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Composite Backward Differentiation Formula for the Bidomain Equations.

Xindan Gao1, Craig S Henriquez2, Wenjun Ying1

  • 1School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China.

Frontiers in Physiology
|December 31, 2020
PubMed
Summary

Two new fully implicit methods, backward Euler and composite backward differentiation formula (CBDF2), efficiently solve cardiac electrical activity models. CBDF2 offers high stability and accuracy for bidomain equations.

Keywords:
bidomain equationscardiaccomposite backward differentiation formulafully implicit methodsoperator splitting

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Area of Science:

  • Computational biology
  • Biomedical engineering
  • Numerical analysis

Background:

  • The bidomain equations are crucial for modeling cardiac electrical activity.
  • Implicit numerical methods offer superior stability over explicit methods for these simulations.
  • Solving large nonlinear systems in implicit methods is computationally intensive.

Purpose of the Study:

  • To present two fully implicit time integration methods for the bidomain equations.
  • To develop computationally efficient and stable numerical schemes for cardiac modeling.
  • To reduce the computational burden of implicit methods for bidomain equations.

Main Methods:

  • Implementation of the backward Euler method.
  • Development and implementation of a second-order, L-stable composite backward differentiation formula (CBDF2) scheme.
  • Application of a nonlinear elimination method to solve the resulting nonlinear systems.
  • Utilizing approximate Newton approaches with efficient solvers for the reduced global system.
  • Exploring operator splitting techniques as an alternative solution strategy.

Main Results:

  • The developed implicit methods, particularly CBDF2, significantly enhance stability and accuracy.
  • The nonlinear elimination method results in a smaller, more manageable global system with a symmetric, possibly positive definite Jacobian.
  • The approximate Newton approach efficiently solves the reduced system using standard solvers.
  • Operator splitting offers an alternative efficient solution pathway.

Conclusions:

  • The CBDF2 scheme is an efficient, stable, and accurate time integration method for the bidomain equations.
  • The proposed methods provide a computationally feasible approach for simulating cardiac electrical activity.
  • These advancements contribute to more effective computational modeling in cardiology.