A Novel Parallel Computing Strategy for Compact Difference Schemes with Consistent Accuracy and Dispersion

Abstract

In this paper, based on the boundary approximation approach for parallelization of the compact difference schemes, a novel strategy for the sub-domain boundary approximation schemes is proposed to maintain consistent accuracy and dispersion with the compact scheme in the interior points. In this strategy, not only the order of accuracy of the sub-domain boundary scheme is the same as the interior scheme, but the coefficient of the first truncation error term is also equal to that of the internal scheme. Furthermore, to realize the consistent dispersion performance for a class of high order upwind compact schemes, which usually include two expressions, we modify the opposite expression to be the sub-domain boundary scheme. As an example of application, the present strategy is applied to a fourth-order upwind compact scheme, and its accuracy is verified by a numerical test. The resolution and efficiency of the newly proposed parallel method are examined by four numerical examples, including propagation of a wave-packet, convection of isentropic vortex, Rayleigh–Taylor instability problems, and propagation of Gauss pulse. The results obtained demonstrate that the present strategy for compact difference schemes has the feasibility to solve the flow problems with high accuracy, resolution and efficiency in parallel computation.

Appendices

Appendix A

The following is to apply the compact fourth-order central difference scheme in conjunction with sixth-order central difference scheme for parallel calculation of the first partial derivative in the same way as Sect. 2.2.3.

If Eq. (24) is assumed to be a sixth-order accuracy scheme, a sixth-order sub-domain boundary scheme at $i=1$ can be constructed with the coefficients $a_k$ solved as follow

$$\begin{aligned} \begin{matrix} {{a}_{-3}}=-\frac{{1}}{{60}}, &{} {{a}_{-2}} =\frac{{3}}{{20}}, &{} {{a}_{-1}}=-\frac{{3}}{{4}}, &{} {{a}_{1}}=\frac{{3}}{{4}}, &{} {{a}_{{2}}}=- \frac{{3}}{{20}}, &{} {{a}_{3}}=\frac{1}{{60}}. \\ \end{matrix} \end{aligned}$$

(74)

The expression of truncation errors is

$$\begin{aligned} {{F}_{1}^{\small {6th}}}=\left. \frac{\partial {{u}}}{\partial {{x}}}\right| _1-\frac{{{h}^{6}}}{140}\left. \frac{{{\partial }^{7}}{{u}}}{\partial {{x}}^{7}}\right| _1+O\left( {{h}^{8}} \right) . \end{aligned}$$

(75)

Comparisons of the effects that the sub-domain boundary scheme has on the $L_{\infty }$-norm errors between numerical results and exact results as well as the convergence order are shown in Table 8.

Table 8 Comparisons of the $L_{\infty }$-norm and the convergence order of serial computing method and the parallel computing setup with 3 and 8 processes using sixth-order central difference scheme as sub-domain boundary approximation. “Ratio” refers to the ratio of parallel computing results to serial computing results

In general, the parallel differencing systems with sixth-order sub-domain boundary scheme provide fourth-order accuracy throughout the domain. However, the $L_{\infty }$-norm errors calculated by the parallel algorithm using sixth-order sub-domain boundary scheme is always 1.2 times the serial results, while that of the parallel algorithm using fourth-order sub-domain boundary scheme with consistent coefficient of first truncation error term are almost equal to the serial results. According to the Eq. (21), since $\alpha _{-1}=\alpha _{1}=\frac{1}{4}$ and $C_e=0$, then

$$\begin{aligned} G_a=G+\frac{1}{5} C {{(\text{ j }k)}^{r+1}}{{h}^{r}}+O({{h}^{r+1}}) \end{aligned}$$

(76)

which indicates that the error of the actual approximation near the boundary is at least 20% greater than the serial one. This is exactly verified by the numerical results.

Appendix B

The following is to extended our approach to match the second truncation error terms in addition and see if it improves the result for parallel calculation of the first partial derivative in the same way as Sect. 2.2.3.

