A high-resolution fuzzy transform combined compact scheme for 2D nonlinear elliptic partial differential equations.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Applied Physics

Background:

• Solving two-dimensional nonlinear elliptic partial differential equations (PDEs) is crucial in various scientific fields.
• Existing methods often face challenges with accuracy and computational efficiency for complex nonlinear problems.

Purpose of the Study:

• To propose a new high-resolution fuzzy transform algorithm for solving two-dimensional nonlinear elliptic PDEs.
• To achieve fourth-order accuracy in numerical solutions using approximating fuzzy components.
• To develop a computationally efficient scheme with minimal data storage.

Main Methods:

• Implementation of the approximating fuzzy components method.
• Utilizing triangular basic functions and local linear combinations of solution values at nine points.
• Connecting approximate fuzzy components with exact solution values via a linear system.
• Employing compact approximations leading to a block tridiagonal Jacobi matrix.
• Constructing closed-form approximate solutions using 2D spline interpolation.

Main Results:

• The proposed algorithm achieves fourth-order accuracy for internal mesh points.
• Compact approximations result in a block tridiagonal Jacobi matrix for efficient computation.
• Closed-form approximate solutions can be easily constructed.
• The method demonstrates fourth-order convergence and provides upper bounds for approximation errors.
• Simulations confirm the scheme's usefulness for quantum mechanics and convection-dominated diffusion problems.

Conclusions:

• The developed high-resolution fuzzy transform algorithm effectively solves two-dimensional nonlinear elliptic PDEs.
• The combination of fuzzy transform and compact discretizations provides high-order accuracy.
• The numerical scheme is computationally efficient and requires minimal data storage, making it suitable for practical applications.