An Explicit Implicit Scheme for Cut Cells in Embedded Boundary Meshes

Abstract

We present a new mixed explicit implicit time stepping scheme for solving the linear advection equation on a Cartesian cut cell mesh. We use a standard second-order explicit scheme on Cartesian cells away from the embedded boundary. On cut cells, we use an implicit scheme for stability. This approach overcomes the small cell problem—that standard schemes are not stable on the arbitrarily small cut cells—while keeping the cost fairly low. We examine several approaches for coupling the schemes in one dimension. For one of them, which we refer to as flux bounding, we can show a TVD result for using a first-order implicit scheme. We also describe a mixed scheme using a second-order implicit scheme and combine both mixed schemes by an FCT approach to retain monotonicity. In the second part of this paper, extensions of the second-order mixed scheme to two and three dimensions are discussed and the corresponding numerical results are presented. These indicate that this mixed scheme is second-order accurate in $L^1$ and between first- and second-order accurate along the embedded boundary in two and three dimensions.

Proof of Theorem 1

(Restatement of Theorem 1) The scheme (6) is TVD for the linear advection equation (1) for $0 \le \lambda \le 1$, if the exact solution has compact support.

Proof

We cannot directly apply Harten’s theorem [23, 37] to show that the scheme is TVD, but we imitate its proof in order to show

$$\begin{aligned} \sum _{i} |s_{i+1}^{n+1} - s_i^{n+1} | \le \sum _{i} |s_{i+1}^n - s_i^n |. \end{aligned}$$

(13)

We split the sum on the left hand side in the following way

$$\begin{aligned} \sum _{i} |s_{i+1}^{n+1} - s_i^{n+1} |= & {} \sum _{i\le -3} |s_{i+1}^{n+1} - s_i^{n+1} | + \sum _{i\ge 2} |s_{i+1}^{n+1} - s_i^{n+1} | \nonumber \\&+\, |s_{-1}^{n+1}-s_{-2}^{n+1} | + |s_{0}^{n+1}-s_{-1}^{n+1} | + |s_{1}^{n+1}-s_{0}^{n+1} | + |s_{2}^{n+1}-s_{1}^{n+1} |.\nonumber \\ \end{aligned}$$

(14)

We estimate the sums $\sum _{i\le -3} |s_{i+1}^{n+1} - s_i^{n+1} |$ and $\sum _{i\ge 2} |s_{i+1}^{n+1} - s_i^{n+1} |$ in the first line (whose behavior is dominated by the explicit scheme) and the four terms in the second line (whose behavior is dominated by the implicit scheme) separately.

Terms involving the explicit scheme The MUSCL scheme can be written as

$$\begin{aligned} s_i^{n+1} = s_i^n - C_{i-1/2}^n \left( s_i^n - s_{i-1}^n\right) \end{aligned}$$

with

$$\begin{aligned} C_{i-1/2}^n = \lambda \frac{s_i^n + (1-\lambda ) s_{x,i}^n \frac{\Delta x}{2} - s_{i-1}^n - (1-\lambda ) s_{x,i-1}^n \frac{\Delta x}{2}}{s_i^n - s_{i-1}^n}, \end{aligned}$$

which satisfy $0 \le C_{i-1/2}^n \le 1$ for the chosen limiter. This relation holds on all cells with indices $i \le -2$ or $i \ge 2$ (with $s_{x,-2}^n = s_{x,1}^n = 0$). This implies for $i \le -3$ and $i \ge 2$ the relation

$$\begin{aligned} |s_{i+1}^{n+1} - s_i^{n+1} | \le (1-C_{i+1/2}^n) |s_{i+1}^n - s_i^n | + C_{i-1/2}^n |s_i^n - s_{i-1}^n |. \end{aligned}$$

Taking the compact support into account, there holds

$$\begin{aligned} \sum _{i\le -3} |s_{i+1}^{n+1} - s_i^{n+1} | \le \sum _{i\le -3} |s_{i+1}^n - s_i^n | - C_{-5/2}^n |s_{-2}^n - s_{-3}^n | \end{aligned}$$

(15)

and

$$\begin{aligned} \sum _{i\ge 2} |s_{i+1}^{n+1} - s_i^{n+1} | \le \sum _{i\ge 2} |s_{i+1}^n - s_i^n | + C_{3/2}^n |s_{2}^n - s_{1}^n |. \end{aligned}$$

(16)

