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A Higher Order Ensemble Simulation Algorithm for Fluid Flows

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Abstract

This report presents an efficient, higher order method for fast calculation of an ensemble of solutions of the Navier–Stokes equations. We give a complete stability and convergence analysis of the method for laminar flows and an extension to turbulent flows. For high Reynolds number flows, we propose and analyze an eddy viscosity model with a recent reparameterization of the mixing length. This turbulence model depends on an ensemble mean compatible with the higher order method. We show the turbulence model has superior stability, also demonstrated in numerical tests. We also give tests showing the potential of the new method for exploring flow problems to compute turbulence intensities, effective Lyapunov exponents, windows of predictability and to verify the selective decay principle.

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Authors and Affiliations

  1. Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, 15260, USA

    Nan Jiang

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  1. Nan Jiang
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Correspondence to Nan Jiang.

Additional information

The research of the author described herein was partially supported by NSF Grant DMS 1216465 and Air Force Grant FA 9550-12-1-0191.

Appendix: Proof of Theorem 3

Appendix: Proof of Theorem 3

The true solution\((u_{j}, p_j)\) of the NSE satisfies

$$\begin{aligned}&\left( \frac{3u_{j}^{n+1}-4u_{j}^{n}+u_j^{n-1}}{2 \Delta t}, v_{h}\right) + b^{*}\left( u_{j}^{n+1}, u_{j}^{n+1}, v_{h}\right) + \nu \left( \nabla u_{j}^{n+1}, \nabla v_{h}\right) \nonumber \\&\quad -\, \left( p_{j}^{n+1},\nabla \cdot v_{h}\right) =\left( f_{j}^{n+1}, v_{h}\right) + Intp\left( u_{j}^{n+1};v_{h}\right) , \quad \text {for all } v_{h}\in V_{h}, \end{aligned}$$
(8.1)

where \(Intp\left( u_{j}^{n+1};v_{h}\right) \) is defined as

$$\begin{aligned} Intp\left( u_{j}^{n+1};v_{h}\right) =\left( \frac{3u_{j}^{n+1}-4u_{j}^{n}+u_j^{n-1}}{2\Delta t}-u_{j,t}(t^{n+1}),v_{h}\right) . \end{aligned}$$

Let \(e_{j}^{n}=u_{j}^{n}-u_{j,h}^{n}=\left( u_{j}^{n}-I_{h} u_{j}^{n}\right) +\left( I_{h} u_{j}^{n}-u_{j,h}^{n}\right) =\eta _{j}^{n}+\xi _{j,h}^{n}\), where \(I_{h} u_{j}^{n} \in V_{h} \) is an interpolant of \(u_{j}^{n}\) in \(V_{h}\). Subtracting (5.1) from (8.1) gives

$$\begin{aligned}&\left( \frac{3\xi _{j,h}^{n+1}-4\xi _{j,h}^{n}+\xi _{j,h}^{n-1}}{2\Delta t},v_{h}\right) +b^{*}\left( u_{j}^{n+1},u_{j}^{n+1},v_{h}\right) +\nu \left( \nabla \xi _{j,h}^{n+1},\nabla v_{h}\right) \nonumber \\&\quad -\,b^{*}\left( 2u_{j,h}^{n}-u_{j,h}^{n-1}\!-\!u_{j,h}^{\prime n},u_{j,h}^{n+1},v_{h}\right) \! -\!b^{*}\left( u_{j,h}^{\prime n},2u_{j,h}^{n}-u_{j,h}^{n-1},v_{h}\right) -\left( p_{j}^{n+1},\nabla \cdot v_{h}\right) \nonumber \\&\qquad = -\left( \frac{3\eta _{j}^{n+1}-4\eta _{j}^{n}+\eta _j^{n-1}}{2\Delta t},v_{h}\right) -\nu \left( \nabla \eta _{j}^{n+1},\nabla v_{h}\right) +Intp\left( u_j^{n+1};v_{h}\right) \!\!. \end{aligned}$$
(8.2)

