1 Introduction

With the development of computational fluid mechanics, more and more in-depth and complex issues need to be known about the detail structure and property of the flow field, such as the mechanism of turbulent flow, transition mechanisms, and transition position prediction. Direct numerical simulation has been widely noticed and used as an effective method to study those questions^[1-3]. Along with it, many high order schemes are developed and used to meet the urgent needs of research and engineering. Up to now, high order schemes are successfully used, including the high order compact scheme^[4], the non-oscillatory non-free dissipative (NND) scheme^[5], the weighted essentially non-oscillatory scheme (WENOS)^[6-9], the discontinuous Galerkin (DG) scheme^[10], etc.

When we solve the flow field with a high order scheme, there are two things needed to be considered. One is that how to deal with the boundary so that we can get the satisfactory results. References [11] and [12] provide us with some successful methods to solve the boundary. However, in fact, for example, the direct numerical simulation of the stability and turbulence characteristics of the flow and points near the boundary usually has to be taken by low-order schemes. This is because that high order schemes need more computational points. For the points near the boundary, such as the boundary of the wall, some computational points may be not included in the domain. If high order schemes are still used for most points at the interior of the domain, the low order schemes are inevitably applied to those points near the boundary. This is the so-called reduced-order boundary, e.g., the interior points with the fifth-order scheme and the boundary points with the third-order scheme. The other reason is that when the flow field contains a discontinuous shock wave as the supersonic flow fluid, the order of the scheme at the discontinuity is usually low. This can be called the internal reduced-order.

It is questionable whether the reduced-order boundary and low-order scheme at the shock wave have an effect on the numerical solution and accuracy of the inside flow field calculated with a high order scheme. Does the actual accuracy of the scheme equal the theoretical accuracy? How about the effect on the numerical solution?

In this paper, according to the actual situation of the direct numerical simulation of flow field, we solve two model equations with the exact solutions, including the ordinary differential equation and the wave equation, using different high order schemes at the interior of the domain and reduced-order methods at the boundary and center of the domain. Comparing with the exact solutions, the effects of the reduced-order boundary and low-order scheme at the shock wave on the numerical solution and actual accuracy of the inside flow field are studied. Moreover, considering the actual study of the fluid mechanics, the other two module equations with the exact solutions which are often used, the convection-diffusion equation, and the Burgers equation are also solved and analyzed with the same process to investigate the reduced-order effect.

2 Module equations and computational schemes 2.1 Module equations

To simulate actual cases, including steady flow and unsteady flow, two module equations with exact solutions are chosen. The one-dimensional ordinary differential equation for steady flow is given by

(1)

where x ∈ Ω which is [0, 1]. Its exact solution with the boundary condition u(0)=u(1)=0 is

(2)

The one-dimensional wave equation for the unsteady flow is given by

(3)

where x ∈ Ω which is [0, 1], and t ∈ [0, T]. It also has an exact solution for the pure initial condition u(x, 0)=sin (2πx), which can be expressed by

(4)

In order to investigate the effects of the reduced-order boundary and the low-order scheme at the shock wave on the numerical solution and the accuracy of the inside flow field, the module equations are calculated with different high order schemes, and different reduced-order methods are used at the boundary and interior of the domain Ω, respectively.

2.2 Computational domain and grid

We assume that the domain Ω is covered by a fixed Cartesian mesh for both Eqs.(1) and (3) (see Fig. 1).

We define the node, cells, and cell sizes by

(5)

(6)

2.3 Computational schemes

According to the above grid, we use different high order schemes to solve Eqs.(1) and (3).

2.3.1 Steady flow case

For steady flow, the central difference scheme (CDS) constructed by the Taylor expansion is used to calculate Eq.(1). The fourth-order CDS is given as an example by

(7)

More details about the CDS can be found in Ref.[13].

