Coupled differential equations metropolis algorithm

I would like to simulate the following coupled SDE. $ $ d\Omega=-\Omega H\left(\mathrm{n}\right)d\tau$ $ $ $ dn=-Ad\tau+\sum_{i}B^{\left(i\right)}dW_{i}+\sum_{i}C^{\left(i\right)}dW’_{i}$ $ provided $ $ A=\frac{1}{2}n\left(T-UM\right)\left(I-n\right)+\frac{1}{2}\left(I-n\right)\left(T-UM\right)n$ $

$ $ B_{xy}^{\left(i\right)}=\sqrt{\frac{U}{2}}\sum_{\sigma,\sigma’}\sigma_{\sigma\sigma’}^{z}n_{x,\left(i\sigma’\right)}\left(\delta_{\left(i\sigma\right),y}-n_{\left(i\sigma\right),y}\right)$ $ $ $ \sigma^{z}=\begin{bmatrix}1 & 0\ 0 & -1 \end{bmatrix}$ $

$ $ C_{xy}^{\left(i\right)}=\sqrt{\frac{U}{2}}\sum_{\sigma,\sigma’}\sigma_{\sigma\sigma’}^{z}\left(\delta_{x,\left(i\sigma’\right)}-n_{x,\left(i\sigma’\right)}\right)n_{\left(i\sigma\right),y}$ $

$ $ M_{\left(i\sigma\right),\left(j\sigma’\right)}=\delta_{ij}\sum_{\eta,\eta’}n_{\left(i\eta\right),\left(i\eta’\right)}\left(\sigma_{\sigma\sigma’}^{z}\sigma_{\eta\eta’}^{z}-\sigma_{\sigma\eta’}^{z}\sigma_{\eta\sigma’}^{z}\right)$ $

Initial condition as $ $ n=\frac{1}{2}I$ $ $ I$ is a 2M X 2M matrix.$ M$ is the mode number.

$ $ H(n)=\sum_{x,y=1}^{2M}T_{xy}n_{xy}+\frac{U}{2}\sum_{i=1}^{M}\sum_{\begin{array}{cc} \eta, & \eta’\ \sigma, & \sigma’ \end{array}\begin{array}{c} \ =1 \end{array}}^{2}\left[n_{\left(i\eta\right),\left(i\sigma’\right)}n_{\left(i\sigma\right),\left(i\eta’\right)}-n_{\left(i\eta\right),\left(i\eta’\right)}n_{\left(i\sigma\right),\left(i\sigma’\right)}+n_{\left(i\eta\right),\left(i\sigma’\right)}\delta_{\sigma\eta’}-n_{\left(i\eta\right),\left(i\sigma’\right)}\delta_{\left(i\sigma\right),\left(i\eta’\right)}\right]\sigma_{\eta\eta’}^{z}\sigma_{\sigma\sigma’}^{z}$ $

For example if $ M=2$ initial value of If M=2, then

Initial value of n matrix will be

$ $ n=.5\begin{bmatrix}1 & 0 & 0 & 0\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 0\ 0 & 0 & 0 & 1 \end{bmatrix}$ $

Also $ $ T=\begin{bmatrix}-.1 & -2 & 0 & 0\ -2 & -.1 & 0 & 0\ 0 & 0 & -.1 & -2\ 0 & 0 & -2 & -.1 \end{bmatrix}$ $

$ U/t=2$

Is there a way to solve the above using the successive metropolis method as shown below? d I found this in the following paper. https://arxiv.org/abs/0704.3792