Set \(v_{h}=\xi _{j,h}^{n+1}\in V_{h}\), and rearrange the nonlinear terms, then we have

$$\begin{aligned}&\frac{1}{4\Delta t}\left( \left\| \xi _{j,h}^{n+1}\right\| ^{2}+\left\| 2\xi _{j,h}^{n+1}-\xi _{j,h}^{n}\right\| ^{2}\right) -\frac{1}{4\Delta t}\left( \left\| \xi _{j,h}^{n}\right\| ^{2}+\left\| 2\xi _{j,h}^{n}-\xi _{j,h}^{n-1}\right\| ^{2}\right) \nonumber \\&\quad +\,\frac{1}{4\Delta t}\left\| \xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1}\right\| ^{2} +\nu \left\| \nabla \xi _{j,h}^{n+1}\right\| ^2\nonumber \\&\quad =-b^{*}\left( u_{j}^{n+1},u_{j}^{n+1},\xi _{j,h}^{n+1}\right) +b^{*}\left( 2u_{j,h}^{n}-u_{j,h}^{n-1},u_{j,h}^{n+1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \quad +\,b^{*}\left( u_{j,h}^{\prime n},2u_{j,h}^{n}-u_{j,h}^{n-1}-u_{j,h}^{n+1}, \xi _{j,h}^{n+1}\right) +\left( p_{j}^{n+1},\nabla \cdot \xi _{j,h}^{n+1}\right) \nonumber \\&\quad \quad -\,\left( \frac{3\eta _{j}^{n+1}-4\eta _{j}^{n}+\eta _j^{n-1}}{ 2\Delta t},\xi _{j,h}^{n+1}\right) -\nu \left( \nabla \eta _{j}^{n+1},\nabla \xi _{j,h}^{n+1}\right) +Intp\left( u_j^{n+1};\xi _{j,h}^{n+1}\right) .\nonumber \\ \end{aligned}$$
(8.3)

Now we bound the right hand side of Eq. (8.3). First, for the nonlinear term, adding and subtracting \(b^{*}(u_{j}^{n+1},u_{j,h}^{n+1} ,\xi _{j,h}^{n+1})\), \(b^{*}(2u^{n}_j-u^{n-1}_j, u_{j,h}^{n+1}, \xi _{j,h}^{n+1})\) and \(b^{*}(u_{j,h}^{\prime n},2u_{j}^{n}-u_{j}^{n-1}-u_j^{n+1},\xi _{j,h}^{n+1})\), we have

$$\begin{aligned}&-b^{*}\left( u_{j}^{n+1},u_{j}^{n+1},\xi _{j,h}^{n+1}\right) +b^{*}\left( 2u_{j,h}^{n}-u_{j,h}^{n-1}, u_{j,h}^{n+1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad +\,b^{*}\left( u_{j,h}^{\prime n},2u_{j,h}^{n}-u_{j,h}^{n-1}-u^{n+1}_{j,h},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad =-b^{*}\left( u_{j}^{n+1},\eta _{j}^{n+1},\xi _{j,h}^{n+1}\right) -b^{*}\left( u_{j}^{n+1} -\left( 2u_{j}^{n}-u_j^{n-1}\right) ,u_{j,h}^{n+1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \quad -\,b^*\left( 2\eta ^n_j-\eta ^{n-1}_j, u_{j,h}^{n+1}, \xi _{j,h}^{n+1}\right) -b^*\left( 2\xi ^n_{j,h}-\xi ^{n-1}_{j,h}, u_{j,h}^{n+1}, \xi _{j,h}^{n+1}\right) \nonumber \\&\quad \quad -\,b^{*}\left( u_{j,h}^{\prime n},2\xi _{j,h}^{n} -\xi _{j,h}^{n-1}-\xi _{j,h}^{n+1},\xi _{j,h}^{n+1}\right) -b^{*}\left( u_{j,h}^{\prime n},2\eta _j^{n}-\eta _j^{n-1}-\eta _j^{n+1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \quad +\,b^{*}\left( u_{j,h}^{\prime n},2u_{j}^{n}-u_{j}^{n-1}-u^{n+1}_{j},\xi _{j,h}^{n+1}\right) . \end{aligned}$$
(8.4)

