An efficient method to solve the fuzzy Crank–Nicolson scheme with application to the groundwater flow problem

Abstract

In this paper, we propose a comprehensive methodological framework for solving the fuzzy groundwater flow problem in a simpler and faster way based on numerical analysis. In particular, a novel simplified matrix explicit inverse formula is proposed as an efficient method to solve the fuzzy Finite Difference numerical scheme, called Crank–Nicolson implicit scheme. The main advantage of the proposed method is that it offers an efficient and simpler solution to the algebraic tridiagonal system of equations that appeared in the fuzzy Crank–Nicolson scheme, without affecting the accuracy of the results. Without doubt, the simulation of an unconfined aquifer flow, using the Finite Difference Method, is often time-consuming due to the costly calculations required to solve the implicit Finite Difference schemes. This process becomes even more difficult when the physical problem is solved in its fuzzy form where the calculations become more complex and greatly increase. However, problem uncertainties are not considered negligible and must be included in the final calculations for a more rational management of the water body. This creates the need to find a new and faster way to solve the fuzzy implicit schemes, as proposed in this work. The simplified matrix explicit inverse formula was applied to the fuzzy Crank–Nicolson scheme to solve the fuzzy partial differential equation of Boussinesq and, hence, the simulation of the physical problem. Its results were compared with the corresponding results of the Thomas algorithm, which until today, is the most common method of solving the tridiagonal system of equations. The case under consideration refers to a sudden rise in the lake's water level, thus resulting in the aquifer recharging from the lake. The performance of the proposed method was more than satisfactory in terms of calculation accuracy and reliability where the numerical results are matched with the differences appearing from the 12 decimal point onward between the two examined methods. The proposed method tested also considered the running times, achieving much better times compared with the Thomas algorithm, where the differences ranged from a few seconds to several hours depending on the examined time and space steps. HIGHLIGHTS This study proposes a novel solution method of the fuzzy Crank–Nicolson scheme.; It provides better running times for the fuzzy numerical schemes.; It compares the proposed simplified explicit inverse formula with the Thomas algorithm.; Faster conclusions for real physical problems contain fuzziness.; This study provides a strong advantage to engineers for efficient water management planning.;

Keywords

• depth profiles
• gh-derivative
• matrix inverse
• parabolic equations
• water volume