An Essentially Non-oscillatory Crank–Nicolson Procedure for the Simulation of Convection-Dominated Flows

Abstract

The Crank–Nicolson (CN) time-stepping procedure incorporating the second-order central spatial scheme is unconditionally stable and strictly non-dissipative for linear convection flows; however, its numerical solution in practice can be oscillatory for nonsmooth solutions. This article studies variants of the CN method for the simulation of linear convection-dominated diffusion flows, in which the explicit convection part is approximated by an upwind scheme, to effectively suppress nonphysical oscillations. The second-order essentially non-oscillatory scheme incorporated in the CN procedure (ENO-CN) has been found effective for a non-oscillatory numerical solution of minimum numerical dissipation. A stability analysis is provided for ENO-CN, which turns out to be unconditionally stable for problems of nonzero diffusion. However, for purely convective flows, it is stable only when the CFL condition is satisfied. Numerical results are presented to demonstrate its stability and accuracy.

Appendix: The Strongly Implicit Procedure (SIP)

The problem (15) in the time level n can be written as the following algebraic system of the form

$$\begin{aligned} A\mathbf{u}=\mathbf{b}, \end{aligned}$$

(41)

where the superscript n is omitted for a simpler presentation. To solve it, we adopt the strongly implicit procedure (SIP) suggested by Stone [41]. For completeness of the paper, we briefly describe the basic idea of SIP.

As for other iterative methods, SIP is based on a regular splitting, $A=M-N,$ with M being an incomplete LU (ILU) factorization;

$$\begin{aligned} M=L_IU_I=A+N, \end{aligned}$$

(42)

where $L_I$ and $U_I$ are respectively the lower and upper triangular components of an ILU factorization of A, where the entries of the main diagonal of $U_I$ are all one. Note that the iteration corresponding to the splitting (42) is formulated as

$$\begin{aligned} L_IU_I\mathbf{u}^k=N\mathbf{u}^{k-1}+\mathbf{b}, \end{aligned}$$

(43)

or, equivalently,

$$\begin{aligned} \begin{array}{ll} \hbox {(a)} &{}\displaystyle \mathbf{r}^{k-1}=\mathbf{b}-A\mathbf{u}^{k-1}, \\ \hbox {(b)} &{}\displaystyle L_IU_I\varvec{\delta }^{k-1}=\mathbf{r}^{k-1}, \\ \hbox {(c)} &{}\displaystyle \mathbf{u}^k=\mathbf{u}^{k-1}+\varvec{\delta }^{k-1}, \end{array} \end{aligned}$$

(44)

where ILU-factorization is carried out by Stone’s SIP [41] .

To make the iteration (43) converge fast, we will choose elements of $L_I$ and $U_I$ in a way that N is as small as possible. Let us begin with a 2D problem in a rectangular mesh, where the grid points are ordered in the row-wise manner and the five-point FD scheme is applied. Then, the ILU factorization is in the form shown in Fig. 7 and the row of M corresponding to the $(\ell ,m)$-th grid point is

$$\begin{aligned} \begin{array}{lcl} M_{S}^{\ell ,m} &{}=&{} L_{S}^{\ell ,m}, \\ M_{SE}^{\ell ,m} &{}=&{} L_{S}^{\ell ,m}U_{E}^{\ell ,m-1}, \\ M_{W}^{\ell ,m} &{}=&{} L_{W}^{\ell ,m}, \\ M_{C}^{\ell ,m} &{}=&{} L_{S}^{\ell ,m}U_{N}^{\ell ,m-1} +L_{W}^{\ell ,m}U_{E}^{\ell -1,m}+L_{C}^{\ell ,m}, \\ M_{E}^{\ell ,m} &{}=&{} L_{C}^{\ell ,m}U_{E}^{\ell ,m}, \\ M_{NW}^{\ell ,m} &{}=&{} L_{W}^{\ell ,m}U_{N}^{\ell -1,m}, \\ M_{N}^{\ell ,m} &{}=&{} L_{C}^{\ell ,m}U_{N}^{\ell ,m}. \end{array} \end{aligned}$$

(45)

Here the subscripts S, W, E, N, and C denote respectively south, west, east, north, and center; which show the relationship of the center point to the adjacent grid points.

