Modeling The Fluid Flow for Seepage Problem: A Crank-Nicolson Approach

The two-dimensional Laplace equation (which models the fluid flow of the seepage problem) was solved by a finite difference method, which is the Crank-Nicolson Method (CNM) subject to some boundary and initial conditions. We compared the numerical results of CNM to the Alternating Direction Explicit Finite Difference Scheme (ADES). First, we derived the finite differential form of the implicit, explicit, and Crank-Nicolson methods for the given model and then presented an algorithm for each method. The resulting systems of linear algebraic equations for the CNM solved using the MATLAB software. The solutions were presented graphically in three dimensions and interpreted. We also analyzed the numerical stability of the CNM by the matrix method and found it to be unconditionally stable. We observed that the seepage flow decreases with distance from the source (dam) and also the smaller the mesh size, the finer the decrease in the fluid seepage.