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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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Initializing implicit linear multistep methods
The standard approach is to use a self-starting time-marching algorithm with sufficiently small timestep (such that the order of accuracy is not spoiled) and compute the 5 non-initial value previous s …