CARP provides several solution methods to solve the cardiac bidomain and monodomain equations. The default numerical scheme is based on the following linear transformation:

where and are the extracellular potential and transmembrane voltage respectively; represents the ionic current variables; and are conductivity tensors of intracellular and extracellular spaces respectively; is the isotropic conductivity of the fluid in which the heart is immersed (bath and cavities); is the capacitance per unit area and is surface to volume ratio; and g model ionic currents and specify the cell membrane model.
At the tissue-bath interface, continuity of the normal component of the extracellular current and continuity of are enforced. The normal component of the intracellular current vanishes at all tissue boundaries whereas the normal component of the extracellular current vanishes at the boundaries of the bath.

Numerically, the bidomain equations are decoupled by an operator splitting approach that leads to a three step scheme involving the solution of a parabolic PDE, an elliptic PDE and a non-linear system of ODEs at each time step. In the simplest case both the parabolic PDE and the non-linear ODE systems are solved via the explicit forward-Euler scheme

where is the discretized operator; is the time step; , and are the temporal discretizations of , and , respectively, for time equal to . For meshes of finer spatial discretization the parabolic PDE is solved employing a implicite Crank-Nicholson approach.

The system of equations is solved using the finite element method, employing linear elements and lumped mass matrices.
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