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Implicit methods are timestepping methods that use an inversion at every timestep. This allows for much better stability properties than explicit methods, though it comes with a serious speed penalty in some cases. Examples of implicit methods include Backward Euler and Crank-Nicholson.

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Is it possible to solve nonlinear PDEs without using Newton-Raphson iteration?

I assume, that you have conducted a space discretization, so that you are about solving the (vector-valued) ODE $$ \dot u_h(t) = F_h(t,u_h(t)), \text{ on [0,T] }, u_h(0) = \alpha. $$ via a numerica …
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