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ExplicitTimeIntegratorThe most basic time-stepper is Forward Euler, an explicit method. For the Monodomain Equations, we have:  To update the variables at time-step (t+h), we can explicitly compute the right-hand side of the equations (hence the term "explicit method"). This makes the time-stepper extremely easy to program and very fast to compute. However, it is only O(h) accurate and thus may require very small time-step sizes and hence very many time-step updates to compute. Note that with some preprocessing, we could do the two matrix vector products, tmp=Mi*vm and inv(A)*tmp, in one step, thus reducing the amount of computation required by each time-step. For the Bidomain Equations, the explicit method requires a second phase. Following the work of Brad Roth (NeedReferences), we pose the Bidomain system as:  To update the variables at time-step (t+h), we can explicitly compute the right-hand side of the first two equations and update q and vm. We then use the new value of vm to compute a source term for the final equation. This last equation then becomes a linear system which we must solve for ve. Note, however, that we are solving directly against the elliptic operators. These operators generally produce matrices with bad condition numbers and bad convergence properties. |