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Cures for the Expansion Shock and the Shock Instability of the Roe Scheme

A common defect of the Roe scheme is the production of non-physical expansion shock and shock instability. An improved method with several advantages was presented to suppress the shock instability. However, this method cannot prevent expansion shock and is incompatible with the traditional curing method for expansion shock. Therefore, the traditional curing mechanism is analyzed. The discussion explains the effectiveness of the traditional curing method and identifies several defects, one of which leads to incompatibility between curing the shock instability and expansion shock. Consequently, a new improved Roe scheme is proposed in this study. This scheme is concise, easy to implement, low computational cost, and robust. More importantly, the scheme can simultaneously cure the shock instability and expansion shock without additional costs.


Introduction
The Roe scheme [1] is one of the most famous and important shock-capturing schemes because of its high accuracy. This scheme has undergone considerable development, such as its extension to incompressible flows [2][3] [4], and has been extensively used for flow computation, such as in Euler flows [5], LES [6] [7], and 2 cavitation [8]. However, the Roe scheme also suffers from a few shortcomings, such as shock instability and expansion shock [9].
Shock instability is a well-known defect of supersonic flows with different performances, such as carbuncle, kinked Mach stem, and odd-even decoupling. Several methods were proposed to cure shock instability; the cures were achieved by adding an entropy fix [10], combining a dissipative scheme [9], increasing the basic upwind dissipation [11] [12], and considering multi-dimensional characteristics [13] [14].
The expansion shock is another defect of the Roe scheme, which is an unphysical solution that violates the entropy condition. Moreover, this defect often yields unacceptable values, such as negative pressure and density, and leads to the divergence of computation for the highly energetic flow. The entropy fix is often adopted to overcome this drawback, but this approach has limited effects while introducing large numerical dissipation and unfavorable empirical parameters. Another considerably common curing method introduces a slight modification by redefining the numerical signal velocities with improved results [12] [15] [16].
In Ref. [17], the momentum interpolation mechanism in the Roe scheme [18] [19] [20] is considered the most important reason for shock instability. Thus, a new improved Roe scheme is proposed [17] by removing the momentum interpolation mechanism for the non-linear flow. This improvement cures the shock instability while removing the problem-independence empirical parameters and decreasing the numerical dissipation. However, in this paper several numerical results show an unexpected defect, wherein the expansion shock becomes serious and the traditional curing method is 3 invalid. To broaden the range of applications of the improved Roe scheme, the current study aims to cure the expansion shock by identifying the reason for the deterioration and further elucidating the mechanism that produces the expansion shock, and thus propose an ideal scheme.
The rest of this paper is organized as follows. Chapter 2 provides the governing equations and the improved Roe scheme for curing shock instability [17]. Chapter 3 analyzes the mechanism of the traditional method of curing the expansion shock.
Chapter 4 provides a new approach to cure the expansion shock while maintaining all the advantages of the improved Roe scheme. Chapter 5 concludes this paper.

Governing Equations
The governing three-dimensional Navier-Stokes equations can be written as follows: ,, x y z , respectively.

Roe Scheme and Improvement
The classical Roe scheme can be expressed in the following general form as the sum of a central term and a numerical dissipation term: where c F is the central term and d F is the numerical dissipation term. For a cell face of the finite volume method,  According to Ref. [21], a scale uniform framework for the shock-capturing scheme is proposed [22]. This framework is simple and easy to analyze and improve with low computation cost.  5 where the first term on the right side  is the basic upwind dissipation; the term p p  is the pressure-difference-driven modification for the cell face pressure, For the classical Roe scheme, where c is the sound speed, and the eigenvalues of the system are defined as follows: Based on Eqs. (11)-(13), Eqs. (7)- (10) can also be further simplified as: Ref. [17] suggests that the shock instability is mainly attributed to the term p U  6 and can be cured by multiplying Eq. (17) to the two functions 1 s and 2 s as follows: where M is the Mach number. The function 2 s , which is a shock detector and can be obtained in Ref.
[17], is not presented in this paper because it is relatively complicated and probably unnecessary for general cases. This improvement is simple and effective.

