A Stable and High Accurate Numerical Simulator for Convection-Diffusion Equation

Abstract

This study presents a simulator to obtain numerical solution of convection-diffusion equation. It includes explicit, fully implicit and semi-implicit time discretization techniques. Although the explicit method is efficient for some simple conduction problems, it needs stability criteria. Otherwise, it is unstable especially for high Courant number and large time step size. When using the explicit method, the time step size is chosen with care to get well-posed numerical solution. This requirement sets a serious constraint for the explicit method because choosing small time step size causes the simulation time to become quite long. Even though the implicit method is stable for large time step size and high Courant number, it leads to numerical dispersion like the explicit method. On the other hand, second-order accurate semi-implicit method significantly reduces numerical dispersion. In addition to time discretization techniques, this simulator contains several space discretization methods such as first-order upstream and UMIST (University of Manchester Institute of Science and Technology) techniques. The proposed numerical simulator is suitable for easily using the different combinations of time and space discretization methods. Secondly, this study is to propose the use of semi-implicit time discretization technique with UMIST space discretization method to minimize numerical dispersion and suppress unphysical oscillation. In spite of the fact that the UMIST method suppresses unphysical oscillation, it causes a small and undesired oscillation at flood front for very large Courant number. Third objective of this study is to propose minor modification on the UMIST method to eliminate this unphysical oscillation.