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Runge-kutta works when you’re given an initial condition and the derivative is with respect to just one variable (so you’re given f(t) just at t=0). What do you do when you’re given a boundary condition and the derivative is with respect to many variables? This.

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I mean, you can also just do finite differences. Of course Galerkin opens up the very “rich” design space of finite elements methods.

Finite differences are simple in cubes and domains that can be mapped to one. Not so much in an arbitrary tetrahedral mesh.

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