Decoupled, Unconditionally Stable, Higher Order Discretizations for MHD Flow Simulation

Abstract

We propose, analyze, and test a new MHD discretization which decouples the system into two Oseen problems at each timestep yet maintains unconditional stability with respect to the time step size, is optimally accurate in space, and behaves like second order in time in practice. The proposed method chooses a parameter $\theta \in [0,1]$, dependent on the viscosity $\nu$ and magnetic diffusivity $\nu _m$, so that the explicit treatment of certain viscous terms does not cause instabilities, and gives temporal accuracy $O(\Delta t^2 + (1-\theta )|\nu -\nu _m|\Delta t)$. In practice, $\nu$ and $\nu _m$ are small, and so the method behaves like second order. When $\theta =1$, the method reduces to a linearized BDF2 method, but it has been proven by Li and Trenchea that such a method is stable only in the uncommon case of $\frac{1}{2}< \frac{\nu }{\nu _m} < 2$. For the proposed method, stability and convergence are rigorously proven for appropriately chosen $\theta$, and several numerical tests are provided that confirm the theory and show the method provides excellent accuracy in cases where usual BDF2 is unstable.

Appendix

We prove here a conditional stability result for the full second order method, i.e. when $\theta =1$, which does not assume $\frac{1}{2}<Pr_m<2$. The condition is that $\Delta t \le \frac{h^2(\nu +\nu _m-|\nu -\nu _m|)}{C_i(\nu -\nu _m)^2}$, where $C_i$ an the inverse inequality constant, and thus if $\nu -\nu _m$ is not small (which is equivalent to $Pr_m$ near 1), this can be a severe timestep restriction when fine meshes are used.

Lemma 6.1

Consider Algorithm 3.1 with $\theta =1$ (the full second order method). If the mesh is sufficiently regular so that the inverse inequality holds (with constant $C_i$) and the time step is chosen to satisfy

$$\begin{aligned} \Delta t \le \frac{h^2(\nu +\nu _m-|\nu -\nu _m|)}{C_i(\nu -\nu _m)^2}, \end{aligned}$$

then the method is stable and solutions satisfy

$$\begin{aligned}&\Vert v_h^M\Vert ^2+\Vert w_h^M\Vert ^2+\frac{(\nu +\nu _m-|\nu -\nu _m|)\Delta t}{2}\sum _{n=1}^{M-1}\left( \Vert \nabla v_h^{n+1}\Vert ^2+\Vert \nabla w_h^{n+1}\Vert ^2\right) \\&\quad \,\le C(\nu ,\nu _m,v_h^0, v_h^1, w_h^0, w_h^1, f_1, f_2). \end{aligned}$$

Proof

Choose $\theta =1,\chi _h=v_h^{n+1}\in V_h$ and $l_h=w_h^{n+1}\in V_h$ in Algorithm 3.1, (3.1)–(3.2). This vanishes the nonlinear and pressure terms, and leaves

$$\begin{aligned}&\frac{1}{{2\Delta }t}\left( 3v_h^{n+1}-4v_h^n+v_h^{n-1},v_h^{n+1}\right) +\frac{\nu +\nu _m}{2}\Vert {\nabla }v_h^{n+1}\Vert ^2+\frac{\nu -\nu _m}{2}\nonumber \\&\quad \times ({\nabla }(2w_h^n-w_h^{n-1}),{\nabla }_+z_h^{n+1}) =(f_1^{n+1},v_h^{n+1}), \end{aligned}$$

(6.1)

$$\begin{aligned}&\frac{1}{{2\Delta }t}(3w_h^{n+1}-4w_h^n+w_h^{n-1},w_h^{n+1}) +\frac{\nu +\nu _m}{2}\Vert {\nabla }w_h^{n+1}\Vert ^2+\frac{\nu -\nu _m}{2} \nonumber \\&\quad \times \left( {\nabla }(2v_h^n-v_h^{n-1}\right) ,{\nabla }w_h^{n+1}) =(f_2^{n+1},w_h^{n+1}). \end{aligned}$$

(6.2)