We estimate the nonlinear terms using (2.2), (2.3) and Young’s inequality as follows.

$$\begin{aligned}&b^{*}\left( u_{j}^{n+1},\eta _{j}^{n+1},\xi _{j,h}^{n+1}\right) \le C\left\| \nabla u_{j}^{n+1}\right\| \left\| \nabla \eta _{j}^{n+1}\right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\quad \le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+C\nu ^{-1}\left\| \nabla u_{j}^{n+1}\right\| ^{2}\left\| \nabla \eta _{j}^{n+1}\right\| ^{2}. \end{aligned}$$
(8.5)
$$\begin{aligned}&b^{*}\left( u_{j}^{n+1}-\left( 2u_{j}^{n}-u_j^{n-1}\right) ,u_{j,h}^{n+1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \le C\left\| \nabla \left( u_{j}^{n+1}-2u_{j}^{n}+u_j^{n-1}\right) \right\| \left\| \nabla u_{j,h}^{n+1}\right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\quad \le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+C\nu ^{-1}\Delta t^3 \left( \int _{t^{n-1}}^{t^{n+1}}\left\| \nabla u_{j,tt} \right\| ^{2} dt\right) \left\| \nabla u_{j,h}^{n+1}\right\| ^{2}. \end{aligned}$$
(8.6)
$$\begin{aligned}&b^*\left( 2\eta ^n_j-\eta ^{n-1}_j, u_{j,h}^{n+1}, \xi _{j,h}^{n+1}\right) \le C\left\| \nabla \left( 2\eta ^n_j-\eta ^{n-1}_j\right) \right\| \left\| \nabla u_{j,h}^{n+1}\right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\quad \le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+C\nu ^{-1}\left( \left\| \nabla \eta ^n_j \right\| ^{2}+\left\| \nabla \eta ^{n-1}_j \right\| ^{2}\right) \left\| \nabla u_{j,h}^{n+1}\right\| ^{2}. \end{aligned}$$
(8.7)
$$\begin{aligned}&2b^*\left( \xi ^n_{j,h}, u_{j,h}^{n+1}, \xi _{j,h}^{n+1}\right) \le C\left\| \nabla \xi ^n_{j,h} \right\| ^{\frac{1}{2}} \left\| \xi ^n_{j,h}\right\| ^{\frac{1}{2}}\left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\quad \le C\left( \epsilon \left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+\frac{1}{\epsilon }\left\| \nabla \xi ^n_{j,h} \right\| \left\| \xi ^n_{j,h} \right\| \right) \nonumber \\&\quad \le C\left( \epsilon \left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+\frac{1}{\epsilon }\left( \delta \left\| \nabla \xi ^n_{j,h} \right\| ^2+\frac{1}{\delta }\left\| \xi ^n_{j,h} \right\| ^2\right) \right) \nonumber \\&\quad \le \left( \frac{\nu }{64} \left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+\frac{\nu }{16} \left\| \nabla \xi ^n_{j,h} \right\| ^2\right) +C\nu ^{-3}\left\| \xi ^n_{j,h} \right\| ^2 . \end{aligned}$$
(8.8)

Similarly,

$$\begin{aligned} b^*\left( \xi ^{n-1}_{j,h}, u_{j,h}^{n+1}, \xi _{j,h}^{n+1}\right) \le \left( \frac{\nu }{64} \left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+\frac{\nu }{16} \left\| \nabla \xi ^{n-1}_{j,h} \right\| ^2\right) +C\nu ^{-3}\left\| \xi ^{n-1}_{j,h} \right\| ^2. \end{aligned}$$
(8.9)

