A stable r-adaptive mesh technique to analyze the advection-diffusion equation

Abstract

This paper offers a study of the moving mesh method employed in one-dimensional linear and nonlinear advection-diffusion equations with different boundary and initial conditions. Advection and diffusion appear in the crux of the physical processes, where the transport of heat or other physical variables evolves. The aim is to present an accurate, stable moving finite-difference meshing scheme with its convergence. The velocity-profile of the considered cases is non-linear; therefore, the difference scheme needs mesh refinement. The moving mesh method analyzes the problem physics and adjusts the mesh according to the problem as it moves nodes in the region of more fluctuations. The approximate numerical results are estimated using four moving mesh partial differential equations with varying numbers of nodes. The moving mesh method is an r-adaptive technique that uses a fixed number of mesh nodes and moves the grids where error reduction is needed. The numerical solutions obtained are compared with the analytical solutions cited from the literature. The study presents five cases dealing with linear and non-linear examples in detail to understand the physics of the problem. The presented difference scheme is considerably more efficient than the numerical methods given in the literature.

Investigating the ability of PINNs to solve Burgers’ PDE near finite-time blowup

“…At the very outset, we note that variants of Burgers equation continue to be explored via ever more sophisticated methods not using deep-learning [51,52]. But in such studies, neither are the setups chosen corresponding to the challenging edge-case of finite-time blow-up as we consider here nor is there a study of the accuracy of any theoretical error bound in implementations.…”