Using the usual BDF2 identity on the time derivative terms and adding the equations yields

$$\begin{aligned}&\frac{1}{4\Delta t}\big (\Vert v_h^{n+1}\Vert ^2-\Vert v_h^n\Vert ^2+\Vert 2v_h^{n+1} -v_h^n\Vert ^2\nonumber \\&\quad -\,\Vert 2v_h^n-v_h^{n-1}\Vert ^2+\Vert w_h^{n+1}\Vert ^2- \Vert w_h^n\Vert ^2+\Vert 2w_h^{n+1}-w_h^n\Vert ^2\nonumber \\&\quad -\,\Vert 2w_h^n-w_h^{n-1}\Vert ^2+\Vert v_h^{n+1}-2v_h^n+v_h^{n-1}\Vert ^2+\Vert w_h^{n+1} -2w_h^n+w_h^{n-1}\Vert ^2\big )\nonumber \\&\quad +\,\frac{\nu +\nu _m}{2} \big (\Vert {\nabla }v_h^{n+1}\Vert ^2+\Vert {\nabla }w_h^{n+1}\Vert ^2\big )\nonumber \\&\quad +\,\frac{\nu -\nu _m}{2}({\nabla }(2w_h^n-w_h^{n-1}),{\nabla }v_h^{n+1})\nonumber \\&\quad +\,\frac{\nu -\nu _m}{2}({\nabla }(2v_h^n-v_h^{n-1}),{\nabla }w_h^{n+1}) =(f_1^{n+1},v_h^{n+1})+(f_2^{n+1},w_h^{n+1}), \end{aligned}$$

(6.3)

and then adding and subtracting the term $\frac{\nu -\nu _m}{2}\left( \nabla v_h^{n+1},\nabla w_h^{n+1}\right)$ provides

$$\begin{aligned}&\frac{1}{4\Delta t}\big (\Vert v_h^{n+1}\Vert ^2-\Vert v_h^n\Vert ^2+\Vert 2v_h^{n+1}-v_h^n\Vert ^2-\Vert 2v_h^n-v_h^{n-1}\Vert ^2\nonumber \\&\quad +\,\Vert w_h^{n+1}\Vert ^2-\Vert w_h^n\Vert ^2+\Vert 2w_h^{n+1}-w_h^n\Vert ^2\nonumber \\&\quad -\,\Vert 2w_h^n-w_h^{n-1}\Vert ^2+\Vert v_h^{n+1}-2v_h^n+v_h^{n-1}\Vert ^2+\Vert w_h^{n+1}-2w_h^n\nonumber \\&\quad +\,w_h^{n-1}\Vert ^2\big )+\frac{\nu +\nu _m}{2}\big (\Vert {\nabla }v_h^{n+1}\Vert ^2+\Vert {\nabla }w_h^{n+1}\Vert ^2\big )\nonumber \\&\quad -\,\frac{\nu -\nu _m}{2}({\nabla }(v_h^{n+1}-2v_h^n+v_h^{n-1}),{\nabla }w_h^{n+1})\nonumber \\&\quad -\,\frac{\nu -\nu _m}{2}({\nabla }(w_h^{n+1}-2w_h^n+w_h^{n-1}),{\nabla }v_h^{n+1})\nonumber \\&\quad +\,\frac{\nu -\nu _m}{2}(\nabla w_h^{n+1},\nabla v_h^{n+1})+\frac{\nu -\nu _m}{2}(\nabla v_h^{n+1},\nabla w_h^{n+1})\nonumber \\&\quad =\,(f_1^{n+1},v_h^{n+1})+(f_2^{n+1},w_h^{n+1}). \end{aligned}$$

(6.4)

