A HYBRID METHOD FOR STIFF REACTION-DIFFUSION EQUATIONS.

Area of Science:

• Numerical analysis
• Computational mathematics
• Chemical kinetics

Background:

• Stiff reaction-diffusion equations pose challenges for numerical solvers.
• Implicit integration factor (IIF2) offers stability but struggles with time-dependent reactions.
• Implicit exponential time differencing (iETD2) handles time-dependent reactions but is computationally expensive.

Purpose of the Study:

• To develop a novel numerical method combining the strengths of IIF2 and iETD2.
• To achieve second-order accuracy and stability for stiff reaction-diffusion systems with time-dependent reactions.
• To improve computational efficiency and handle non-homogeneous boundary conditions.

Main Methods:

• A hybrid approach (hIFE2) applying IIF2 to non-time-dependent reactions and iETD2 to time-dependent reactions.
• Utilizing a transformation to manage non-homogeneous boundary conditions.
• Extending the method for higher spatial dimensions using compact and array representations.

Main Results:

• The hIFE2 method demonstrates superior stability, accuracy, and efficiency compared to existing methods.
• It maintains second-order temporal accuracy with larger time-steps.
• The method effectively handles both linear and nonlinear reaction terms and non-homogeneous boundary conditions.

Conclusions:

• The hIFE2 method offers a robust and efficient solution for stiff reaction-diffusion equations with complex reaction terms.
• This hybrid approach provides a significant advancement in numerical methods for such systems.
• The method's versatility makes it applicable to a wide range of scientific and engineering problems.