An Efficient Adaptive Rescaling Scheme for Computing Moving Interface Problems

Abstract

In this paper, we present an efficient rescaling scheme for computing thelong-timedynamics of expanding interfaces. The idea is to design an adaptive time-space mapping such that in the new time scale, the interfaces evolves logarithmically fast at early growth stage and exponentially fast at later times. The new spatial scale guarantees the conservation of the area/volume enclosed by the interface. Compared with the original rescaling method in [J. Comput. Phys. 225(1) (2007) 554–567], this adaptive scheme dramatically improves the slow evolution at early times when the size of the interface is small. Our results show that the original three-week computation in [J. Comput. Phys. 225(1) (2007) 554–567] can be reproduced in about one day using the adaptive scheme. We then present the largest and most complicated Hele-Shaw simulation up to date.

Nonlinear limiting dynamics of a shrinking interface in a Hele-Shaw cell

“…2007; Zhao et al. 2016, 2017). Introduce a new frame such that where the space scaling captures the size of the interface, is the position vector of the scaled interface and parameterizes the interface.…”

“…To overcome these numerical issues, we have developed a rescaled boundary integral scheme (Zhao et al 2018); the method is briefly described here. The idea is to map the original time and space (x, t) into new coordinates (x,t) such that the interface can evolve at an arbitrary speed in the new rescaled frame (see also Li et al 2007;Zhao et al 2016Zhao et al , 2017. Introduce a new frame (x,t) such that…”

Computation of a Shrinking Interface in a Hele-Shaw Cell

“…To overcome these difficulties, we develop a spectrally accurate boundary integral method in which a new time and space rescaling is implemented. The rescaling idea [29,49,50] is to map the original time and space (x, t) into new coordinates (\= x, \= t) such that the interface can evolve at an arbitrary speed in the new rescaled frame. In particular, for the shrinking interface problem, we choose (1) the space scaling function R( \= t) so that the shrinking interface is always mapped back to its initial size, i.e., the interface does not shrink in the rescaled frame; (2) the time scaling function \rho ( \= t) to slow down the motion of the interface, especially at later times when the interface becomes very small and shrinks extremely rapidly.…”

“…After the peak point Q D , although the time step \Delta \= t continues to decrease, the shape factor decays monotonically to a circular shape as the fingers are smoothed out (Figure 4(a)). We note that one may use another time scale function to speed up the calculation in this time period to gain more efficiency, e.g., following the idea in [50].…”

“…The time scaling function \rho (t) = \= \rho ( \= t) maps the original time t to the new time \= t and \rho (t) has to be positive and continuous. The evolution of the interface in the scaled frame can be accelerated [50,49] or decelerated by choosing a different \rho (t). A straightforward calculation shows the normal velocity in the new frame…”