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首页> 外文期刊>Journal of Computational Physics >Towards a compact high-order method for non-linear hyperbolic systems. I: The Hermite Least-Square Monotone (HLSM) reconstruction
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Towards a compact high-order method for non-linear hyperbolic systems. I: The Hermite Least-Square Monotone (HLSM) reconstruction

机译：迈向非线性双曲系统的紧凑高阶方法。 I：Hermite最小二乘单调（HLSM）重建

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摘要

A new Hermite Least-Square Monotone (HLSM) reconstruction to calculate accurately complex flows on non-uniform meshes is presented. The coefficients defining the Hermite polynomial are calculated by using a least-square method. To introduce monotonicity conditions into the procedure, two constraints are added into the least-square system. Those constraints are derived by locally matching the high-order Hermite polynomial with a low-order TVD or ENO polynomial. To emulate these constraints only in regions of discontinuities, data-depending weights are defined; those weights are based upon normalized indicators of smoothness of the solution and are parameterized by a O(1) quantity. The reconstruction so generated is highly compact and is fifth-order accurate when the solution is smooth; this reconstruction becomes first-order in regions of discontinuities. By inserting this reconstruction into an explicit finite-volume framework, a spatially fifth-order non-oscillatory method is then generated. This method evolves in time the solution and its first derivative. In a one-dimensional context, a linear spectral analysis and extensive numerical experiments make it possible to assess the robustness and the advantages of the method in computing multi-scales problems with embedded discontinuities.
机译：提出了一种新的Hermite最小二乘单调（HLSM）重构，可精确计算非均匀网格上的复杂流。通过使用最小二乘法计算定义Hermite多项式的系数。为了将单调性条件引入到过程中，在最小二乘系统中添加了两个约束。这些约束是通过将高阶Hermite多项式与低阶TVD或ENO多项式进行局部匹配而得出的。为了仅在不连续区域模拟这些约束，定义了取决于数据的权重;这些权重基于解决方案平滑度的归一化指标，并通过O（1）量进行参数化。这样生成的重构非常紧凑，当求解光滑时，重构精度达到五阶。这种重建在不连续区域成为一阶。通过将这种重构插入明确的有限体积框架中，可以生成空间五阶非振荡方法。该方法会随着时间的推移演化解决方案及其一阶导数。在一维情况下，线性光谱分析和广泛的数值实验使评估该方法在计算具有嵌入式不连续性的多尺度问题时的鲁棒性和优势成为可能。

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