Scheme 2

The second scheme is defined using the linearized Crank-Nicolson Galerkin method, which consists of finding $U^n, V^n \in \mathcal{V}_{m_1}$ and $Z^n, R^n \in \mathcal{V}_{m_2}$ such that

\[\begin{align*} \begin{aligned} & \big(\varphi,\bar{\partial}V^n\big) + \alpha^{n-\frac{1}{2}}\Big[ \big(\nabla\varphi,\nabla\widehat{U}^n\big) - \big(\varphi,\widehat{R}^n)_{\Gamma_1} + \big(\varphi,g(V^{\ast n})\big)_{\Gamma_1}\Big] + \big(\varphi,f(U^{\ast n})\big) = \big(\varphi,f_1^{n-\frac{1}{2}}\big), \quad\forall\varphi\in \mathcal{V}_{m_1}, \\ & \big(\phi,q_1\bar{\partial}R^n + q_2\widehat{R}^n + q_3\widehat{Z}^n + q_4\widehat{V}^n\big)_{\Gamma_1} = \big(\phi,f_2^{n-\frac{1}{2}}\big)_{\Gamma_1}, \quad\forall\phi\in \mathcal{V}_{m_2}, \\ & \bar{\partial}U^n = \widehat{V}^n,\quad \bar{\partial}Z^n = \widehat{R}^n, \end{aligned} \end{align*}\]

for $n = \text{“1,0''},\,1,\,2,\,\ldots$, with $U^0, V^0 \in \mathcal{V}_{m_1}$ and $Z^0, R^0 \in \mathcal{V}_{m_2}$ given as approximations of the initial conditions $u_0, v_0, z_0$, and $r_0$, defined in the same way as in Scheme 1.

Note

The first two time steps in \eqref{def:approx2} constitute a single-step predictor-corrector initialization.
In the prediction step (case $n=\text{“1,0''}$), temporary approximations $U^{\text{“1,0''}}$, $V^{\text{“1,0''}}$, $Z^{\text{“1,0''}}$, and $R^{\text{“1,0''}}$ at $t_1$ are computed using the initial solution $U^0$, $V^0$, $Z^0$, and $R^0$.
In the correction step (case $n=1$), the temporary approximations from the prediction step, together with the initial solutions, are used to obtain the definitive approximations $U^1$, $V^1$, $Z^1$, and $R^1$.
For $n \geq 2$, the approximations are generated using information from the two previous time steps at the nonlinear terms.

Notation

In addition to the operators $\displaystyle\bar{\partial}w^n=\frac{w^n - w^{n-1}}{\tau}$ and $\displaystyle\widehat{w}^n = \frac{w^n + w^{n-1}}{2}$, consider

\[\bar{\partial}w^{\text{“1,0''}} = \frac{w^{\text{“1,0''}} - w^0}{\tau},\quad \widehat{w}^{\text{“1,0''}} = \frac{w^{\text{“1,0''}} + w^0}{2}, \quad\text{and}\quad w^{*n} = \begin{cases}\displaystyle w^0, & \text{if } n = \text{“1,0''}, \\[10pt] \displaystyle \frac{w^{\text{“1,0''}}+w^0}{2}, & \text{if } n = 1, \\[10pt] \displaystyle \frac{3w^{n-1}-w^{n-2}}{2}, & \text{if } n \geq 2. \end{cases}\]

Matrix Formulation

\[\begin{align*} \begin{aligned} & M^{m_1\times m_1}\bar{\partial}v^n + \alpha^{n-\frac{1}{2}}\Big[ K^{m_1\times m_1}\widehat{d}^n - M^{m_1\times m_2}\widehat{r}^n + G^{m_1}(v^{\ast n})\Big] + F^{m_1}(d^{\ast n}) = \mathcal{F}^{m_1}(f_1^{n-\frac{1}{2}}), \\ & M^{m_2\times m_2}\big[ q_1\bar{\partial}r^n + q_2\widehat{r}^n + q_3\widehat{z}^n] + q_4M^{m_2\times m_1}\widehat{v}^n = \mathcal{F}^{m_2}(f_2^{n-\frac{1}{2}}), \\ & \bar{\partial}d^n = \widehat{v}^n,\quad \bar{\partial}z^n = \widehat{r}^n. \end{aligned} \end{align*}\]

Solving the Algebraic Systems

\[Q(n) \widehat{v}^n = L(n,v^{*n},d^{*n}).\]

Once $\hat{v}^n$ has been determined, we compute $\hat{r}^n$ via

\[\hat{r}^n = - \frac{\tau q_4}{q_5}\hat{v}_{1:m_2}^n + \frac{2q_1}{q_5} r^{n-1} - \frac{\tau q_3}{q_5} z^{n-1} + \frac{\tau}{q_5} \Big(M^{m_2\times m_2}\Big)^{-1} \mathcal{F}^{m_2}(f_2^{n-\frac{1}{2}}),\]

and the remaining quantities $d^n$, $v^n$, $z^n$, and $r^n$ by

\[\begin{align*} & v^n = 2\hat{v}^n - v^{n-1}, \quad r^n = 2\hat{r}^n - r^{n-1}, \quad d^n = \tau\hat{v}^n + d^{n-1}, \quad z^n = \tau\hat{r}^n + z^{n-1}. \end{align*}\]

