in a more efficient and more accurate numerical scheme.