关于求解三维退化Kawarada方程的半自适应LOD方法的正性、单调性和稳定性
On the positivity, monotonicity, and stability of a semi-adaptive LOD method for solving three-dimensional degenerate Kawarada equations
AI总结：
本文研究三维退化Kawarada方程的数值求解，提出半自适应局部一维积分器结合弧长监测机制，获得了保持正性单调性的准则并证明了数值稳定性。
摘要：
本文研究三维退化Kawarada方程的数值求解。这些偏微分方程具有高度非线性的源项，并表现出强烈的淬火奇异性，这给高可靠方案的设计与分析带来了严峻挑战。本研究全程考虑任意固定的非均匀空间网格，这些网格不一定是对称的。数值解通过半自适应局部一维（LOD）积分器推进。时间自适应通过合适的弧长监测机制实现。研究并获得了保持正性和单调性的准则。在谱范数下，从von Neumann意义证明了该分裂方法的数值稳定性。提出并研究了扩展的稳定性期望。
英文摘要：
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.
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