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| 一维Hamilton-Jacobi方程的高阶混合WENO格式 |
| A High-Order Hybrid WENO Scheme for One-Dimensional Hamilton–Jacobi Equations |
| 投稿时间：2026-01-05 修订日期：2026-03-15 |
| DOI： |
| 中文关键词: Hamilton-Jacobi方程 混合WENO格式 数值振荡 高分辨率 |
| 英文关键词: Hamilton-Jacobi equation hybrid WENO scheme numerical oscillations high resolution |
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| 中文摘要: |
| 为了解决Hamilton-Jacobi方程因其解出现奇异性而导致数值格式的高精度与稳定性难以兼顾的问题,本文基于Unequal-Sized WENO格式(US-WENO)的核心思想,设计出一种混合五阶有限差分加权本质无振荡(WENO)数值方法。该方法区别于传统方法在于:在重构过程中引入一阶差商替代点值,使导数信息成为预警信号,从而提前触发非线性权重的自适应调整机制。数值实验结果表明,该方法在解的光滑区域能保持高阶精度,在间断附近可自动切换至小模板以确保高分辨率,有效抑制了数值振荡,提升了计算稳定性与精度。 |
| 英文摘要: |
| To solve the problem that high accuracy and stability of numerical schemes for Hamilton-Jacobi equations cannot be balanced due to solution singularities, a hybrid fifth-order finite difference Weighted Essentially Non-Oscillatory (WENO) numerical method is proposed based on the core idea of the Unequal-Sized WENO (US-WENO) scheme. Unlike traditional methods, this method replaces point values with the first-order difference quotient in the reconstruction process, taking derivative information as a warning signal to trigger the adaptive adjustment mechanism of nonlinear weights in advance. Numerical experiments demonstrate that the method maintains high-order accuracy in the smooth regions of the solution and switches automatically to a small stencil near discontinuities to ensure high resolution, thus effectively suppressing numerical oscillations and improving computational stability and accuracy. |
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