A linearly implicit scheme and fast multigrid solver for 3D Fitzhugh-Nagumo equation

In this work, a second-order finite difference (FD) scheme for three-dimensional (3D) nonlinear Fitzhugh-Nagumo (FN) equation with the nonlinear term treated with semi-implicitly technique is proposed. The existence and uniqueness of the difference scheme is proved, and the stability and convergence of numerical solution in L-infinity-norm are also shown. Then, we employ an efficient extrapolation cascadic multigrid (EXCMG) method to solve the large linear system arising from the proposed second-order FD discretization for the 3D FN equation. Numerical results are presented to verify our theoretical findings of the difference scheme and the efficiency of the EXCMG method. The EXCMG method can also be extended to solve other kinds of time-dependent nonlinear partial differential equations.

Automated summary

This work proposes a second-order finite difference (FD) scheme for the three-dimensional (3D) nonlinear Fitzhugh-Nagumo (FN) equation, with the nonlinear term treated using a semi-implicit technique. The existence and uniqueness of the difference scheme are proven, and the stability and convergence of the numerical solution are demonstrated. Additionally, an efficient extrapolation cascadic multigrid (EXCMG) method is employed to solve the large linear system arising from the FD discretization. Numerical results confirm the theoretical findings of the difference scheme and the efficiency of the EXCMG method, which can also be extended to solve other types of time-dependent nonlinear partial differential equations.