On the positivity, monotonicity, and stability of a semi-adaptive LOD method for solving three-dimensional degenerate Kawarada equations
Joshua L. Padgett, Qin Sheng
TL;DR
This work tackles the numerical solution of three-dimensional degenerate Kawarada equations with strong quenching singularities on nonuniform grids. It introduces a semi-adaptive Local One-Dimensional (LOD) splitting scheme that employs arc-length-based temporal adaptation and a $[1/1]$ Padé-based exponential splitting to advance the solution, discretizing the $v'(t)=\sum_{\sigma=1}^3 M_{\sigma} v+g(v)$ system via Kronecker-structured matrices. The authors establish positivity, monotonicity, and stability properties under precise grid- and step-size conditions, including von Neumann stability for the linearized (frozen) case and extensions to the nonlinear regime with bounded Jacobians. The results provide a robust framework for reliable simulations of Kawarada-type quenching on complex grids and point to future work in high-performance computing and higher-order splitting techniques.
Abstract
This paper concerns the numerical solution of three-dimensional degenerate Kawarada equations. These partial differential equations possess highly nonlinear source terms, and exhibit strong quenching singularities which pose severe challenges to the design and analysis of highly reliable schemes. Arbitrary fixed nonuniform spatial grids, which are not necessarily symmetric, are considered throughout this study. The numerical solution is advanced through a semi-adaptive Local One-Dimensional (LOD) integrator. The temporal adaptation is achieved via a suitable arc-length monitoring mechanism. Criteria for preserving the positivity and monotonicity are investigated and acquired. The numerical stability of the splitting method is proven in the von Neumann sense under the spectral norm. Extended stability expectations are proposed and investigated.