Quasi-implicit and two-level explicit finite-difference procedures for solving the one-dimensional advection equation

Abstract

A variety of explicit and implicit algorithms has been studied dealing with the solution of the one-dimensional advection equation. These schemes are based on the weighted finite difference approximations. This solution approach proves to be very effective. This procedure is carried out in many disciplines including hydrology, oceanography and meteorology. The main idea behind the finite-difference methods for obtaining the solution of a given partial differential equation is to approximate the derivatives appearing in the equation by a set of values of the function at a selected number of points. The most usual way to generate these approximations is through the use of Taylor series. The basis of analysis of the finite-difference equations considered here is the modified equivalent partial differential equation approach. It is worth noting that from the truncation error of the modified equivalent equation, it is possible to eliminate the dominant error terms associated with the finite-difference equations that contain weights, thus leading to more accurate techniques. Quasi-implicit and two-level explicit finite-difference techniques are considered in this article to approximately solve the one-dimensional advection problem. Stability of the techniques are investigated using the von Neumann stability analysis. The results of a numerical experiment are presented, and the accuracy and central processor unit (CPU) time needed are discussed and compared. Numerical tests show that the new third-order or fourth-order methods are more accurate than either of the second-order techniques.