I'll start by doing this in MATLAB.
A Standard Brownian Motion $dX_t$ can be approximated by a scaled random walk through $\triangle{X}=Z\sqrt{\triangle t}$. Analogously the drift of a Super-Brownian Motion can be approximated as $\dot{W}(t,x)=Z_{x,t}\sqrt{\triangle t}\sqrt{\triangle x}$ where $Z_{x,t}\sim N(0,1)$ or a random variable with variance 1. For example, a Bernoulli random variable taking 1 and -1 each with probability $\frac{1}{2}$
A Super-Brownian Motion is a continuous stochastic process satisfying the differential equation
$$\frac{\partial X}{\partial t}=\frac{X''(t,x)}{2}+\sqrt{X}\dot{W}$$
By Feyman-Kac's formula and Ito's Isometry
$$E[[\int_0^t\int_{\mathbb{R}}\sqrt{X(s,x)}dW(s,x)]^2]=E[\int_0^t\int_{\mathbb{R}}X(s,x)dxds]$$
Thus we may discretize the solution by $$E[\sum_{s