Kinetic schemes for assessing stability of traveling fronts for the Allen-Cahn equation with relaxation
For researchers in numerical analysis and reaction-diffusion systems, this work provides numerical confirmation of theoretical stability conditions, though it is an incremental extension of the authors' prior theoretical work.
The paper develops kinetic-based finite volume schemes for the hyperbolic Allen-Cahn equation and uses numerical experiments to validate previously established theoretical stability results for traveling waves, demonstrating their validity beyond formal assumptions.
This paper deals with the numerical (finite volume) approximation of reaction-diffusion systems with relaxation, among which the hyperbolic extension of the Allen--Cahn equation represents a notable prototype. Appropriate discretizations are constructed starting from the kinetic interpretation of the model as a particular case of reactive jump process. Numerical experiments are provided for exemplifying the theoretical analysis (previously developed by the same authors) concerning the stability of traveling waves, and important evidence of the validity of those results beyond the formal hypotheses is numerically established.