Two Classical Numerical Tests
Two classical numerical examples are available to test the expansion shock by the proposed scheme. One is a specific one-dimensional shock tube and the other is the supersonic corner problem. The initial condition of the shock tube is given as 3 The initial condition is given as 7.04108 uv  , and 1 R p  at the x-axis position of 0.05. In this study, the mesh grids are 200 for the shock tube, and 400*400 for the supersonic corner. For time discretization, the four-stage Runge-Kutta scheme is adopted. For space discretization, the first-order accuracy is adopted (unless otherwise specified) to discuss the schemes themselves.

Traditional Curing Method
To avoid the expansion shock, the tradition curing method redefines the physical signal velocities (i.e., non-linear eigenvalues 4  and 5  ). For example, Ref. [15] proposed that: To precisely obtain a contact discontinuity, Ref. [12] suggested that only U in 4  and 5  should be improved: Eqs.

Performances of the Schemes
Figs. 1 and 2 show the results by the classical Roe scheme, as described by Eqs. (6) and (14) Fig. 2(a). No evident expansion shock was observed, but the shock instability was expectedly strong. The traditional curing method produces similar results (see Fig. 2(b)).   Fig. 3(b)). The shock instability becomes substantially weak and is nearly cured. However, a strong expansion shock occurs. In the iterative calculation process, the density occasionally becomes negative and the following limitation is necessary to prevent computational divergence:

Y-Axis
where  is a small positive value.
The traditional curing method for expansion shock in Eqs. (22)-(23) can also be 10 integrated into the improved Roe scheme. For the shock tube, this method can produce results as Fig. 1(b). However, this approach is invalid for the supersonic corner and even substantially increases the expansion shock. The computation diverges because of negative density even when using Eq. (24). This unexpected problem seems confusing and may hinder the possible extensive application of the improved Roe scheme because of concerns regarding computational robust. Therefore, in the following sections the mechanism of preventing expansion shock is further analyzed and the new method satisfies the stringent requirement of simultaneously curing the expansion shock and the shock instability without additional costs.

Analysis of the Schemes
To develop the new method, the mechanism of the traditional curing method is first further analyzed. Eqs. (22)-(23) can be decomposed into the following five conditions: (1) Uc  : (3) Uc  and L Uc  and R Uc  : By considering Eqs. (25)-(34) can be summarized as follows: where       2  and  and  1 sign max 0, otherwise 2 2  and  and  1 sign max 0, otherwise 2

   
Therefore, Eqs. (7)-(10) become: Therefore, for the subsonic expansion flows, (2) Uc  and Consequently, (3) Uc  and L Uc  and Therefore, the increment terms are unsatisfactory because they are not equal to zero and not smooth between Conditions (1) and (2).
The other two conditions of Uc  are not discussed for simplicity because they produce the same conclusions as the conditions of Uc  .

Further Analysis of the Curing Expansion Shock Mechanism
The preceding discussion reveals a few unsatisfactory features of the traditional curing method in Eqs. (22)- (23). Moreover, the discussion provides clues regarding the mechanism of expansion shock suppression. Two inspirations are obtained as follows.
(1) An increment factor is designed as follows: where L UU  in Eq. (38) and The present scheme is significantly concise and easy to implement; the computational cost only has a negligible increase as well. Compared with the scheme of Eqs. (14)- (16) and (18)

Conclusions
The performance of several Roe-type schemes is discussed in terms of expansion shock, and the mechanism of curing expansion shock is analyzed based on the traditional method. Several unfavorable features of the traditional curing method are discovered, and the possible curing mechanism is not completely utilized. Therefore, an improved method is proposed to overcome these problems. The present scheme is 19 substantially concise, easy to implement, and robust with a low computational cost. This scheme is particularly well compatible with the improvement to cure the shock instability. Therefore, the present scheme is simultaneously free from the problems of shock instability and expansion shock without additional expense.