Algorithm 2 Centroid space-time discretization algoriithm—lattice:

The field is u(r, t) with local dynamics ${\partial }_{tu}=\mathcal{F}\left[u\right]$, discretize space over nodes i = 1, …, M and time $t_k$ = t0 + $k_h$. Input: {$u_i^0$}, operator $\mathcal{F}$h includes local and neighboring nodes, step h, and number of steps N. For each time layer k → k + 1 and each node I, we have the following [58,59]: (a) Euler/RK: $u_i^{k+1}=u_i^k+h{\mathcal{F}}_h(u_{i-1}^k,u_i^k,u_{i+1}^k)$; (b) Centroid (implicit) version. We solve the following equation:                 $u_i^{k+1}=u_i^k+{\mathcal{F}}_h({\displaystyle \frac{u_{i-1}^k+u_{i-1}^{k+1}}{2}},{\displaystyle \frac{u_i^k+u_i^{k+1}}{2}},{\displaystyle \frac{u_{i+1}^k+u_{i+1}^{k+1}}{2}})$ by applying iterations until convergence $\Vert u_i^{\left(n+1\right),k+1}-u_i^{\left(n\right),k+1}\Vert \le tol;$ 3.Output: {$u_i^k$}, for $k=1\dots N$. Discrete cyclic vertices and lattice step are obtained with a continuous model and the vertex map. The flow ${\varphi }^t{:\mathbb{R}}^3\to {\mathbb{R}}^3$ generated by the Lorenz system has the following form:         $$\dot{x}=\sigma \left(y-x\right), \dot{y}=\left(\rho -z\right), \dot{z}=xy-\beta z,$$ (21) with classical parameters $\sigma >0, \rho >0, \beta >0$. Let $X(t; X_0)=\left(x\right(t), y(t), z(t\left)\right)$ be a solution with initial condition X0.