Using Cauchy–Schwarz and Young’s inequalities we have that

$$\begin{aligned}&\frac{1}{4\Delta t}\big (\Vert v_h^{n+1}\Vert ^2-\Vert v_h^n\Vert ^2+\Vert 2v_h^{n+1}-v_h^n\Vert ^2-\Vert 2v_h^n-v_h^{n-1}\Vert ^2\nonumber \\&\quad +\Vert w_h^{n+1}\Vert ^2-\Vert w_h^n\Vert ^2+\Vert 2w_h^{n+1}-w_h^n\Vert ^2\nonumber \\&\quad -\,\Vert 2w_h^n-w_h^{n-1}\Vert ^2+\Vert v_h^{n+1}-2v_h^n+v_h^{n-1}\Vert ^2+\Vert w_h^{n+1}\nonumber \\&\quad -2w_h^n+w_h^{n-1}\Vert ^2\big )+\frac{\nu +\nu _m}{2}\big (\Vert {\nabla }v_h^{n+1}\Vert ^2+\Vert {\nabla }w_h^{n+1}\Vert ^2\big )\nonumber \\&\quad \le \,\frac{|\nu -\nu _m|}{2}\Vert {\nabla }\big (w_h^{n+1}-2w_h^n+\,w_h^{n-1}\big )\Vert \Vert {\nabla }v_h^{n+1}\Vert \nonumber \\&\quad +\frac{|\nu -\nu _m|}{2}|\Vert \nabla (v_h^{n+1}-2v_h^n+v_h^{n-1})\Vert \Vert {\nabla }w_h^{n+1}\Vert \nonumber \\&\quad +\,|\nu -\nu _m|\Vert {\nabla }w_h^{n+1}\Vert \Vert {\nabla }v_h^{n+1}\Vert +\Vert f_1^{n+1}\Vert _{-1}\Vert {\nabla }v_h^{n+1}\Vert \nonumber \\&\quad +\,\Vert f_2^{n+1}\Vert _{-1}\Vert {\nabla }w_h^{n+1}\Vert . \end{aligned}$$

(6.5)

Young’s inequality provides the following bounds on the last five terms in (6.5):

$$\begin{aligned}&|\nu -\nu _m|\Vert {\nabla }v_h^{n+1}\Vert \Vert {\nabla }w_h^{n+1}\Vert \le \frac{|\nu -\nu _m|}{2}\Vert {\nabla }\\&\quad v_h^{n+1}\Vert ^2+\frac{|\nu -\nu _m|}{2}\Vert {\nabla }w_h^{n+1}\Vert ^2,\\&\Vert f_1^{n+1}\Vert _{-1}\Vert \nabla v_h^{n+1}\Vert \le \frac{\nu +\nu _m-|\nu -\nu _m|}{8}\Vert {\nabla } v_h^{n+1}\Vert ^2\\&\quad +\,\frac{2}{\nu +\nu _m-|\nu -\nu _m|}\Vert f_1^{n+1}\Vert _{-1}^2,\\&\Vert f_2^{n+1}\Vert _{-1}\Vert {\nabla }w_h^{n+1}\Vert \le \frac{\nu +\nu _m-|\nu -\nu _m|}{8}\Vert {\nabla }w_h^{n+1}\Vert ^2\\&\quad +\,\frac{2}{\nu +\nu _m-|\nu -\nu _m|}\Vert f_2^{n+1}\Vert _{-1}^2,\\&\frac{|\nu -\nu _m|}{2}|\Vert {\nabla }(w_h^{n+1}-2w_h^n+w_h^{n-1})\Vert \Vert {\nabla }v_h^{n+1}\Vert \le \frac{\nu +\nu _m-|\nu -\nu _m|}{4}\Vert {\nabla }v_h^{n+1}\Vert ^2\\&\quad +\,\frac{(\nu -\nu _m)^2}{4(\nu +\nu _m-|\nu -\nu _m|)}\Vert \nabla (w_h^{n+1}-2w_h^n+w_h^{n-1})\Vert ^2,\\&\frac{|\nu -\nu _m|}{2}|\Vert \nabla (v_h^{n+1}-2v_h^n+v_h^{n-1})\Vert \Vert {\nabla }w_h^{n+1}\Vert \le \frac{\nu +\nu _m-|\nu -\nu _m|}{4}\Vert {\nabla }w_h^{n+1}\Vert ^2\\&\quad +\,\frac{(\nu -\nu _m)^2}{4(\nu +\nu _m-|\nu -\nu _m|)}\Vert \nabla (v_h^{n+1}-2v_h^n+v_h^{n-1})\Vert ^2. \end{aligned}$$

