Towards a Simple, Fast, Oscillation-free, High-resolution Algorithm for Convection-Dominated Flows: The Position-State Separation Method

A simple, fast, oscillation-free, high-resolution algorithm for convection-dominated flows is proposed. The stability, accuracy and time complexity of this method called the position-state separation method (POSS) are discussed. Several classical numerical experiments in different dimensions and coordinate systems are conducted to demonstrate the excellent performance of the POSS regarding computational cost, robustness, and high-resolution. It is shown that the step time of POSS is not restricted by the CFL condition and that the time complexity of the method when dealing with the convection-dominated flows increases linearly with the number of unknowns.