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Which numerical libraries / math software can allow me to solve partial differential equations in spherical coordinates? (my system consists of N degrees of freedom, each degree lives in $\mathbb{C}^{2}$ and the whole system is $(\mathbb{C}^{2})^{N}$).

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You can look into MudPack to see if this helps for your problem here: MUDPACK

Hope this helps a bit.

Good Luck.

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