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

Convergence of the Finite Volume Method on Unstructured Meshes for a 3D Phase Field Model of Solidification

“…The two predominant classes of results on the numerical analysis of the PFM are adaptations of [22] and the use of approximative sequences of functions and compact embedding that show existence of the weak solution and convergence at the same time [19,10,5]. One can find examples of the first approach in our previous work [40] or [14] and the use of compact embedding techniques is demonstrated in [8,18,33,20]. In our previous work [40], we took the first mentioned approach and derived estimates that show convergence rates of the FVM applied to a simplified phase field (Allen-Cahn) equation on an unstructured mesh .…”

“…One can find examples of the first approach in our previous work [40] or [14] and the use of compact embedding techniques is demonstrated in [8,18,33,20]. In our previous work [40], we took the first mentioned approach and derived estimates that show convergence rates of the FVM applied to a simplified phase field (Allen-Cahn) equation on an unstructured mesh . In this paper, we have chosen the second of the two mentioned approaches.…”

Local radial basis function collocation method preserving maximum and monotonicity principles for nonlinear differential equations

“…At present, most of the numerical methods for the maximum principle of Allen-Cahn equation, the monotonicity principle of the blow-up model and the pattern formation generation of Gray-Scott model are finite difference method [34][35][36][37][38][39][40], finite element method [41][42][43][44], and finite volume method [45,46]. There are few studies on the preserving physical properties of partial differential equations using the radial basis configuration method.…”