Elsevier

Computers & Fluids

Volume 208, 15 August 2020, 104625
Computers & Fluids

A new 6th-order WENO scheme with modified stencils

https://doi.org/10.1016/j.compfluid.2020.104625Get rights and content

Highlights

  • A new 6th-order weighted essentially non-oscillatory (WENO) scheme is developed.

  • A reference smoothness indictor and two modified stencils are used.

  • The scheme achieves the optimal order of convergence at the first-order critical points.

  • The scheme has higher resolution compared with previous 6th-order WENO schemes.

Abstract

In this article, a new 6th-order weighted essentially non-oscillatory (WENO) scheme is developed. As with previous 6th-order central-upwind WENO schemes, the present scheme is a convex combination of four candidate linear reconstructions. The difference is that the most upwind and downwind stencils use four cell values, while the inner two stencils nominally use three cell values but the original quadratic reconstructions are modified to be 4th-order approximations by adding cubic correction terms involving the five cell values of the classical 5th-order WENO scheme. Sixth-order accuracy of the new scheme in smooth regions including critical points is achieved by using a reference smoothness indicator. Several numerical examples show that the new scheme has higher resolution compared with the recently developed 6th-order WENO schemes.

Introduction

The detailed simulation of complex shock waves and fine scale structures in compressible flows requires robust shock capturing methods with small dissipation and dispersion errors. High-order numerical methods for hyperbolic conservation laws provide an effective means for the simulation of complex compressible flows. Many high-order high-resolution methods have been developed in the past four decades, such as piecewise parabolic method (PPM) [1], essentially non-oscillatory schemes (ENO) [2], [3], weighted ENO schemes (WENO) [4], [5], discontinuous Galerkin methods [6], [7], monotonicity-preserving schemes [8], and so on. These methods have been shown to be capable of producing satisfactory numerical results, and there have been a lot of subsequent studies on them for improving computational efficiency, robustness, accuracy, and reducing dissipation and dispersion errors.

The classical 5th-order WENO scheme [5] has attracted a great deal of attention due to the merit of easy implementation, high order accuracy in smooth regions, and essentially non-oscillatory property near discontinuities. However, the scheme is a bit dissipative for the simulation of small scale structures in smooth regions. Up to now, four different approaches have been developed for reducing numerical dissipation of the classical WENO scheme. The first approach, e.g., [9], [10], [11], is to hybridize a low-dissipation scheme which is dominant in smooth regions and a WENO scheme which is dominant in non-smooth regions. The second approach, e.g., [12], [13], is to modify the weights to cure accuracy degeneration at critical points and distribute a little more weights to the less smooth stencils. There are multiple strategies for this approach, e.g., a mapping function (WENO-M [12]), a global reference smoothness indicator (WENO-Z [13], [14]), and smoothness indicators based on L1-norm (WENO-P [15]). The third approach is to modify the stencils and corresponding weights, like P-WENO [16], WENO-ZQ [17] and WENO-MS [18] schemes. The above three approaches use three candidate stencils over the same five-point global stencil as the classical 5th-order WENO scheme. The fourth approach is to add an additional downwind candidate stencil to the classical 5th-order upwind WENO scheme. This approach started from the 6-point WENO-SYMOO and WENO-SYMBO schemes in [19], which are designed based on the idea of optimal order of accuracy and optimal bandwidth-resolving efficiency, respectively. However, the order of the WENO-SYMBO scheme degenerates even in smooth regions, and the order-optimized WENO-SYMOO scheme is unstable near contact discontinuities even when only moderate discontinuities are involved. Ref. [20] defined a smoothness indicator for the downwind stencil and devised a reference smoothness indicator τ similar to that in the WENO-Z scheme [13] for achieving the optimal order of accuracy. However, it is found that this scheme needs extra artificial dissipation to maintain the numerical stability. Hu et al. [21] developed a 6th-order central-upwind WENO scheme which is analogous to the scheme [20] but with a different smoothness indicator for the downwind stencil and no additional artificial dissipation is needed.

