High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems

Abstract

Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical $L^2$ projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.

Introduction

In this paper, high-order Runge-Kutta discontinuous Galerkin (RKDG) methods [8], [9], [10], [12] with new multi-resolution WENO limiters [58] are applied to solve steady Euler equations

$$\{\begin{array}{c}f{(u)}_x+g{(u)}_y=0,\hfill \\ u(x,y)=u_0(x,y),\hfill \end{array}$$

on structured meshes. One way to get a numerical solution of (1.1) is to solve the associated unsteady Euler equations

$$\{\begin{array}{c}{u}_t+f{(u)}_x+g{(u)}_y=0,\hfill \\ u(x,y,0)=u_0(x,y),\hfill \end{array}$$

and then drive the numerical residual to zero. High-order DG methods are applied to discretize the spatial variables and explicit, nonlinearly stable high-order Runge-Kutta methods [47], [13] are adopted to discretize the temporal variable. Our main objective of this paper is to design a new troubled cell indicator to precisely detect the cells that need further limiting procedures and then adopt the arbitrary high-order spatial limiting procedures [58] for the RKDG methods to solve two-dimensional steady-state problems.

If one confirms that the numerical residual of the unsteady Euler equations (1.2) is small enough, ideally at or close to the level of machine zero, the numerical solution of the steady Euler equations (1.1) is acceptable. The appearance of strong discontinuities in the simulation of (1.1) and (1.2) is the main difficulty. If the numerical solution has strong shocks or contact discontinuities, its physical variables change abruptly. Many high-resolution numerical schemes have been designed with the aim of controlling the oscillations by the use of artificial viscosities [24], [25] or limiters [20], [24], [50]. Jameson et al. [23], [26] proposed a third-order finite volume discretization method with dissipative terms and applied a Runge-Kutta time discretization method for solving the steady Euler equations. However, the main drawback of such schemes is that one often needs to adjust certain parameters in the artificial viscosity to maintain sharp shock transitions and to suppress oscillations near strong shocks. If limiters are used in designing numerical schemes, such numerical schemes could be very efficient in computing supersonic flows including strong shocks and contact discontinuities [20]. Yet the application of total variation diminishing (TVD) type limiters will degrade the accuracy of the numerical scheme to first-order near local smooth extrema [41], and the lack of sufficient smoothness of the numerical fluxes with the application of such limiters often results in the numerical residual not converging close to machine zero. Yee et al. [53] designed an implicit stable high-resolution TVD scheme and applied it to compute steady-state problems. Yee and Harten [52] designed TVD schemes to solve multi-dimensional hyperbolic conservation laws and steady-state problems in curvilinear coordinates. In 1996, Jiang and Shu [27] designed a fifth-order finite difference WENO scheme. When the classical high-order WENO schemes [27] are used to solve for the steady-state problems, their numerical residual often hangs at a truncation error level without settling down close to machine zero even after a long time iteration. Serna et al. [45] proposed a new limiter to reconstruct the numerical flux and improve the convergence of the numerical solution to steady states. Zhang et al. [57] found that slight post-shock oscillations would propagate from the region near the shocks downstream to the smooth regions and result in the numerical residual hanging at a high truncation error level rather than converging to machine zero. Zhang et al. [54] designed an upwind-biased interpolation technique to improve the convergence of high-order WENO scheme for steady-state problems. But the numerical residual computed by such new schemes still could not converge close to machine zero for some two-dimensional steady-state problems [54]. In 2016, a novel high-order fixed-point sweeping WENO method [51] was proposed to simulate steady-state problems and could obtain better convergence property. However, the numerical residual could not settle down close to machine zero for some benchmark steady-state tests as before.

Now let us first review the history of the development of discontinuous Galerkin (DG) methods. In 1973, Reed and Hill [44] designed the first DG method in the framework of neutron transport. Due to its desirable properties, DG methods were also used extensively in different fields [14], [21], [29], [33], [40]. The hybrid DG/FV methods [15], [16], [36], [55], [56] which combine the advantageous features of both have become popular. Luo et al. [36], [38] designed a new DG method for solving the compressible equations with a Taylor basis. If unsteady or steady-state problems are not smooth enough, their numerical solutions might contain oscillations near strong discontinuities and result in nonlinear instability in nonsmooth regions. One possible methodology to suppress oscillations is to apply nonlinear limiters to the high-order RKDG methods. A major development of the DG method with a classical minmod type total variation bounded (TVB) limiter was carried out by Cockburn et al. in a series of papers [8], [9], [10], [11], [12]. One type of limiters is based on slope modification, such as classical minmod type limiters [8], [9], [10], [12], the Barth-Jespersen limiter [2], the Venkatakrishnan limiter [50], the moment based limiter [3], and an improved moment limiter [5]. Such limiters belong to the slope type limiters and they could suppress oscillations at the price of possibly degrading numerical accuracy at smooth extrema. Another type of limiters is based on the essentially non-oscillatory (ENO), weighted ENO (WENO), and Hermite WENO (HWENO) methodologies [1], [17], [18], [22], [27], [32], [34], [35], [37], [39], which can achieve uniform high-order accuracy in smooth regions and keep essentially non-oscillatory property near strong discontinuities. However, it is very difficult to implement RKDG methods with the applications of WENO limiters for solving steady-state problems. When such high-order RKDG methods are applied to compute steady Euler equations, the numerical residual could not converge close to machine zero and would hang at a higher truncation error level.

Likewise, when the classical fifth-order finite difference WENO scheme [27], [46] with a third-order TVD Runge-Kutta time discretization [47] is used to solve for the steady-state problems, the numerical residual often hangs at the truncation error level instead of converging to machine zero even after a long time iteration. Yet the numerical residual of the new high-order finite difference and finite volume multi-resolution WENO schemes [60] could converge close to machine zero without introducing any slight post-shock oscillations on structured meshes. With the application of a series of unequal-sized spatial stencils, the multi-resolution WENO schemes could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities. We think this is the most important reason that the numerical residual of the classical fifth-order finite difference WENO scheme [27], [46] could not convergence to a tiny number, since its spatial approximation could not degrade to the first-order accuracy with the application of equal-sized three-point spatial stencils. So we extend high-order RKDG methods with high-order multi-resolution WENO limiters [58] to solve for the steady Euler equations with the application of a new troubled cell indicator on structured meshes. This new troubled cell indicator is very simple and works well for precisely detecting the cells that need further limiting procedure. To the best of our knowledge, it is the first type of high-order RKDG methods with WENO limiters that could confirm the numerical residual to converge close to machine zero for two-dimensional steady-state problems containing strong shocks at the boundary.

This paper is organized as follows. In Section 2, we give a brief review of the RKDG methods. In Section 3, we propose a new troubled cell indicator to detect the cells needing further limiting procedures and design arbitrary high-order limiting procedures using second-order, third-order, fourth-order, and fifth-order multi-resolution WENO limiters for steady-state computations as examples. In Section 4, several standard steady-state problems including sophisticated wave structures, both inside the computational fields and passing through the boundaries of the computational domain, are presented to demonstrate the good performance of the numerical residual converging close to machine zero. Concluding remarks are given in Section 5.