Fig. 7

Systematic presentation of $L_IU_I=M$. The subscripts S, W, E, N, and C denote respectively south, west, east, north, and center. Note that diagonals of M marked by subscripts SE and NW are not found in A

The $(\ell ,m)$-th component of $N\mathbf{x}$ is

$$\begin{aligned} \begin{array}{lcl} (N\mathbf{x})_{\ell ,m} &{}=&{} N_C^{\ell ,m}x_{\ell ,m} +N_S^{\ell ,m}x_{\ell ,m-1} +N_W^{\ell ,m}x_{\ell -1,m} +N_E^{\ell ,m}x_{\ell +1,m} \\ &{} &{} +N_N^{\ell ,m}x_{\ell ,m+1} +M_{SE}^{\ell ,m}x_{\ell +1,m-1} +M_{NW}^{\ell ,m}x_{\ell -1,m+1}. \end{array} \end{aligned}$$

(46)

By utilizing the approximations

$$\begin{aligned} \begin{array}{lcl} x_{\ell +1,m-1}&{}\approx &{}\alpha \left( x_{\ell ,m-1}+x_{\ell +1,m}-x_{\ell ,m}\right) , \\ x_{\ell -1,m+1}&{}\approx &{}\alpha \left( x_{\ell ,m+1}+x_{\ell -1,m}-x_{\ell ,m}\right) , \end{array} \quad 0<\alpha \le 1, \end{aligned}$$

(47)

we can rewrite (46) as

$$\begin{aligned} \begin{array}{lcl} (N\mathbf{x})_{\ell ,m} &{}\approx &{} \left( N_C^{\ell ,m}-\alpha M_{SE}^{\ell ,m} -\alpha M_{NW}^{\ell ,m}\right) x_{\ell ,m} \\ &{} &{} +\left( N_S^{\ell ,m}+\alpha M_{SE}^{\ell ,m}\right) x_{\ell ,m-1} +(N_W^{\ell ,m}+\alpha M_{NW}^{\ell ,m}) x_{\ell -1,m} \\ &{} &{} +\left( N_E^{\ell ,m}+\alpha M_{SE}^{\ell ,m}\right) x_{\ell +1,m} +(N_N^{\ell ,m}+\alpha M_{NW}^{\ell ,m}) x_{\ell ,m+1}. \end{array} \end{aligned}$$

(48)

Set each of coefficients in the right-side of (48) to be zero. Then, it follows from (45) that entries of N are presented by those of $L_I$ and $U_I$:

$$\begin{aligned} \begin{array}{l} N_S^{\ell ,m}= -\alpha M_{SE}^{\ell ,m} =-\alpha L_{S}^{\ell ,m}U_{E}^{\ell ,m-1}, \\ N_W^{\ell ,m}= -\alpha M_{NW}^{\ell ,m} =-\alpha L_{W}^{\ell ,m}U_{N}^{\ell -1,m}, \\ N_C^{\ell ,m}= \alpha \left( M_{SE}^{\ell ,m}+M_{NW}^{\ell ,m}\right) =\alpha \left( L_{S}^{\ell ,m}U_{E}^{\ell ,m-1} +L_{W}^{\ell ,m}U_{N}^{\ell -1,m}\right) , \\ N_E^{\ell ,m}= -\alpha M_{SE}^{\ell ,m} =-\alpha L_{S}^{\ell ,m}U_{E}^{\ell ,m-1}, \\ N_N^{\ell ,m}= -\alpha M_{NW}^{\ell ,m} =-\alpha L_{W}^{\ell ,m}U_{N}^{\ell -1,m}. \end{array} \end{aligned}$$

(49)

Now, utilizing $M=A+N$, (45), and (49), one can obtain Stone’s ILU factorization [41]:

$$\begin{aligned} \begin{array}{lcl} L_{S}^{\ell ,m} &{}=&{} A_S^{\ell ,m}/\left( 1+\alpha U_E^{\ell ,m-1}\right) , \\ L_{W}^{\ell ,m} &{}=&{} A_W^{\ell ,m}/\left( 1+\alpha U_N^{\ell -1,m}\right) , \\ L_{C}^{\ell ,m} &{}=&{} A_C^{\ell ,m}+\alpha (L_{S}^{\ell ,m}U_{E}^{\ell ,m-1} +L_{W}^{\ell ,m}U_{N}^{\ell -1,m}) \\ &{} &{} -L_{S}^{\ell ,m}U_{N}^{\ell ,m-1} -L_{W}^{\ell ,m}U_{E}^{\ell -1,m}, \\ U_{E}^{\ell ,m} &{}=&{} \left( A_E^{\ell ,m}-\alpha L_{S}^{\ell ,m}U_{E}^{\ell ,m-1}\right) / L_C^{\ell ,m}, \\ U_{N}^{\ell ,m} &{}=&{} \left( A_N^{\ell ,m}-\alpha L_{W}^{\ell ,m}U_{N}^{\ell -1,m}\right) / L_C^{\ell ,m}. \end{array} \end{aligned}$$

(50)

The SIP (50) deserves the following remarks:

• The approximations in (47) are second-order accurate when $\alpha =1$. But in the case, the algorithm (43) incorporating (50) can be unstable; the parameter $\alpha$ has been chosen between 0.92 and 0.96 for various applications in computational fluid dynamics [9]. Entries of $L_I$ and $U_I$ used in (50) whose indices are outside the index boundaries should be set zero.

• The basic idea of SIP shown for the five-point FD scheme can be applied to nine-point FD schemes, their 3D extensions, and other discretization methods such as finite volume and finite element methods, with minor modifications; although the derivation is so tedious.