More recently, Hu [22] found that the 6th-order WENO scheme [21] generates evident oscillations around discontinuities for CFL number greater than 0.6, and the oscillations grow with increasing grid points. Hu [22] fixed this problem by letting the most downwind stencil to include an upwind point so as to increase numerical stability. A direct use of the Jiang-Shu nonlinear weights makes the resulting scheme achieve only 5th-order accuracy in smooth regions of the solution without critical points. Two strategies were used in [22] to recover the optimal order of accuracy. One is the mapping function [12], [23], [24], another is the reference smoothness indicator [13], [21]. The latter strategy has less computational cost and can obtain nearly identical results to the first one.

In this work, we further improve upon the 6th-order WENO scheme of Hu [22] to enhance the resolution while maintaining the robustness and efficiency. The most upwind stencil is modified to be composed of 4 grid points, while the inner two stencils still use 3 grid points but the reconstructions are made 4th-order accurate by adding cubic correction terms involving the global five grid points of the classical WENO scheme as did in [18]. We compute the smoothness indicator of each stencil according to the Jiang-Shu formula [5]. However, the resulting Jiang-Shu weights can not satisfy the sufficient condition for ensuring the optimal order of convergence, and the scheme achieves only 5th-order accuracy in smooth regions. To recover the optimal order, we use a reference global smoothness indicator to construct a WENO-Z type scheme (called WENO-MSZ6 where MS stands for “Modified Stencil”). Several numerical examples show that the WENO-MSZ6 scheme achieves 6th-order accuracy in smooth regions including the first order critical points and has small dissipation errors while maintaining the robustness and efficiency of the recently developed 6th-order WENO schemes [21], [22].

The organization of this paper is as follows. In Section 2, the recently developed 6th-order WENO schemes are reviewed. In Section 3, we present a new modified 6th-order WENO scheme using a reference smoothness indicator to recover the optimal order. Section 4 presents several benchmark examples to demonstrate the performance of the new scheme. A conclusion is given in Section 5.

Section snippets

The 6th-order central-upwind WENO schemes

In this section, we briefly review two similar and most recent 6th-order central-upwind finite difference WENO schemes, i.e., WENO-CU6 [21] and WENO-Z6 [22] for solving the one-dimensional hyperbolic conservation lawut+f(u)x=0,axb,t>0,where u(x, t) is the conservative variable, f(u(x, t)) is the flux function. Eq.  (1) is solved on a uniform grid defined by the nodes xi=a+(i0.5)Δx,i=1,,N, which are also called cell centers, with cell interfaces given by xi+1/2=xi+Δx/2, where Δx=(ba)/N is

Construction of a new 6th-order WENO scheme

In this section, we present a new 6th-order WENO scheme. As with previous 6th-order WENO schemes [19], [21], [22], we also use four stencils for calculating the numerical flux f^i+1/2 as shown in Fig. 2. The stencils Sk are composed of different numbers of nodes, specifically,S0={i2,i1,i,i+1},S1={i1,i,i+1},S2={i,i+1,i+2},S3={i,i+1,i+2,i+3},where S0 has an extra point i+1. For the two nominally 3-point stencils S1 and S2, we add cubic correction terms to the original 2nd-degree polynomial

Numerical tests

In this section, we provide several numerical examples to demonstrate the performance of the proposed WENO-MSZ6 scheme. The numerical results are compared with the 6th-order WENO-CU6 [21] with C=20 in Eq.  (9) and the WENO-Z6 [22] schemes. ϵ=1040 is used in Eqs.  (9), (22) and (29). The presentation of this section starts with problems for the linear advection equation, followed by problems for the 1D and 2D Euler equations and 2D Navier-Stokes equations. The characteristic decomposition and

Conclusions

A new central-upwind 6th-order WENO scheme (WENO-MSZ6) based on modified candidate stencil approximations is presented. The four stencil approximations achieve 4th-order accuracy thus the sufficient condition for the optimal 6th-order accuracy can be one order lower than that in previous 6th-order WENO schemes. A reference smoothness indicator is used to achieve the optimal order even at critical points. We compare the present scheme with the two recently developed WENO-CU6 and WENO-Z6 schemes.

CRediT authorship contribution statement

Yahui Wang: Methodology, Software, Writing - original draft. Yulong Du: Methodology, Visualization. Kunlei Zhao: Methodology, Visualization. Li Yuan: Supervision, Methodology, Writing - review & editing.

Declaration of Competing Interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.

Acknowledgements

This work is supported by Natural Science Foundation of China (91641107, 91852116), Fundamental Research of Civil Aircraft (MJ-F-2012-04).

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