Combining, we now have that

$$\begin{aligned}&\frac{1}{4\Delta t}\big (\Vert v_h^{n+1}\Vert ^2-\Vert v_h^n\Vert ^2+\Vert 2v_h^{n+1}-v_h^n\Vert ^2\nonumber \\&\quad -\,\Vert 2v_h^n-v_h^{n-1}\Vert ^2+\Vert w_h^{n+1}\Vert ^2-\Vert w_h^n\Vert ^2+\Vert 2w_h^{n+1}-w_h^n\Vert ^2\nonumber \\&\quad -\,\Vert 2w_h^n-w_h^{n-1}\Vert ^2+\Vert v_h^{n+1}-2v_h^n+v_h^{n-1}\Vert ^2+\Vert w_h^{n+1}\nonumber \\&\quad -\,2w_h^n+w_h^{n-1}\Vert ^2\big )+\frac{\nu +\nu _m-|\nu -\nu _m|}{8}\big (\Vert {\nabla }v_h^{n+1}\Vert ^2\nonumber \\&\quad +\,\Vert {\nabla }w_h^{n+1}\Vert ^2\big )\le \frac{(\nu -\nu _m)^2}{4(\nu +\nu _m-|\nu -\nu _m|)}\big (\Vert \nabla (v_h^{n+1}-2v_h^n+v_h^{n-1})\Vert ^2\nonumber \\&\quad +\,\Vert \nabla (w_h^{n+1}-2w_h^n+w_h^{n-1})\Vert ^2\big )\nonumber \\&\quad +\,\frac{2}{\nu +\nu _m-|\nu -\nu _m|}\big (\Vert f_1^{n+1}\Vert _{-1}^2+\Vert f_2^{n+1}\Vert _{-1}^2\big ). \end{aligned}$$

(6.6)

The inverse inequality provides the estimate

$$\begin{aligned}\Vert \nabla (z_h^{n+1}-2z_h^n+z_h^{n-1})\Vert ^2\le C_ih^{-2}\Vert z_h^{n+1}-2z_h^n+z_h^{n-1}\Vert ^2,\end{aligned}$$

which allows Eq. (6.6) to be written as

$$\begin{aligned}&\frac{1}{4\Delta t}\big (\Vert v_h^{n+1}\Vert ^2-\Vert v_h^n\Vert ^2+\Vert 2v_h^{n+1}-v_h^n\Vert ^2-\Vert 2v_h^n\nonumber \\&\quad -\,v_h^{n-1}\Vert ^2+\Vert w_h^{n+1}\Vert ^2-\Vert w_h^n\Vert ^2+\Vert 2w_h^{n+1}-w_h^n\Vert ^2 -\Vert 2w_h^n-w_h^{n-1}\Vert ^2\big )\nonumber \\&\quad +\left[ \frac{1}{4{\Delta }t}-\,\frac{(\nu -\nu _m)^2C_ih^{-2}}{4(\nu +\nu _m-|\nu -\nu _m|)}\right] \Vert w_h^{n+1}-2w_h^n+w_h^{n-1}\Vert ^2\nonumber \\&\quad +\left[ \frac{1}{4{\Delta }t}-\frac{(\nu -\nu _m)^2C_ih^{-2}}{4(\nu +\nu _m-|\nu -\nu _m|)}\right] \Vert v_h^{n+1}-2v_h^n+v_h^{n-1}\Vert ^2\nonumber \\&\quad +\,\frac{\nu +\nu _m-|\nu -\nu _m|}{8}\big (\Vert {\nabla }v_h^{n+1}\Vert ^2+\Vert {\nabla }w_h^{n+1}\Vert ^2\big )\nonumber \\&\quad \le \,\frac{2}{\nu +\nu _m-|\nu -\nu _m|}\left( \Vert f_1^{n+1}\Vert _{-1}^2+\Vert f_2^{n+1}\Vert _{-1}^2\right) . \end{aligned}$$

(6.7)

Now using the assumption on the time step size and applying standard techniques completes the proof. $\square$