Accuracy, temporal performance and stability comparisons of discretization methods for the numerical solution of Partial Differential Equations (PDEs) in the presence of steep moving fronts

Abstract

In this paper, several competitive spatial discretization methods recently developed for the convection term are reviewed and analyzed in terms of accuracy, temporal performance, and stability. This analysis was performed using a prediction–correction DAE integrator within the framework of the MOL (Method Of Lines). The discretization methods are classified by Fixed Stencil (FS), Adaptive Stencil (AS) and Weighted Stencil (WS) approaches and their main characteristics are demonstrated via a number of bench marking tests. Of the 14 discretization methods tested, four have been shown to be the most reliable, when we consider the accuracy of the calculation and the computational time required. Application of commonly-used FS methods to convection-dominated problems containing a moving shock results in a relatively short calculation time. However, almost of the FS methods, except for the first-order upwind FS method, are unstable, providing spurious oscillatory profiles (termed the Gibbs phenomena) near the shock. Employing AS methods such as ENO (Essentially Non-Oscillatory) schemes, this Gibbs phenomena disappears. Therefore, AS methods enhance stability and accuracy but are somewhat prohibitive. WS methods (i.e. Weighted ENO schemes), which use a convex combination of candidate stencils weighted by the ENO idea, can reduce the calculation time, and are inheriting of the non-oscillation nature inherent in the AS methods. The ENO and WENO methods are efficient to track a shock and steep moving front and these methods are essential for numerical schemes which use relatively small number of mesh points.