By skew symmetry and inequalities (2.4) and (2.5), we have

$$\begin{aligned}&b^{*}\left( u_{j,h}^{\prime n},2\xi _{j,h}^{n}-\xi _{j,h}^{n-1}-\xi _{j,h}^{n+1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \le C \left\| \nabla u_{j,h}^{\prime n} \right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \left\| \nabla (\xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1})\right\| ^{1/2}\left\| \xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1}\right\| ^{1/2} \nonumber \\&\quad \le C \left\| \nabla u_{j,h}^{\prime n} \right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \left( h^{-1/2}\right) \left\| \xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1}\right\| \nonumber \\&\quad \le \frac{1}{8\Delta t}\left\| \xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1}\right\| ^{2} +\frac{C_e}{16}\frac{\Delta t}{h}\left\| \nabla u_{j,h}^{\prime n} \right\| ^2\left\| \nabla \xi ^{n+1}_{j,h} \right\| ^{2}. \end{aligned}$$
(8.10)
$$\begin{aligned}&b^{*}\left( u_{j,h}^{\prime n},\eta _{j}^{n+1}-2\eta _{j}^{n}+\eta _j^{n-1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \le C\left\| \nabla u_{j,h}^{\prime n}\right\| \left\| \nabla \left( \eta _{j}^{n+1}-2\eta _{j}^{n} +\eta _j^{n-1}\right) \right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\quad \le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+\frac{C \Delta t^3}{\nu }\left\| \nabla u_{j,h}^{\prime n}\right\| ^{2}\left( \int _{t^{n-1}}^{t^{n+1}}\left\| \nabla \eta _{j,tt}\right\| ^{2} \,dt\right) . \end{aligned}$$
(8.11)
$$\begin{aligned}&b^{*}\left( u_{j,h}^{\prime n},u_{j}^{n+1}-2u_{j}^{n}+u_j^{n-1},\xi _{j,h}^{n+1}\right) \nonumber \\&\quad \le C\left\| \nabla u_{j,h}^{\prime n}\right\| \left\| \nabla \left( u_{j}^{n+1}-2u_{j}^{n}+u_j^{n-1}\right) \right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\quad \le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+C \nu ^{-1} \Delta t^3\left\| \nabla u_{j,h}^{\prime n}\right\| ^{2}\left( \int _{t^{n-1}}^{t^{n+1}}\left\| \nabla u_{j,tt}\right\| ^{2}\,dt\right) . \end{aligned}$$
(8.12)

Next, consider the pressure term. Since \(\xi _{j,h}^{n+1}\in V_{h}\) we have

$$\begin{aligned} \left( p_{j}^{n+1},\nabla \cdot \xi _{j,h}^{n+1}\right)&= \left( p_{j}^{n+1}-q_{j,h}^{n+1}, \nabla \cdot \xi _{j,h}^{n+1}\right) \nonumber \\&\le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+C \nu ^{-1}\left\| p_{j}^{n+1}-q_{j,h}^{n+1}\right\| ^{2} , \quad \forall q_{j,h}^{n+1} \in Q_h. \end{aligned}$$
(8.13)

The other terms, are bounded as

$$\begin{aligned} \left( \frac{3\eta _{j}^{n+1}-4\eta _{j}^{n}+\eta _j^{n-1}}{ 2\Delta t}, \xi _{j,h}^{n+1}\right)&\le C\left\| \frac{3\eta _{j}^{n+1}-4\eta _{j}^{n} +\eta _j^{n-1}}{ 2\Delta t}\right\| \, \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\le C \nu ^{-1}\left\| \frac{3\eta _{j}^{n+1}-4\eta _{j}^{n}+\eta _j^{n-1}}{ 2\Delta t}\right\| ^{2}+\frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}\nonumber \\&\le \frac{C}{\nu \Delta t}\int _{t^{n-1}}^{t^{n+1}}\left\| \eta _{j,t}\right\| ^{2}\; dt+\frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}. \end{aligned}$$
(8.14)
$$\begin{aligned} \nu \left( \nabla \eta _{j}^{n+1},\nabla \xi _{j,h}^{n+1}\right)&\le C\nu \left\| \nabla \eta _{j}^{n+1}\right\| ^{2}+ \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}. \end{aligned}$$
(8.15)

