Error estimate of the finite volume scheme for the Allen-Cahn equation
The authors consider the numerical solution of the initial-boundary value problem for the Allen-Cahn equation \[ \xi \frac{\partial p}{\partial t} = \xi \Delta p + \frac{1}{\xi} f_0(p) + c_0W \text{ in } (0,T) \times \Omega, \] where \(\Omega \subset \mathbb{R}^d\) is a bounded polygonal domain with \(d \in \{2,3\}\), \(f_0(p):=p(1-p)(p-1/2)\), \(\xi>0\) is a constant parameter and \(c_0W\) is a constant forcing term. The authors study only the isotropic case. \vskip 0.8em For the numerical solution the method of lines is used with finite volumes as space discretization. Finite volumes turn out to be more suitable than finite differences because they produce edge-symmetric stencils. A first-order error estimate is proved for the semidiscrete finite volume problem. The proof is based on a maximum principle also provided by the authors. Numerical results are shown for a 2D and a 3D example, in which the finite volume scheme uses a uniform rectangular mesh and the time integration is carried out by the fourth-order Runge-Kutta-Merson solver.
scientific article; zbMATH DE number 6894733
scientific article