BIT Numerical Mathematics

, Volume 58, Issue 2, pp 489–507

# Error estimate of the finite volume scheme for the Allen–Cahn equation

Article

## Abstract

The Allen–Cahn equation originates in the phase field formulation of phase transition phenomena. It is a reaction-diffusion ODE with a nonlinear reaction term which allows the formation of a diffuse phase interface. We first introduce a model initial boundary-value problem for the isotropic variant of the equation. Its numerical solution by the method of lines is then considered, using a finite volume scheme for spatial discretization. An error estimate is derived for the solution of the resulting semidiscrete scheme. Subsequently, sample numerical simulations in two and three dimensions are presented and the experimental convergence measurement is discussed.

## Keywords

Allen–Cahn equation Error estimate Finite volume method Phase field Semidiscrete scheme Solidification

## Mathematics Subject Classification

65M08 65M15 80A22 74N05

## Notes

### Acknowledgements

This work has been supported by the project of the Czech Science Foundation no. 14-36566G Multidisciplinary research centre for advanced materials.

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