# Efficient Discretization of the Laplacian: Application to Moving Boundary Problems
## 2 Materials and Methods
### Application to the Numerical Solution of the Stefan Problem
#### 2.5.5 Numerical Results  
Our model problem represents the physical scenario of a melting volume with heat applied from the bottom. This is modeled using positive Dirichlet boundary conditions on the fixed bottom of \(\Omega_{1,t}\). Additionally, negative Dirichlet boundary conditions are applied to the top surface of \(\Omega_{0,t}\) to account for the outward heat flow due to a negative temperature gradient.  
The initial condition and the corresponding initial grid points are shown in Figure 6. The regions and grid point corresponding to positive temperature are depicted with red  
Figure 5: The steps of shifting the interface \(\Gamma^{n}\) to obtain the new one \(\Gamma^{n+1}\).  
and those with negative temperature with blue color. The discretization of the interface \(\Gamma_{t}\) is defined by the green grid points.  
In practical scenarios involving the modeling of ice layer melting, the characteristic horizontal dimension is often significantly larger than their average thickness. In this way, their interface becomes flat and horizontal [23], corresponding to the numerical experiments in this manuscript.  
Elapsing the time, the material in \(\Omega\) is melting such that the zero temperature level is moving upwards. This is also shown in the numerical simulations in Figure 7. Since we applied homogeneous Neumann boundary conditions, starting from the left and right boundary, the front will be flattened. Also, by the warm-up both from the bottom and the top boundary, corresponding to the boundary conditions in (14) and (15), the \"cold\" region shrinks significantly.  
The simulation data are summarized as follows.  
* Governing equations and the corresponding initial and boundary data are given in Sections 2.4.1 and 2.4.2, respectively.  
Figure 6: Initial condition (**left**, with a heat map) and the corresponding initial grid (**right**) for the simulation of the problem in (11) and (12). Bottom region: \(\Omega_{1,t}\), depicted with red color (**left**) and red grid points (**right**); top region: \(\Omega_{0,t}\), depicted with blue color (**left**) and blue grid points (**right**). The common interface and the corresponding grid points are colored with green. The horizontal distribution of grid points is uniform, while the vertical distribution becomes more dense towards the common interface.  
Figure 7: The result of the simulation procedure for the Stefan problem after 7 time steps (**left**) and 17 time steps (**right**), respectively. The corresponding parameters are given in the list below. Again, blue color represents negative temperature, while red indicates positive temperature.  
- \(D_{0}=2.02\cdot 10^{-1}\) on the top domain;
- \(D_{1}=0.143\cdot 10^{-2}\) on the bottom domain.  
- \(Ny=48\)--vertical: \(24+24\) interior points taking twice the common interface.
* Time step: \(\delta=0.08\).
* Number of time steps: 30.
* Simulation time of one step: 1.05 s.
* Software and hardware: Python code on a Laptop with Intel i7-6700 processor with 2.6 GHz.