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To help get around the convergence properties of the elliptic operators themselves, we can pose a semi-implicit method for both the mono- and bi-domains. For the monodomain case, we split the elliptic (matrix) term into two pieces, one at time-step (t+h) and one at time-step t (scaled by theta and (1-theta), respectively):

We then collect the terms at time-step (t+h) on the left-hand side:

In particular, note that the matrix we are solving against now has an extra term added to the main diagonal. This terms goes as 1/h and thus, as h decreases, the main diagonal of the matrix gets larger. As the matrix gets more and more diagonally dominant, it becomes easier to solve.

For the semi-implicit bidomain, we get the following (in block form):

Now, the main diagonal again increases as 1/h, but so do the vi to ve sub-blocks.