Terms involving the implicit scheme To estimate the remaining four terms in the second line of Eq. (14) in terms of data at time $t^{n}$ we exploit the convexity of the implicit scheme. We write the updates for $s_{-1}^{n+1}$ and $s_{0}^{n+1}$ as

$$\begin{aligned} s_{-1}^{n+1} = \lambda _{-1} s_{-1}^n + (1-\lambda _{-1}) s_{-2}^n \quad \text {and} \quad s_0^{n+1} = \lambda _{0} s_{0}^n + (1-\lambda _{0}) s_{-1}^{n+1} \end{aligned}$$

with $\lambda _{-1} = 1/(1+\lambda )$ and $\lambda _0 = 1/(1+\lambda /\alpha )$, $\lambda _{-1}, \lambda _{0} \in (0,1]$. Then we have the following estimates: for the difference $|s_2^{n+1} - s_1^{n+1} |$, we get

$$\begin{aligned} |s_2^{n+1} - s_1^{n+1} |&= |(1-C_{3/2}^n)s_2^n + C_{3/2}^n s_1^n - (1-\lambda ) s_1^n - \lambda s_0^{n+1} | \\&\le (1-C_{3/2}^n) |s_2^n - s_1^n | + \lambda |s_1^n - s_0^{n+1} |. \end{aligned}$$

For the difference $|s_1^{n+1} - s_0^{n+1} |$, there holds

$$\begin{aligned} |s_1^{n+1} - s_0^{n+1} | = |(1-\lambda ) s_1^n + \lambda s_0^{n+1} - s_0^{n+1} | = (1-\lambda ) |s_1^n - s_0^{n+1} | \end{aligned}$$

with

$$\begin{aligned} |s_1^n - s_0^{n+1} | = |s_1^n - \lambda _0 s_0^n - (1-\lambda _0) s_{-1}^{n+1} | \le |s_1^n - s_0^n | + (1-\lambda _0) |s_0^n - s_{-1}^{n+1} |. \end{aligned}$$

For the difference $|s_0^{n+1} - s_{-1}^{n+1} |$, we get

$$\begin{aligned} |s_0^{n+1} - s_{-1}^{n+1} | = |\lambda _0 s_0^n + (1-\lambda _0) s_{-1}^{n+1} - s_{-1}^{n+1} | = \lambda _0 |s_0^n - s_{-1}^{n+1} |. \end{aligned}$$

This implies

$$\begin{aligned}&|s_2^{n+1} - s_1^{n+1} | + |s_1^{n+1} - s_0^{n+1} | + |s_0^{n+1} - s_{-1}^{n+1} | \\&\qquad \qquad \le (1-C_{3/2}^n) |s_2^n - s_1^n | + |s_1^n - s_0^{n+1} | + \lambda _0 |s_0^n - s_{-1}^{n+1} |\\&\qquad \qquad \le (1-C_{3/2}^n) |s_2^n - s_1^n | + |s_1^n - s_0^n | + |s_0^n - s_{-1}^{n+1} |. \end{aligned}$$

Finally, there holds

$$\begin{aligned} |s_0^n - s_{-1}^{n+1} | = |s_0^n - \lambda _{-1} s_{-1}^n - (1-\lambda _{-1}) s_{-2}^n | \le |s_0^n - s_{-1}^n | + (1-\lambda _{-1}) |s_{-1}^n - s_{-2}^n | \end{aligned}$$

as well as

$$\begin{aligned} |s_{-1}^{n+1} - s_{-2}^{n+1} |&= |\lambda _{-1} s_{-1}^n + (1-\lambda _{-1}) s_{-2}^n - (1-C_{-5/2}^n) s_{-2}^n - C_{-5/2}^n s_{-3}^n |\\&\le \lambda _{-1} |s_{-1}^n - s_{-2}^n | + C_{-5/2}^n |s_{-2}^n-s_{-3}^n |. \end{aligned}$$

To summarize, we get

$$\begin{aligned}&|s_2^{n+1} - s_1^{n+1} | + |s_1^{n+1} - s_0^{n+1} | + |s_0^{n+1} - s_{-1}^{n+1} | + |s_{-1}^{n+1} - s_{-2}^{n+1} | \\&\quad \le (1-C_{3/2}^n) |s_2^n - s_1^n | + |s_1^n - s_0^n | + |s_0^n - s_{-1}^n | + |s_{-1}^n - s_{-2}^n | + C_{-5/2}^n |s_{-2}^n-s_{-3}^n |. \end{aligned}$$

Therefore, we have estimated all terms in (14). Putting the results in Eqs. (15) and (16) as well as our results for the implicit terms together implies the claim (13).$\square$