Add tests for central differences for a 2nd order operator about diffeqoperators.jl HOT 8 CLOSED

Agreed, but we please write those equations in the weak PDE form so it's easier to do the translation. To keep things simple, it's just the heat equation $ u_t = \sigma(x)u_{xx} $. There are some heat equation tests already:

https://github.com/JuliaDiffEq/DiffEqOperators.jl/blob/master/test/heat_eqn.jl

but it never hurts to add more.

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Shouldn't that be $\sigma(x)$ for the variance (i.e. still linear PDE)?

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Yes thanks. Fixed.

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Maybe we can put in comments in the tests that mention both the PDE and stochastic process forms. Economists/finance people will find it much easier to think in terms of stochastic processes than the resulting PDEs. Also, don't forget that those are frequently NOT the PDEs that would come out of the discretized operators (e.g. backwards equations are different and the variational-inequalities that come out of stopping problems are also very different from that).

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Note, given the grid x_grid to find the discretized $\sigma(x) d W_t$ operator, you should do something like sigma(x_grid)' * DerivativeOperator(...) * sigma(x_grid) where the parameters for the derivative operator are the same as the one without the changing variance.

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Correction, you need the u in there to apply the operator.

sigma(x_grid)' * DerivativeOperator(...) * u * sigma(x_grid)

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Note: only test the absorbing barrier with a value of 0 for now.

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Again, come back to this after #51 is complete.

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