Finally,

$$\begin{aligned} Intp\left( u_{j}^{n+1};\xi _{j,h}^{n+1}\right)&= \left( \frac{3u_{j}^{n+1}-4u_{j}^{n}+u_j^{n-1}}{2\Delta t}-u_{j,t}(t^{n+1}),\xi _{j,h}^{n+1}\right) \nonumber \\&\le C\left\| \frac{3u_{j}^{n+1}-4u_{j}^{n}+u_j^{n-1}}{2\Delta t}-u_{j,t}(t^{n+1})\right\| \left\| \nabla \xi _{j,h}^{n+1}\right\| \nonumber \\&\le \frac{\nu }{64}\left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}+\frac{C\Delta t^3}{\nu } \int _{t^{n-1}}^{t^{n+1}}\left\| u_{j,ttt}\right\| ^{2} dt . \end{aligned}$$
(8.16)

Combining, we now have the following inequality

$$\begin{aligned}&\frac{1}{4\Delta t}\left( \left\| \xi _{j,h}^{n+1}\right\| ^{2}+\left\| 2\xi _{j,h}^{n+1} -\xi _{j,h}^{n}\right\| ^{2}\right) -\frac{1}{4\Delta t}\left( \left\| \xi _{j,h}^{n}\right\| ^{2}+\left\| 2\xi _{j,h}^{n}-\xi _{j,h}^{n-1}\right\| ^{2}\right) \nonumber \\&\quad +\,\frac{1}{8\Delta t}\left\| \xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1}\right\| ^{2}+\frac{\nu }{16} \left( \left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2}-\left\| \nabla \xi _{j,h}^{n}\right\| ^{2}\right) \nonumber \\&\quad +\,\frac{\nu }{16}\left( \left( \left\| \nabla \xi _{j,h}^{n+1}\right\| ^{2} +\left\| \nabla \xi _{j,h}^{n}\right\| ^{2}\right) -\left( \left\| \nabla \xi _{j,h}^{n}\right\| ^{2} +\left\| \nabla \xi _{j,h}^{n-1}\right\| ^{2}\right) \right) \nonumber \\&\quad +\,\left( \frac{\nu }{16}-\frac{C_e}{16}\frac{\Delta t}{h}\left\| \nabla u_{j,h}^{\prime n} \right\| ^2\right) \left\| \nabla \xi ^{n+1}_{j,h} \right\| ^{2} \le C\nu ^{-3}\left( \left\| \xi ^n_{j,h} \right\| ^2+\left\| \xi ^{n-1}_{j,h} \right\| ^2\right) \nonumber \\&\quad +\,C\nu ^{-1}\left\| \nabla u_{j}^{n+1}\right\| ^{2}\left\| \nabla \eta _{j}^{n+1}\right\| ^{2} +\frac{C\Delta t^3}{\nu }\left( \int _{t^{n-1}}^{t^{n+1}}\left\| \nabla u_{j,tt} \right\| ^{2} dt\right) \left\| \nabla u_{j,h}^{n+1}\right\| ^{2}\nonumber \\&\quad +\,C\nu ^{-1}\left( \left\| \nabla \eta ^n_j \right\| ^{2}\!+\!\left\| \nabla \eta ^{n-1}_j \right\| ^{2}\right) \left\| \nabla u_{j,h}^{n+1}\right\| ^{2}\!+\!\frac{C \Delta t^3}{\nu }\left\| \nabla u_{j,h}^{\prime n}\right\| ^{2}\left( \int _{t^{n-1}}^{t^{n+1}}\left\| \nabla \eta _{j,tt}\right\| ^{2} \;dt\right) \nonumber \\&\quad +\,\frac{C \Delta t^3}{\nu }\left\| \nabla u_{j,h}^{\prime n}\right\| ^{2} \left( \int _{t^{n-1}}^{t^{n+1}}\left\| \nabla u_{j,tt}\right\| ^{2} \;dt\right) +C \nu ^{-1}\left\| p_{j}^{n+1}-q_{j,h}^{n+1}\right\| ^{2}\nonumber \\&\quad +\,\frac{C}{\nu \Delta t}\int _{t^{n-1}}^{t^{n+1}}\left\| \eta _{j,t}\right\| ^{2}\;dt +C\nu \left\| \nabla \eta _{j}^{n+1}\right\| ^{2}+\frac{C\Delta t^3}{\nu } \int _{t^{n-1}}^{t^{n+1}}\left\| u_{j,ttt}\right\| ^{2} dt . \end{aligned}$$
(8.17)