An explicit central difference scheme based on eight-point stencils is taken into account here. For instance, when $i=1$ it can be expressed as

$$\begin{aligned} {{F}_{1}}=\frac{1}{h}\sum \limits _{n=-4}^{4}{{{a}_{n}}{{u}_{1+n}}}. \end{aligned}$$

(77)

If Eq. (77) is assumed to be a fourth-order accuracy scheme and the approximation $F_1$ has the same coefficient of first and second truncation error term as Eq. (22), the coefficients $a_n$ can be solved that

$$\begin{aligned}&\begin{matrix} {{a}_{-4}}=\frac{1}{432},&{{a}_{-3}}=-\frac{1}{36},&{{a}_{-2}}=\frac{37}{216},&{{a}_{-1}}=-\frac{83}{108}, \end{matrix}\nonumber \\&\begin{matrix} {{a}_{0}}=0, &{} {{a}_{1}}=\frac{83}{108}, &{} {{a}_{2}}=-\frac{37}{216}, &{} {{a}_{3}}=\frac{1}{36}, &{} {{a}_{4}}=-\frac{1}{432}. \\ \end{matrix} \end{aligned}$$

(78)

Comparisons of the effects that the sub-domain boundary scheme has on the $L_{\infty }$-norm errors between numerical results and exact results as well as the convergence order are shown in Table 9.

Table 9 Comparisons of the $L_{\infty }$-norm and the convergence order of serial computing method and the parallel computing setup with 3 and 8 processes using eight-stencil fourth-order central difference scheme as sub-domain boundary approximation. “Ratio” refers to the ratio of parallel computing results to serial computing results

Compared to Table 1, the extension to match the second truncation error terms does improve the results, especially when the grids size is small. This shows that for the case of small number of grid points per process, in order to minimize the discrepancies and improve the results, if enlarging the size of sub-boundary scheme stencils is available, we can expand our approach to match the second truncation error term or even more.

Appendix C

If the explicit schemes is employed as the sub-domain boundary schemes for the parallelization of CCU45, in order to meet the proposition in Sect. 2.1 for consistent accuracy and to have DRP property, the explicit schemes based on a seven-point stencil is required. Consider the following explicit schemes:

• (a) EC7 scheme:

  $$\begin{aligned} {F}_{bound\ i}=\frac{1}{h}\sum _{n=-3}^{3}a_{n}u_{i+n}. \end{aligned}$$

  (79)

• (b) E7U4 scheme:

  $$\begin{aligned} {F}_{bound\ i}=\frac{1}{h}\sum _{n=-4}^{2}b_{n}u_{i+n}. \end{aligned}$$

  (80)

• (c) E7U5 scheme:

  $$\begin{aligned} {F}_{bound\ i}=\frac{1}{h}\sum _{n=-5}^{1}c_{n}u_{i+n}. \end{aligned}$$

  (81)

The integrated error $E(d_0)$ is defined as follows.

$$\begin{aligned} E\left( d_0 \right) =\int _{0}^{r\pi }{{\left| \omega -{{k}_{i}}\left( d_0 \right) \right| }^{2}} {W\left( \omega \right) } d\omega \end{aligned}$$

(82)

where r is a factor to determine the optimization integration range, $d_0$ represents $b_0$ or $c_0$, and $W(\omega )$ is a weighting function.

In fact, the EC7 scheme is a central difference scheme and its dispersion relation can not be optimized by changing $a_0$. One can impose the conditions that E7U4 and E7U5 schemes are accurate to fourth-order and keep the coefficient of first truncation error term consistent with Eq. (49). These leave one of the coefficients, note $b_0$ or $c_0$, as a free parameter. This parameter can then be chosen to minimize the error integral E of Eq. (82). After carrying out the constrained minimization, the coefficients can be found, which are shown in Tables 10, 11 and 12.

Table 10 Coefficients of Eq. (79) for the EC7 scheme

Table 11 Coefficients of Eq. (80) for the E7U4 scheme

Table 12 Coefficients of Eq. (81) for the E7U5 scheme

With the use of Fourier transform, the modified wavenumber defined in Eq. (50) $k_e\left( \omega \right) =k_{r}\left( \omega \right) +\text{ j } k_{i}\left( \omega \right)$ of the schemes can be obtained. The profiles of the imaginary part $k_{i}$ of each scheme as well as the CCU45 scheme are shown in Fig. 4. Under the consideration of not employing too many halo points for data transfer, we find that it is difficult to improve the resolution characteristics of the explicit sub-domain boundary scheme by optimization with one or two extra halo points.