A brief review on the convergence to steady state solutions of Euler equations with high-order WENO schemes

Weighted essentially non-oscillatory (WENO) schemes are a class of high-order shock capturing schemes which have been designed and applied to solve many fluid dynamics problems to study the detailed flow structures and their evolutions. However, like many other high-order shock capturing schemes, WENO schemes also suffer from the problem that it can not easily converge to a steady state solution if there is a strong shock wave. This is a long-standing difficulty for high-order shock capturing schemes. In recent years, this non-convergence problem has been studied extensively for WENO schemes. Numerical tests show that the key reason of the non-convergence to steady state is the slight post shock oscillations, which are at the small local truncation error level but prevent the residue to settle down to machine zero. Several strategies have been proposed to reduce these slight post shock oscillations, including the design of new smoothness indicators for the fifth-order WENO scheme, the development of a high-order weighted interpolation in the procedure of the local characteristic projection for WENO schemes of higher order of accuracy, and the design of a new type of WENO schemes. With these strategies, the convergence to steady states is improved significantly. Moreover, the strategies are applicable to other types of weighted schemes. In this paper, we give a brief review on the topic of convergence to steady state solutions for WENO schemes applied to Euler equations.