2.3.2 Unsteady flow case

For unsteady flow, we need to deal with the spatial and temporal term of Eq.(3) sequentially. First of all, the WENOS developed by the essentially non-oscillatory scheme (ENOS) is used to make this spatial term discrete. We assume that the semi-discrete approximation of Eq.(3) is denoted by

(8)

where L is a spatial discrete operator, and is the numerical approximation of the flux f_i and can be split into and . In order to illustrate the basic idea of the WENOS, the general process that we calculate (denoted by ) is given as an instruction below.

For the spatial point , (i) choose a stencil for the order of the accuracy r, i.e.,

(9)

(ii) use the Lagrange interpolation polynomial construct numerical flux on this stencil S_k^r(i) to obtain the polynomial q_k^r(x) shown as follows:

(10)

(iii) take a convex combination of the polynomial q_k^r(x) to reconstruct the numerical flux so that we can have higher order of accuracy and computational efficiency. Then, we have

(11)

where ω_k given by Shu and Osher^[6-7] is the weight function about the smoothness of the stencil S_k^r(i).

Considering the symmetry, , and can be got similarly. The assumption of Eq.(8) is then achieved, and the third-order explicit Runge-Kutta method with the total variation diminishing (TVD) characteristic is introduced to solve the time derivative term of Eq.(8) accurately as follows:

(12)

where we assume that uⁿ has been obtained for the time level tⁿ, and then uⁿ⁺¹ can be calculated by Eq.(12) for the time level tⁿ⁺¹. Especially, in order to avoid the effect for the calculation accuracy, the time step is given by Δt=N_C · (Δx)^1.7, where N_C is the courant number. The error caused by the time step is O(Δt)³ ≈ O((Δx)^1.7)³ ≈ O(Δx)⁵, which is the same with the highest spatial error we have.

Finally, we summarize that the wave equation (3) can be solved with the WENOS and the Runge-Kutta method together. The readers can see Refs.[6]-[9] and [13] for more details for both of them.

2.4 Accuracy formula

After that, we calculate the numerical accuracy of the scheme by the accuracy formula to directly describe the effect of the reduced-order boundary and low-order scheme at the shock wave on the numerical solution and accuracy of the flow field inside. It is shown that

(13)

(14)

where

(15)

In the above equations, N_x is the number of the used space grid points, D_E1 and D_E2 are errors for different N_x, h₁ and h₂ are the corresponding space steps, is the numerical accuracy of the scheme, L¹ and L^∞ are the norms of different definition and express errors of different means, and a(i) and a_e(i) indicate the numerical solution and the exact solution on the point i, respectively.

Specifically, when we calculate the numerical accuracy of the scheme by Eqs.(13) and (14), we assume that the used space grid points are the interior points, which are calculated by a high order scheme to study the reduced-order effect on the actual accuracy of the inside the flow field accurately. Namely, if there are m reduced-order points at the domain, including N space grid points, then the number of the used points is

3 Calculation results and analysis

In this section, we solve the differential equation (1) and the wave equation (3) by the above schemes with different outside boundary situations which take the value of the ghost points outside the domain and different reduced-order methods, and analyze the obtained results of the calculation.

3.1 Verification of formulas and programs

Firstly, we consider solving Eqs.(1) and (3) by the WENOS and the CDS, respectively, with the periodic outside boundary condition and the non-reduced-order method to verify the validity of the relevant formulas and programs (see Table 1). Especially, for Eq.(3), we assume that the Courant number is 0.2, and the results are shown at t=0.4.

Then, we obtain the numerical accuracy by using Eqs.(13) and (14). The curve of the absolute error of the exact solution and numerical solution for the fourth-order CDS and the fifth-order WENOS is only shown as a reference when the numerical accuracy almost equal the theoretical accuracy (see Table 2, Table 3, and Fig. 2).

We can see from Table 2, Table 3, and Fig. 2 that, whether with the CDS or the WENOS, the numerical accuracy and solution almost equal the theoretical values. Therefore, it is obvious that the relevant formulas and programs, including the WENOS, the CDS, the third-order explicit Runge-Kutta method with the TVD characteristic, the accuracy formula, and the time step, are right.

