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I would like to solve heat equation

$$\frac{\partial W}{\partial u}=\frac{\partial^{2} W}{\partial x^{2}}$$ such that $W(0,x)=\delta(x)$. This is very easy and has explicit solution. However, I want to solve it numerical, say explicit scheme. I would like to know how to discretize the grid points for delta function for input to give the results. Of course I think I can approximate of $\delta(x)$ as $\lim_{t\rightarrow 0}\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^{2}}{2t}}$. However, there are many function which can tends to dirac function. Which one should I choose? Even for the Gaussian case I stated here, which $t$ should I choose? How does it affect the space grid for input?

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