Numerical scheme for the one-phase 1D Stefan problem using curvilinear coordinates

Abstract

Abstract. In this paper we present a new approach to solving a one-dimensional, one-phase Stefan problem. The proposed method is based on choosing (a) suitable curvilinear space coordinate/s for the heat-flow equation and the finite difference method. In the final part of this paper, examples of numerical calculations are shown.

Resonant Sensors for Low-Cost, Contact-Free Measurement of Hydrolytic Enzyme Activity in Closed Systems

“…This PDE is governed by the following boundary conditions: (1) at time zero, the enzyme concentration throughout the entire substrate gel is zero, (2) at all times the flux at the O-ring interface ( r = R b in Figure a) is zero, and (3) the concentration of enzyme at the reactive interface ( R i ( t ) in Figure a) is equal to the concentration of enzyme in the center well. The third boundary condition is actually a Stefan boundary condition because its location is time-dependent (as the gel digests, the reactive interface moves toward the O-ring). − The third boundary condition is further complicated because the center well enzyme concentration is also depleting with time due to diffusion of the enzyme into the gel. This problem cannot be solved analytically, and thus was modeled using numerical methods, namely, centered finite differences in the spatial domain with a grid resolution of 10 μm and fourth order Runge–Kutta solver (Matlab ode45) in the time domain (see SI 9).…”

The developments of multi-level computational methodologies for discrete element modelling of granular materials

“…This problem can be formulated in non-dimensional variables for a finite sheet 0 ≤ x ≤ l (l is a standard length) by assuming constant thermal values and using a simple scaling [277].…”