3.2 Reduced-order calculation

Now, we turn to the calculations of Eqs.(1) and (3) with different reduced-order methods by the WENOS and the CDS, respectively, in order to study the reduced-order effect for steady flow and unsteady flow.

3.2.1 Discussion of calculation

Firstly, when we calculate Eqs.(1) and (3) with the above high order scheme, we apply different reduced-order methods at the boundary and center of the domain, respectively, to simulate the situations of the reduced-order boundary and the low-order scheme at the shock wave. Then, considering real calculation of the flow field, the initial and boundary conditions for Eq.(3) are changed to be u(x, 0)=0 and u(0, t)=sin (2πt), respectively. The outside boundary of the inlet and outlet is used for the reduced-order calculation to take the value of the ghost points outside the domain.

Moreover, the value of the points at the inlet and outlet are given by the exact solution (2) for Eq.(1). It is proved in the non-reduced-order condition that the numerical solution and accuracy almost equal the theoretical values. However, because Eq.(3) waves from left to right (see Fig. 3), the exact solution (4) is unsuited for the boundary of the outlet for Eq.(3). Moreover, we find that, comparing with the boundary of the inlet with the exact solution (4) for Eq.(3), the obtained numerical solution in Table 1 can make the numerical accuracy close to the theoretical accuracy rapidly without a very small time step in the non-reduced-order condition. Above all, for the outside boundary of Eq.(3), we define that the value of the points at the inlet is given by the solved numerical solution in Table 1, and the extrapolation boundary of the second-order Lagrange interpolation is used to approximate the value of the points at the outlet.

In order to avoid the effect of the mesh and study the reduced-order effect effectively, we set the number of the space grid as a consistent value for all the reduced-order situations.

3.2.2 Calculation of steady flow

Here, we calculate Eq.(1) with different reduced-order methods and different order CDSs so as to study the effect for the steady flow (see Table 4).

Then, in order to understand the reduced-order effect on the actual order of the interior points of the domain for the steady flow, we calculate the numerical accuracy by Eqs.(13) and (14). Especially, because the results of the numerical accuracy of No.1 and No.2 in Table 4 are similar, we only list the biggest value of the results of No.1 in Table 4 (see Table 5) to save the space that can be got by the real calculation finally.

We can see from Table 5 that the numerical accuracy of the reduced-order at the boundary of the domain almost equals the theoretical value. However, the numerical accuracy of the reduced-order at the center of the domain does not. Therefore, we can say that the numerical accuracy of the interior points of the domain does not change basically for all the reduced-order situations without the numerical accuracy of the reduced-order at the center of domain.

Moreover, the curves of the solution and error are plotted to analyze the effect of the reduced-order on the real solution of the interior points of the domain for the steady flow. Considering the similarity and space, the results of the numerical solution and the error of No.1 in Table 5 are given when the number of the space grid is 80 (see Figs. 4 and 5). Specially, Fig. 4 shows the curves of the numerical solution with different reduced-order situations and the exact solution (2) when N is 80. Figure 5 shows the curve of the absolute error of the numerical solution with different reduced-order situations in Table 4 and the periodic boundary condition in Table 1 when N is 80.

We can see from Fig. 4 and Fig. 5 that, comparing with the numerical solution with the periodic boundary condition in Table 1 for Eq.(1), no matter whether the reduced-order is at the boundary or at the center of the domain, the error of the numerical solution of the interior points of the domain enlarges. The biggest value of the error is at the reduced-order points. Therefore, we can say that, for steady flow, all reduced-order situations indeed make the error of the numerical solution of the interior points of domain enlarge.

3.2.3 Calculation of unsteady situation

We now consider calculating Eq.(3) with the WENOS to study the effect for the unsteady situation as steady situation. Especially, we assume that the Courant number is 0.02, and the results are shown at t=2.0 (see Table 6).

Then, the numerical accuracy is calculated by Eqs.(13) and (14) to learn the reduced-order effect on the actual order of the interior points of the domain for the unsteady flow. Since the results of the numerical accuracy of No.1 and No.2 in Table 6 are also similar, only the biggest value of the results of No.2 in Table 6 is listed (see Table 7) to save the space that can be got by the real calculation finally.

