Eighth Mississippi State - UAB Conference on Differential Equations and
Computational Simulations.
Electron. J. Diff. Eqns., Conference 19 (2010), pp. 189-196.

Title: Comparison of time stepping schemes on the cable equation

Authors: Chuan Li  (Univ. of Tennessee, Knoxville TN, USA)
         Vasilios Alexiades (Univ. of Tennessee, Knoxville TN, USA)
 
Abstract: 
 Electrical propagation in  excitable tissue, such as nerve
 fibers and heart muscle, is described by a parabolic PDE
 for the transmembrane voltage $V(x,t)$, known as the cable equation,
 $$
 \frac{1}{r_a}\frac{\partial^2V}{\partial x^2} =
 C_m\frac{\partial V}{\partial t} + I_{\rm ion}(V,t)
 + I_{\rm stim}(t)
 $$
 where $r_a$ and $C_m$ are the axial resistance and membrane
 capacitance. The source term $I_{\rm ion}$ represents the total ionic
 current across the membrane, governed by the Hodgkin-Huxley or other
 more complicated ionic models. $I_{\rm stim}(t)$ is an applied stimulus
 current.

 We compare the performance of various low and high order
 time-stepping numerical schemes, including DuFort-Frankel
 and adaptive Runge-Kutta, on the 1D cable equation.

Published September 25, 2010.
Math Subject Classifications: 65M08, 35K57, 92C37.
Key Words: Explicit schemes; super time stepping; adaptive Runge Kutta;
           Dufort Frankel; action potential; Luo-Rudy ionic models.