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Third-order Energy Stable WENO scheme

机译：三阶能量稳定WENO方案

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A new third-order Energy Stable Weighted Essentially Non-Oscillatory (ESWENO) finite difference scheme for scalar and vector hyperbolic equations with piecewise continuous initial conditions is developed. The new scheme is proven to be linearly stable in the energy norm for both continuous and discontinuous solutions. In contrast to the existing high-resolution shock-capturing schemes, no assumption that the reconstruction should be total variation bounded (TVB) is explicitly required to prove stability of the new scheme. We also present new weight functions which drastically improve the accuracy of the third-order ESWENO scheme. Based on a truncation error analysis, we show that the ESWENO scheme is design-order accurate for smooth solutions with any number of vanishing derivatives, if its tuning parameters satisfy certain constraints. Numerical results show that the new ESWENO scheme is stable and significantly outperforms the conventional third-order WENO scheme of Jiang and Shu in terms of accuracy, while providing essentially non-oscillatory solutions near strong discontinuities. (C) 2009 Elsevier Inc. All rights reserved.
机译：针对具有分段连续初始条件的标量和向量双曲方程，开发了一种新的三阶能量稳定加权本质非振荡（ESWENO）有限差分方案。事实证明，该新方案对于连续和不连续解在能量范数上都是线性稳定的。与现有的高分辨率震荡捕获方案相比，没有明确要求重建应为总变化有界（TVB）的假设来证明新方案的稳定性。我们还提出了新的权重函数，这些函数极大地提高了三阶ESWENO方案的准确性。基于截断误差分析，我们表明，如果调整参数满足某些约束条件，则对于任何数量的消失导数的平滑解决方案，ESWENO方案都是设计顺序准确的。数值结果表明，新的ESWENO方案是稳定的，并且在准确性方面明显优于Jiang和Shu的常规三阶WENO方案，同时在强不连续点附近提供了基本的非振荡解决方案。 （C）2009 Elsevier Inc.保留所有权利。

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