We can see from Table 7 that the numerical accuracy of all conditions does not equal the theoretical value and the values of p¹ and p^∞ at the same row have large differences for the reduced-order at the boundary. According to the analysis, we find that it is caused by the extrapolation of the second-order Lagrange interpolation. Then, we obtain the new result by removing those points of the Lagrange interpolation to avoid this (see Table 8).

We can see from Table 8 that the numerical accuracy of the above situations does not equal the theoretical value of a high order scheme without the numerical accuracy of the reduced-order at the outlet of the domain. Therefore, we can say that the numerical accuracy of the interior points of the domain does not equal the theoretical value for all the reduced-order situations at the points including the reduced-order points and the downstream points of the reduced-order points.

Moreover, it is similar to the steady flow that the curves of the solution and the error for different reduced-order situations and time are plotted below to study the effect of the reduced-order on the real solution of the interior points of the domain for the unsteady flow. Considering the similarity and space, the results of the numerical solution and the error of No.2 in Table 6 are only given when the number of the space grid is 80 (see Figs. 6, 7, and 8). Among them, Fig. 6 indicates the curves of the numerical solution with different reduced-order situations and the exact solution (4) when N is 80 and t is 2.0, Fig. 7 shows the curves of the absolute error of the numerical solution with different reduced-order situations in Table 6 and the periodic boundary condition in Table 1 when N is 80 and t is 2.0, and Fig. 8 shows the curves of the absolute error of the numerical solution with the reduced-order at the inlet of the domain in Table 6 and the periodic boundary condition in Table 1 when N is 80 in different time.

We can see from Figs. 6, 7, and 8 that, comparing with the numerical solution with the periodic boundary condition in Table 1 for Eq.(3), for all the reduced-order situations, the error of the numerical solution of the interior points, including the reduced-order points and the downstream points of the reduced-order points, enlarges. The error of the numerical solution of the points of the Lagrange interpolation also enlarges. Moreover, the error of the numerical solution almost remains constant with time. Therefore, we can say that, for unsteady flow, all reduced-order situations make the error of the numerical solution of the interior points, including the reduced-order points and the downstream points of the reduced-order points, enlarge without time.

4 Application for other module equations

In this section, we extend the reduced-order problem to some module equations with the exact solutions, which are often used in the actual study of fluid mechanics. According to the calculation results, the reduced-order effect is analyzed for those module equations.

4.1 Module equations

We consider solving two module equations with exact solutions. The one-dimensional inviscous Burgers equation is given by

(16)

where x ∈ Ω which is [-1, 1], and t ∈ [0, T]. For the periodic boundary condition and the initial condition with , there is an exact solution which can be calculated by the iterative method as follows:

(17)

where . Then, the one-dimensional linear convection-diffusion equation is

(18)

where c and μ are constant which can be set to 1.0 and 0.01, respectively, and x ∈ Ω which is [0, 2π], and t ∈ [0, T]. The exact solution, for the periodic boundary condition and the initial condition where u(x, 0)=sin (x), is expressed by

(19)

4.2 Calculation results and analysis

Now, we calculate those module equations (16) and (18), respectively, with the same methods and situations introduced above.

4.2.1 Calculation of Burgers equation

Actually, according to the calculation condition shown in Table 6, Eq.(16) can be solved by the WENOS. Especially, we assume that the Courant number is 0.01. Considering the similarity and space, the results of the condition corresponding to the one of No.2 in Table 6 are only given at the time 0.3.

Then, the numerical accuracy is given by Eqs.(13) and (14). We can see from Table 9 that it is obvious that it is very close to the results of the numerical accuracy of the inviscous Burgers equation and the wave equation. Therefore, we can say that for the inviscous Burgers equation, the reduced-order effect for the accuracy is similar to the one on the wave equation.

Moreover, the curves of the solution and the error for different reduced-order situations are plotted when the number of the space grid is 80 (see Figs. 9 and 10). Specifically, Fig. 9 indicates the curves of the numerical solution with different reduced-order situations and the exact solution (17), while Fig. 10 shows the curves of the absolute error of the numerical solution with different reduced-order situations and the periodic boundary condition.

