Abstract
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.
Highlights
Over the past few decades, computational fluid dynamics (CFD) has become an important tool to study flow structures and aerodynamic forces through solving the compressible Euler or Navier-Stokes equations
In [15], Zhang and Shu studied the mechanism of the non-convergence of the fifth-order weighted essentially non-oscillatory (ENO) (WENO) scheme, and found that there is a relationship between the smoothness indicators and the post-shock oscillations
2.2 Local characteristic decomposition for conservative law systems In the previous subsection, we have described the WENO procedure for obtaining numerical fluxes for finite difference WENO schemes for solving scalar conservation laws
Summary
Over the past few decades, computational fluid dynamics (CFD) has become an important tool to study flow structures and aerodynamic forces through solving the compressible Euler or Navier-Stokes equations. Almost all high-order shock capturing schemes suffer from the problem that they can not converge to steady state solutions for compressible flow if there is a strong shock wave. In [15], Zhang and Shu studied the mechanism of the non-convergence of the fifth-order WENO scheme, and found that there is a relationship between the smoothness indicators and the post-shock oscillations. In [13], Zhang et al studied the non-convergence of WENO schemes of higher orders of accuracy They found that there is some relationship between the local characteristic projection and the appearance of post shock oscillations. Zhu et al [16–22] proposed new types of WENO schemes with the application of a series of unequal-sized spatial stencils Such new WENO schemes have very nice convergence property to steady state solutions. 2 Methodology of the WENO schemes we give a brief overview of the finite difference WENO schemes including the numerical fluxes, the smoothness indicators, and the discretization of the time derivative
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