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 $$E[\sum_{s<t}\sum_{s\in\mathbb{R}}\sqrt{X(s,x)}Z_{x,s}\sqrt{\triangle t}\sqrt{\triangle x}]\approx E[\sum_{s<t}\sum_{x\in\mathbb{R}}X(s,x)Z_{x,s}^2\triangle t\triangle x]$$ To simulate the stochastic differential equation, we may have $$\frac{X''(t,x)}{2}\approx \frac{1}{2}X(t,x+\triangle x)+\frac{1}{2}X(t,x-\triangle x)-X(t,x)$$ and $$\frac{\partial X}{\partial t}\approx X(t+\triangle t,x)-X(t,x)$$ For more about simulating Super-Brownian Motion, see Achim Klenke: http://www.aklenke.de.