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Sparsity II: Explicit stages #115

@rckirby

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@rckirby

When implicit methods have explicit first or last stages, one can solve a linear system with one fewer stage, but our logic has to be updated accordingly

  • advance has to start and/or end with the relevant explicit updates
  • form manipulation has to produce the reduced linear system on a smaller number of stages
  • we have to make sure we don't break anything along the way.

Additionally, certain methods with explicit first stage (e.g. LobattoIIIA, aka Crank-Nicolson) have a no-op explicit first stage. Currently, doing Crank-Nicolson in Irksome is just not competitive since one solves for two stages instead of one (Implicit midpoint/GL(1) is equivalent for linear problems and works fine for nonlinear and doesn't have this problem).

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