Details

Rewriting the second equation in terms of $\hat{r}^n$ and substituting it into the first yields:

\[\begin{align*} & M^{m_1\times m_1}\bar{\partial}v^n + \alpha^{n-\frac{1}{2}}\Big[ K^{m_1\times m_1}\widehat{d}^n + G^{m_1}(v^{\ast n})\Big] + F^{m_1}(d^{\ast n}) \\[5pt] &\qquad + \frac{\alpha^{n-\frac{1}{2}}}{q_5} \begin{bmatrix} \tau q_4M^{m_2\times m_2} \widehat{v}_{1:m_2}^n - M^{m_2\times m_2}\big( 2q_1r^{n-1} - \tau q_3z^{n-1} \big) - \tau \mathcal{F}^{m_2}(f_2^{n-\frac{1}{2}}) \\[5pt] 0^{(m_1-m_2)} \end{bmatrix} = \mathcal{F}^{m_1}(f_1^{n-\frac{1}{2}}). \end{align*}\]

Applying the identities

\[\bar\partial v^n = \frac{2}{\tau}\widehat{v}^n - \frac{2}{\tau}v^{n-1}, \quad \widehat{d}^n = \frac{\tau}{2} \widehat{v}^n + d^{n-1},\]

we obtain

\[\begin{align*} & M^{m_1\times m_1} \big(\frac{2}{\tau}\widehat{v}^n-\frac{2}{\tau}v^{n-1}\big) + \alpha^{n-\frac{1}{2}}\Big[ K^{m_1\times m_1} \big(\frac{\tau}{2}\widehat{v}^n+d^{n-1}\big) + G^{m_1}(v^{\ast n})\Big] + F^{m_1}(d^{\ast n}) \\[10pt] &\qquad + \frac{\alpha^{n-\frac{1}{2}}}{q_5} \begin{bmatrix} \tau q_4M^{m_2\times m_2} \widehat{v}_{1:m_2}^n - M^{m_2\times m_2}\big( 2q_1r^{n-1} - \tau q_3z^{n-1} \big) - \tau \mathcal{F}^{m_2}(f_2^{n-\frac{1}{2}}) \\[5pt] 0^{(m_1-m_2)} \end{bmatrix} = \mathcal{F}^{m_1}(f_1^{n-\frac{1}{2}}). \end{align*}\]

Isolating $\widehat{v}^n$:

\[\begin{align*} & \Big( 2M^{m_1\times m_1} + \frac{\tau^2}{2}\alpha^{n-\frac{1}{2}} K^{m_1\times m_1} + \frac{\tau^2q_4}{q_5}\alpha^{n-\frac{1}{2}} \begin{bmatrix} M^{m_2\times m_2} & 0^{m_2\times(m_1-m_2)}\\[5pt] 0^{(m_1-m_2)\times m_2} & 0^{(m_1-m_2)\times(m_1-m_2)} \end{bmatrix} \Big) \widehat{v}^n \\[10pt] &\qquad = -\tau\alpha^{n-\frac{1}{2}} G^{m_1}(v^{\ast n}) - \tau F^{m_1}(d^{\ast n}) + 2M^{m_1\times m_1}v^{n-1} - \tau \alpha^{n-\frac{1}{2}}K^{m_1\times m_1} d^{n-1} + \tau \mathcal{F}^{m_1}(f_1^{n-\frac{1}{2}}) \\[10pt] &\qquad + \frac{\tau}{q_5}\alpha^{n-\frac{1}{2}} \begin{bmatrix} M^{m_2\times m_2}\big( 2q_1r^{n-1} - \tau q_3z^{n-1} \big) + \tau \mathcal{F}^{m_2}(f_2^{n-\frac{1}{2}}) \\[5pt] 0^{(m_1-m_2)} \end{bmatrix} \end{align*}\]

Matrix and vector definitions

\[\begin{align*} Q = & 2M^{m_1\times m_1} + \frac{\tau^2}{2}\alpha^{n-\frac{1}{2}}K^{m_1\times m_1} + \frac{\tau^2q_4}{q_5}\alpha^{n-\frac{1}{2}} \begin{bmatrix} M^{m_2\times m_2} & 0^{m_2\times(m_1-m_2)}\\[5pt] 0^{(m_1-m_2)\times m_2} & 0^{(m_1-m_2)\times(m_1-m_2)} \end{bmatrix} \\[30pt] L =& -\tau\alpha^{n-\frac{1}{2}} G^{m_1}(v^{\ast n}) - \tau F^{m_1}(d^{\ast n}) + 2M^{m_1\times m_1}v^{n-1} - \tau \alpha^{n-\frac{1}{2}}K^{m_1\times m_1} d^{n-1} + \tau \mathcal{F}^{m_1}(f_1^{n-\frac{1}{2}}) + \frac{\tau}{q_5}\alpha^{n-\frac{1}{2}} \begin{bmatrix} M^{m_2\times m_2}\big( 2q_1r^{n-1} - \tau q_3z^{n-1} \big) + \tau \mathcal{F}^{m_2}(f_2^{n-\frac{1}{2}}) \\[5pt] 0^{(m_1-m_2)} \end{bmatrix} \end{align*}\]