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Squark comments on Open Thread for February 11 - 17 - Less Wrong
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It shouldn't be that hard to find code that solves a non-linear PDE. Google search reveals http://einsteintoolkit.org/ an open source that does numerical General Relativity.
However, QFT is not a PDE, it is a completely different object. The keyword here is lattice QFT. Google reveals this gem: http://xxx.tau.ac.il/abs/1310.7087
Nonperturbative string theory is not completely understood, however all known formulations reduce it to some sort of QFT.