Analysis of three dimensional advection-diffusion type models using wavelet scheme with error estimate
Muhammad Awais Fiaz, Abdul Ghafoor, Emel Khan · Physica Scripta · 2025
Abstract Solving high-dimensional partial differential equations (PDEs) is an essential problem in various fields of science and engineering. Analytical solutions for high-dimensional PDEs are often intricate and rare, as very few problems possess such solutions. Therefore, numerical methods play a pivotal role in overcoming these challenges. This work proposes an efficient numerical method for solving three-dimensional advection-diffusion equations. The present approach employs Haar wavelets to estimate the solution and its spatial derivatives, while the time derivative is approximated using a finite difference scheme. The collocation technique is then applied, transforming the three-dimensional PDEs into coupled linear algebraic equations with unknown wavelet coefficients. Solving this linear system provides the unknown coefficients, which refine the solution and its derivatives sequentially at each time step. Additionally, the error and stability analysis of the proposed numerical scheme are examined, linking the resolution level with the error and the spectral radius with stability. The method is implemented for the solution of three-dimensional advection-diffusion equations. To asses the attainments and performance of the method, L ∞ , L 2, and L RMS , error norms along with the relative error associated with infinity and L 2 norms are presented in tabulated form. Two- and three-dimensional solution profiles are also displayed to demonstrate the efficiency of the suggested technique. Simulations confirm that the proposed numerical method is a powerful tool for the numerical approximations of three-dimensional advection-diffusion type models.