2.1 Variable Order Method of Lines

The solution procedure of the incompressible Navier-Stokes equations (1) and (2) is based on the fractional step approach on the collocated grid system.

In the method of lines approach, the spatial derivatives arc discretized by the appropriate scheme, so that the partial differential equations (PDEs) in space and time are reduced to the system of ordinary differential equations (ODEs) in time. The resulting ODEs are integrated by the Runge-Kutta type time integration scheme.

In the spatial discretization, the convective terms are approximated by the variable order proper convective scheme [2], because of the consistency of the discrete continuity equation, the conservation property, and the variable order of spatial accuracy. This scheme is the extension of the proper convective scheme proposed by Morinishi [4] to the variable order. The variable order proper convective scheme can be described by

  (4)

where M denotes the order of spatial accuracy, and the operators in eq.(4) are defined by

${\overline{f}}^{\ell \prime x_j}{|}_{\mathbf{x}}=\frac{1}{2}(f_{{\mathbf{x}}_j+\ell \prime /2}+f_{{\mathbf{x}}_j-\ell \prime /2}),$

${\overline{u_j}}^{x_j}{|}_{{\mathbf{x}}_j\pm \ell \prime /2}=\underset{m=1}{\overset{M/2}{\mathrm{\Sigma }}}\frac{C_{m\prime }}{2}[u_j{|}_{{\mathbf{x}}_j\pm \{\ell \prime +m\prime \}/2}+u_j{|}_{{\mathbf{x}}_j\perp \{\ell \prime -m\prime \}/2}],$

$\frac{{\delta }_{\ell \prime }{u}_i}{{\delta }_{\ell \prime }{x}_j}{|}_{{\mathbf{x}}_j\pm \ell \prime /2}=\frac{\pm 1}{\ell \prime {\mathrm{\Delta }x}_j}(u_i{|}_{{\mathbf{x}}_j\pm \ell \prime }-u_i{|}_{{\mathbf{x}}_j}),$

where m’ = 2m − 1. In this technique, the arbitrary order of spatial accuracy can be obtained automatically by changing only one parameter M. The coefficients c_l′ and c_m′ are the weighting coefficients and Δx_j denotes the grid spacing in the x_j direction.

On the other hand, the diffusion terms are discretized by the modified differential quadrature (MDQ) method [5] as

$\frac{{\partial }^2{u}_i}{\partial {x_i}^2}{|}_{\mathbf{x}}=\underset{m-M/2}{\overset{M/2}{\mathrm{\Sigma }}}{{\mathrm{\Phi }}_m}^{"}\left(\text{X}\right)u_i{|}_{{\text{x}}_i+m\mathrm{.}}$

where φ_m″(x) is the second derivative of the function φ_m(x) defined by

${\mathrm{\Phi }}_m\left(\text{x}\right)=\frac{\mathrm{\Pi }\left(\text{x}\right)}{(\text{x}-{\text{x}}_{i+m})\mathrm{\Pi }\prime \left({\text{x}}_{i+m}\right)},$

$\mathrm{\Pi }\left(\text{x}\right)=(\text{x}-{\text{x}}_{i-M/2})(\text{x}-{\text{x}}_i)(\text{x}-{\text{x}}_{i+M/2})\mathrm{.}$

The coefficients of the variable order proper convertive scheme, c_ℓ′ can be computed automatically by using the MDQ coefficients. Then, the incompressible Navier-Stokes equations are reduced to the system of ODEs in time. This system of ODEs is integrated by the Runge-Kutta type scheme.

2.2 Variable Order Multigrid Method

The pressure equation described by

  (11)

is solved by the variable order multigrid method [3]. In eq.(11), $u_i^{*}$ denotes the fractional step velocity, α is the parameter determined by the time integration scheme. The overbar denotes the interpolation from the collocated location to the staggered location. In the variable order multigrid method, the unsteady term is added to the pressure equation. Then, the pressure (elliptic) equation is transformed to the parabolic equation in space and pseudo-time, τ.

  (12)

Equation (12) can be solved by the variable order method of lines. The spatial derivatives are discretized by the aforementioned MDQ method, so that eq.(12) is reduced to the system of ODEs in pseudo-time.

  (13)

This system of ODEs in pseudo-time is integrated by the rational Runge-Kutta (RRK) scheme [6]. because of its wider stability region. The RRK scheme can be written by

$p^{m+1}=p^m+\frac{2\overrightarrow{g1}(\overrightarrow{g1},\overrightarrow{g3})\overrightarrow{g3}(\overrightarrow{g1},\overrightarrow{g1})}{(\overrightarrow{g3},\overrightarrow{g3})}\mathrm{.}$

$\overrightarrow{g_1}=\mathrm{\Delta }\tau \overrightarrow{L}\left(\overrightarrow{p^m}\right),\overrightarrow{g_2}=\mathrm{\Delta }\tau \overrightarrow{L}(\overrightarrow{p^m}+c_2\overrightarrow{g_1}),\overrightarrow{g_3}=b_1\overrightarrow{g_2},$

where ∆τ and m are; the pseudo-time step and level, and the operators such as $(\overrightarrow{g_1},\overrightarrow{g_3})$ denote inner product of vectors $\overrightarrow{g_1}$ and $\overrightarrow{g_3}$. Also the coefficients b₁, b₂, and c₂ satisfy the relations b₁ + b₂ = 1, c₂ = −1/2.

In addition, the multigrid technique [7] is incorporated into the method in order to accerelate the convergence. Then, the same order of spatial accuracy as the momentum equations can be specified.

3.1 Multigrid Property

Figure 2 shows the work unit until convergence. The comparison of computational time and work unit is shown in Table 1. The computation is executed with the second order of spatial accuracy. The present three relaxation schemes indicate the mulatigrid convergence. In these relaxation schemes, the RRK scheme gives the good multigrid property with respect to the work unit and computational time. In order to check the influence of spatial accuracy, the comparison of work unit is shown in Fig.3. In both relaxation schemes, the multigrid convergence; can be obtained and the RRK scheme has the independent property of spatial accuracy.

3.2 Parallel Property

In order to implememt to a parallel platform, the domain decomposition technique; is used. In this work, 16 sub-domains are assigned to 16 PEs. Figure 4 shows the parallel efficiency defined by

$Efficiency=\frac{T_{\sin gle}}{N\cdot T_{parallel}}\times 100(\%)$

where N denotes number of PEs, T_single and T_parallel are the CPU time on single PE and N PEs, respectively. In both relaxation schemes, as the order of spatial accuracy becomes higher, the parallel efficiency becomes higher. The RRK scheme gives the higher efficiency than the checkerboard SOR method. In 2D multigrid method, from fine grid to coarse grid, the operation counts become 1/4, but the data transfer becomes 1/2. Then, the parallel efficiency on coarser grid will be lower, so that, the total parallel efficiency is not too high.

In order to improve parallel efficiency, we consider the restriction of PEs on coarser grid level. Figure 5 shows the parallel efficiency with the restriction of PEs on coarser than 64 × 32 grid level. In Fig.5, version 1 and version 2 denote the parallel efficiency without and with restriction of PEs on coarser grid level, respectively. It is clear that the parallel efficiency of version 2 can be about 10% higher than version 1.