# On the positivity, monotonicity, and stability of a semi-adaptive LOD method for solving three-dimensional degenerate Kawarada equations

## TL;DR

This paper develops a semi-adaptive LOD method for solving complex 3D degenerate Kawarada equations, ensuring positivity, monotonicity, and stability despite nonlinearities and singularities.

## Contribution

It introduces a semi-adaptive LOD scheme with a novel arc-length based temporal adaptation and stability analysis for challenging degenerate PDEs.

## Key findings

- Positivity and monotonicity criteria are established.
- Stability of the splitting method is proven in the von Neumann sense.
- The method effectively handles highly nonlinear source terms and singularities.

## 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.