An Implicit Boundary Integral Method for Interfaces Evolving by Mullins-Sekerka Dynamics

Abstract

This paper proposes an algorithm for simulating interfacial motions in Mullins-Sekerka dynamics using a boundary integral equations formulated on implicit surfaces. The proposed method is able to simulate the dynamics on unbounded domains, and as such will provide a tool to better understand the dynamics obtained in these situations. As pointed out in Gurtin's paper [23], the behavior of the dynamics are far more interesting on domains that are unbounded. However, it seems that there is no other computational method that can simulate the nonlinear interfacial motion of Mullins-Sekerka dynamics in three dimensions on unbounded domains, allowing the interfaces to merge, break up, without the need of finding explicit representations.The Mullins-Sekerka flow is a Stefan-type free boundary problem involving an nonlocal interfacial motion dynamically controlled by the solution of Laplace's equation with appropriate boundary conditions obtained on both sides of the interface. As a result, it is reasonable to consider boundary integral methods, especially for exterior domains, combined with level set methods [40][20], for easy tracking and for being able to handle topological changes.Various numerical methods have been proposed to solve elliptic problems on irregular domains. We mention here some of the few "usual suspects" that use fi-

An extrapolative approach to integration over hypersurfaces in the level set framework

“…For integration of singularities such as 1/ √ x in the interval [0, 1], one typically needs to require that the step size h = h(x) decreases sufficiently fast as x tends to 0, otherwise, the resulting quadrature will have a significant drop in the order of accuracy. However, as Table 5 shows, the relative errors computed with φ (1) decrease very slowly, but they are all under 1%; the relative errors computed with φ (2) decrease steadily as the mesh refines.…”

“…Analytically, the gradients of φ (1) and φ (2) exist almost everywhere. So in our computation, we globally define the gradients to be |∇φ (1) i,j | = √ 2 and |∇φ (2) i,j | = 8|φ (2) i,j |. Let f (r) = 1/ √ r, for r = 0 and f (0) = 10 9 .…”

“…where δ = δ 0.1,∞,1 . In Table 5, we present numerical errors of our computations, where is adjusted for each kernel so that the same number of points are used in the narrow bands {0.1 < |φ (1) i,j | < } and {0.1 < |φ (2) i,j | < }. This is an example that suggests the potential of the proposed extrapolative approach in computing integrands involving integrable singularities.…”

Implicit boundary integral methods for the Helmholtz equation in exterior domains

“…The exact integral formulation we use in this work was first proposed in [11] and extended in [12]. It is also generalized in [4] to simulate the Mullins-Sekerka dynamics of complicated interface geometry in unbounded domains.…”

Numerical Preservation of Velocity Induced Invariant Regions for Reaction–Diffusion Systems on Evolving Surfaces

“…Among the applications of RDSs on surfaces we mention brain growth [26], cell migration [3], chemotaxis [12], developmental biology [28], electrodeposition [24] and phase field modeling [42]. The growing interest toward PDEs on evolving surfaces has stimulated the development of several numerical methods for such problems, among which we mention (but not limited to) embedding methods [2], kernel methods [18], implicit boundary integral methods [5,35], surface finite element methods (SFEM) [10] and some of their recent variations and extensions [13,16,17,20,23,40].…”