In this paper, we introduce a novel high-order shock tracking method and provide a proof of concept. Our method leverages concepts from implicit shock tracking and extended discontinuous Galerkin methods, primarily designed for solving partial differential equations featuring discontinuities. To address this challenge, we solve a constrained optimization problem aiming at accurately fitting the zero iso-contour of a level set function to the discontinuities. Additionally, we discuss various robustness measures inspired by both numerical experiments and existing literature. Finally, we showcase the capabilities of our method through a series of two-dimensional problems, progressively increasing in complexity.

翻译：本文提出一种新型高阶激波追踪方法并给出概念验证。该方法融合隐式激波追踪与扩展间断伽辽金方法的核心理念，专为求解含间断的偏微分方程设计。为应对该挑战，我们求解一个约束优化问题，旨在将水平集函数的零等值线精确拟合至间断处。此外，基于数值实验与现有文献的启发，我们探讨了多种鲁棒性增强策略。最后，通过一系列复杂度逐步递增的二维问题，展示了该方法的能力。