Under (5.2), \((\frac{\nu }{16}-\frac{C_e}{16}\frac{\Delta t}{h}\left\| \nabla u_{j,h}^{\prime n} \right\| ^2)\) is nonnegative and thus can be eliminated from the LHS of (8.17). We sum (8.17) from \(n=1\) to \(n=N-1\), multiply through by \(2\Delta t\) and apply interpolation inequalities and the discrete Gronwall inequality (Girault and Raviart [17, p. 176]). This yields

$$\begin{aligned}&\frac{1}{2}\left\| \xi _{j,h}^{N}\right\| ^{2}+\frac{1}{2}\left\| 2\xi _{j,h}^{N}-\xi _{j,h}^{N-1}\right\| ^{2} +\frac{1}{4}\sum _{n=1}^{N-1}\left\| \xi _{j,h}^{n+1}-2\xi _{j,h}^{n}+\xi _{j,h}^{n-1}\right\| ^{2}\nonumber \\&\qquad +\,\frac{\nu \Delta t}{4}\left\| \nabla \xi _{j,h}^{N}\right\| ^{2} +\frac{\nu \Delta t}{8}\left\| \nabla \xi _{j,h}^{N-1}\right\| ^{2}\nonumber \\&\quad \le exp\left( \frac{C N\Delta t}{\nu ^2}\right) \Big \lbrace \frac{1}{2}\left\| \xi _{j,h}^{1}\right\| ^{2}+\frac{1}{2}\left\| 2\xi _{j,h}^{1}-\xi _{j,h}^{0}\right\| ^{2} +\frac{\nu \Delta t}{4}\left\| \nabla \xi _{j,h}^{1}\right\| ^{2}+\frac{\nu \Delta t}{8} \left\| \nabla \xi _{j,h}^{0}\right\| ^{2}\nonumber \\&\qquad +\,C\frac{h^{2k}}{\nu }\left\| \left| \nabla u_j\right| \right\| ^2_{\infty , 0}\left\| \left| u_j\right| \right\| ^2_{2, k+1} +C\frac{\Delta t^4}{\nu }\left\| \left| \nabla u_{j, tt}\right| \right\| ^2_{2,0}\nonumber \\&\qquad +\,C \frac{h^{2k}}{\nu }\left\| \left| \nabla u_{j}\right| \right\| ^{2}_{2,k+1} +C\Delta t^2 h^{2k+1}\left\| \left| \nabla u_{j, tt}\right| \right\| ^2_{2,k} +C\Delta t^3 h \left\| \left| \nabla u_{j, tt}\right| \right\| ^2_{2,0}\nonumber \\&\qquad +\,C\frac{h^{2s+2}}{\nu }\left\| \left| p_j\right| \right\| ^{2}_{2,s+1} + Ch^{2k+2}\nu ^{-1}\left\| \left| u_{t,j}\right| \right\| ^{2}_{2,k+1}\nonumber \\&\qquad +\,C\nu h^{2k}\left\| \left| \nabla u_{j} \right| \right\| _{2,k}^{2}+\frac{C{\Delta t}^{4}}{\nu }\left\| \left| u_{j,ttt}\right| \right\| ^{2}_{2,0}\Big \rbrace . \end{aligned}$$
(8.18)

Using triangle inequality on the error and absorbing constants into a new constant \(C\), we obtain Theorem 3 and Corollary 1.

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Jiang, N. A Higher Order Ensemble Simulation Algorithm for Fluid Flows. J Sci Comput 64, 264–288 (2015). https://doi.org/10.1007/s10915-014-9932-z

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