In this short article, the upwind and central compact finite difference schemes for spatial discretization of the first-order derivative are analyzed. Comparison of the schemes is provided and the best discretization scheme for convection dominated problems is suggested.
With the ever-increasing interest in numerical calculations demanding high accuracy for a wide range of length scales, such as large-eddy simulation and direct numerical simulation of turbulence, high-order numerical methods are desired. Particularly, high-order finite difference, finite volume, and finite element methods have received more attention in handling complex problems. These high-order methods try to achieve high accuracy and avoid spurious oscillations and are usually characterized by their self-adaptive nature. The use of high-order methods is particularly warranted by the need to simulate flows containing discontinuous phenomena, such as fluid interfaces and steep shear layers. The compact high-order finite difference schemes provide an effective way of combining the robustness of finite difference schemes and the accuracy of spectral methods [
In order to examine approximating behaviors of various numerical schemes, the following linear convection equation (also known as one-way wave equation) is considered.
∂ u ∂ t + c ∂ u ∂ x = 0 , c > 0. (1)
The semi discrete form Equation (1) is
∂ u j ∂ t + c ∂ u j ∂ x j = 0. (2)
The solution of Equation (1) represented by u ( x , t ) by a typical Fourier mode is given by:
u ( x , t ) = u ^ k ( t ) e i k x , (3)
u ^ k is the Fourier mode of the wave number k and i = − 1 , the exact spatial differentiation of Equation (3) is represented by;
u ′ = i ( k h ) u ^ k h e i k x , (4)
where the wave number is scaled by the grid size h = l n , where l is the length of
domain and n is the number of grids. By analogy the numerical approximation of the derivative is written as [
u ′ = k e q ( k h ) u ^ k h e i k x = ( k r ( k h ) + i ( k i ) k h ) u ^ k h e i k x . (5)
The exact solution of Equation (1) is u ( x , t ) = e i k ( x − c t ) , and the exact solution
of Equation (2) can be written as u ( x j , t ) = e − k r c t Δ x e i k ( x j − k i k Δ x c t ) , where the modified
wave number k e q = k r + i k i . k i is related to the phase speed in the numerical solution, and k r is related to the numerical damping of a difference scheme. Fourier analysis of different discretization schemes allows us to choose the best scheme.
In this subsection, third and fifth-order upwind compact and upwind explicit schemes are analyzed. For the third-order upwind compact scheme [
2 3 u j + 1 3 u j − 1 = u j + 1 + 4 u j − 5 u j − 1 6 , (6)
that satisfy the relation
u j = 1 6 e i α + 2 3 − 5 6 e − i α 2 3 + 1 3 e − i α u ^ ( t ) e i k x j = cos α + i sin α + 4 − 5 ( cos α − i sin α ) 4 + 2 ( cos α − i sin α ) u ^ ( t ) e i k x j = ( 1 − cos α ) 2 + i sin α ( 8 + cos α ) 5 + 4 cos α u ^ ( t ) e i k x j (7)
where
k r = ( 1 − cos α ) 2 5 + 4 cos α , k i = sin α ( 8 + cos α ) 5 + 4 cos α . (8)
Similarly, for the fifth-order upwind compact scheme [
3 5 u j + 2 5 u j − 1 = − u j + 2 + 12 u j + 1 + 36 u j − 44 u j − 1 − 3 u j − 2 60 , (9)
with
u j = − 1 60 e 2 i α + 12 60 e i α + 36 60 − 44 60 e − i α − 3 60 e − 2 i α 36 60 + 24 60 e − i α u ^ ( t ) e i k x j = 1 117 + 108 cos α ( 24 − 6 cos 3 α − 18 sin 2 α − 18 sin α + i ( 6 sin 3 α + 45 sin α cos α + 180 sin α ) ) u ^ ( t ) e i k x j (10)
where
k r = 2 ( 1 − 3 cos α + 3 cos 2 α − cos 3 α ) 3 ( 13 + 12 cos α ) , k i = sin α ( 2 sin 2 α + 15 cos α + 60 ) 3 ( 13 + 12 cos α ) . (11)
For the explicit third-order upwind scheme [
k r = 1 6 ( 3 − 4 cos α + cos 2 α ) , k i = 1 6 ( 8 sin α − sin 2 α ) , (12)
and for the explicit fifth-order upwind scheme, we have,
k r = 10 − 15 cos α + 6 cos 2 α − cos 3 α 30 , k i = 45 sin α − 9 sin 2 α + sin 3 α 30 . (13)
In this section, various compact finite difference schemes are studied. The family of cell centered central compact schemes given by Lele et al. [
ν u ′ i − 2 + μ u ′ i − 1 + u ′ i + μ u ′ i + 1 + ν u ′ i + 2 = c u i + 3 − u i − 3 6 h + c u i + 2 − u i − 2 4 h + c u i + 1 − u i − 1 2 h (14)
The order of these schemes can be based parameters values as shown in
Taking Fourier transform of Equation (14), we have
| Scheme | Upper limits of wave number | |
|---|---|---|
| kr < 2% | | 1 − k i / α | < 2 % | |
| 5th-order upwind compact | 1.35 | 1.71 |
| 3rd-order upwind compact | 0.91 | 1.61 |
| 5th-order upwind | 1.08 | 1.25 |
| 3rd-order upwind | 0.72 | 0.902 |
| Order of scheme | μ | v | a | b | c |
|---|---|---|---|---|---|
| 2nd order | 0 | 0 | 1 | 0 | 0 |
| 4th order | 0 | 0 | 4 3 | 1 3 | 0 |
| 6th order | 1 3 | 0 | 14 9 | 1 9 | 0 |
| 8th order | 4 9 | 1 36 | 40 27 | 25 54 | 0 |
| 10th order | 1 2 | 1 20 | 17 20 | 191 150 | 1 100 |
k i = a sin α + b 2 sin 2 α + c 3 sin 3 α 1 + 2 μ cos α + 2 ν cos 2 α ; k r = 0. (15)
The different values of k i are given in
The difference between modified wave number and exact wave number is very small, therefore these schemes have spectral like resolution. The comparison of various central compact schemes is presented in
In this subsection, the upwind and central compact schemes are compared based upon the resolution characteristics k i vs α . For this purpose, two upwind compact schemes and two central compact schemes are selected from the previous sections.
The comparison plot for k i vs α is shown in
The comparison of the scheme enables us to find the scheme best suitable from the chosen schemes.
We have analyzed upwind, upwind compact and central compact schemes of
| Order of scheme | ki |
|---|---|
| 2nd order | sinα |
| 4th order | 8 sin α + sin 2 α 6 |
| 6th order | 14 9 sin α + 1 18 sin 2 α 1 + 2 3 cos α |
| 8th order | 40 27 sin α + 25 108 sin 2 α 1 + 8 9 cos α + 1 18 cos α |
| 10th order | 17 12 sin α + 191 300 sin 2 α + 1 300 sin 3 α 1 + cos α + 1 10 cos 2 α |
different order accuracy for numerical investigation of convection equation. It is observed that the use of the upwind compact scheme makes the numerical solution more stable as compared with the central scheme and can be used for convection dominated problems. A comparison is also given with non-compact schemes of the same order of accuracy with almost the same computational cost.
The work of Abdullah Shah is supported by HEC under NRPU No. 7781 and PSF No. 5651.
The authors declare no conflicts of interest regarding the publication of this paper.
Shah, A., Khan, S.A. and Ullah, N. (2019) High Resolution Compact Finite Difference Schemes for Convection Dominated Problems. World Journal of Mechanics, 9, 259-266. https://doi.org/10.4236/wjm.2019.912017