Innovating New Numerical Techniques for Solving Partial Differential Equations
DOI:
https://doi.org/10.65405/3answj98الكلمات المفتاحية:
Adaptive Discretization, Partial Differential Equations (PDEs), Dynamic Error Control, Iterative Weighted Approximation, Numerical Simulation.الملخص
Partial differential equations (PDEs) are basic equations of fluid mechanics and heat transfer, of electromagnetics, of wave propagation and of financial systems, in which complex phenomena are described. But the usual numerical techniques are difficult to apply to very nonlinear, multi-dimensional, and stiff PDEs, resulting in instabilities, a high computational cost, slow convergence, and errors.In this work, an innovative numerical method combining the technique of adaptive discretization, dynamic error control and an iterative weighted approximation method is presented. The method not only is numerically stable, but also is very economical in terms of the number of operations and truncation error.The proposed approach is exhaustively tested for a number of benchmark problems involving PDEs and compared with the existing techniques, such as the Finite Difference Method, Finite Element Method, Spectral Method, and Physics-Informed Neural Networks (PINNs).The numerical experiment shows that the solution error, solution convergence rate and computational efficiency can be significantly improved without compromising accuracy. The results from the stability analysis and convergence studies show good performance for a broad range of discretization parameters and reliability for engineering applications. In general, the framework is a very effective and efficient approach to the solution of difficult PDEs in contemporary scientific computing and simulations.
التنزيلات
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