We can see from Figs. 9 and 10 that the obtained results are the same as those of the wave equation. Therefore, we can conclude that, for the inviscous Burgers equation, the reduced-order effect for the solution is similar to that on the wave equation.

4.2.2 Calculation of convection-diffusion equation

For convection-diffusion equations, because the viscous term exists, we solve Eq.(18) with the calculation conditions shown in Table 6 as the wave equation with the WENOS and the CDS to disperse the convection term and the viscous term, respectively. We assume that the Courant number is 0.01. The results of the condition corresponding to the one of No.2 in Table 6 are given similarly.

Then, we have the numerical accuracy given by Eqs.(13) and (14) when t=10.0. It is seen that the result of the numerical accuracy is similar to the results of the wave equation and the inviscous Burgers equation. Therefore, we can say that, for linear convection-diffusion equations, the reduced-order effect for the accuracy is similar to the effects on the above two module equations.

The curves of the solution and the error for different reduced-order situations and time are plotted in Figs. 11, 12, and 13 when the number of the space grid is 80. Specifically, Fig. 11 indicates the curves of the numerical solution with different reduced-order situations and the exact solution (19) at the time of 10.0. Figure 12 shows the curves of the absolute error of the numerical solution with different reduced-order situations and the periodic boundary conditions at the time of 10.0. Figure 13 shows the curves of the absolute error of the numerical solution with reduced-order at the inlet of the domain and the periodic boundary condition for different time.

We can see from Figs. 11, 12, and 13 that the results are similar to those of the wave equation and the inviscous Burgers equation. Moreover, it is interesting that the absolute error of the numerical solution with the reduced-order situation and periodic boundary condition gradually decreases with time. Therefore, we can say that, for linear convection-diffusion equations, the reduced-order effect for the solution is similar to the effects on the above two module equations, except that the error of the numerical solution gradually decreases with time, which is resulted in from the effect of the viscous term.

5 Conclusions

According to the actual situation of the direct numerical simulation of flow, we solve two model equations with the exact solutions, i.e., an ordinary differential equation and a wave equation, by calculating with different high order schemes and different reduced-order situations. The reduced-order effects on the numerical solution and accuracy of the interior points of the domain are studied by comparing the obtained results with the exact solutions. Moreover, the other two model equations with the exact solutions, which are often used in fluid mechanics, convection-diffusion equation, and Burgers equation, are also studied with the same process for the reduced-order problem here. The conclusions can be summarized as follows.

(i) For steady cases, the numerical accuracy of the interior points of the domain does not change basically for all reduced-order situations without reduced-order at the center of the domain. All reduced-order situations indeed make the error of the numerical solution of all interior points of domain enlarge.

(ii) For unsteady cases, the numerical accuracy and solution of the interior points of the domain are affected by all reduced-order situations at the same points, including the reduced-order points and the downstream points of the reduced-order points. It is there that the numerical accuracy does not equal the theoretical value of the used high order scheme, and the error of the actual solution enlarges without time.

(iii) The boundary of the extrapolation of the interpolation also affects the numerical accuracy and solution of the relevant points similarly.

(iv) For the inviscous Burgers equation, the effect of reduced-order on the numerical solution and accuracy of the interior points of the domain is almost similar to the wave equation, while for the linear convection-diffusion equation, the effect of reduced-order is also similar to the wave equation, except that the error of the actual solution gradually decreases with time, resulting from the effect of the viscous term.

Based on the above conclusions, we know something about the effect of the reduced-order boundary and low-order scheme at the shock wave on the numerical solution and accuracy of the flow field inside. When we calculate some actual flow problems with the high order scheme, such as the real turbulent flow including many small scale structures affected easily and the flow field with reduced-order at the inlet, the relevant effect caused by the reduced-order boundary, shock wave, and extrapolation boundary need to be considered. How to avoid the effect will be studied in the following work.