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首页> 外文期刊>Turkish Journal of Analysis and Number Theory >A New Approximation (ziti's δ-scheme) of the Entropic (Admissible) Solution of the Hyperbolic Problems in One and Several Dimensions: Applications to Convection, Burgers, Gas Dynamics and Some Biological Problems
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A New Approximation (ziti's δ-scheme) of the Entropic (Admissible) Solution of the Hyperbolic Problems in One and Several Dimensions: Applications to Convection, Burgers, Gas Dynamics and Some Biological Problems

机译：一维和双维双曲问题的熵（可容许）解的新逼近（zitiδ方案）：对流，Burgers，气体动力学和一些生物学问题的应用

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摘要

As it is well known in numerical analysis, most of the numerical schemes have undesirable oscillations, especially near the domain's border, or near the physical phenomena (empty region, collapse, boundary layer, among others) (mathematically invisible) eg: Burgers equation(the solution loses its regularity in finite time). In the case where the differential problem solution presents a singularity (shock, blow-up which cannot be numerically detected easily), the classical scheme cannot generally operate correctly and in the best case we are confronted with a very difficult algorithm, especially in several dimensions. Our objective here is to construct a less complicated scheme compared to the classical methods by keeping their advantages and obtained the admissible solution in the most difficult situations without complications obtained from the selected meshing. In this paper, we applied our new method called ziti's δ- scheme which is able to resist to such oscillations near the singularity and enables us to detect a lot of physical phenomena (eg: shock waves, rarefaction waves, conservation of the matter quantity ...). We depict the ziti's δ- scheme for the multidimensional partial differential equations and systems on any meshing with simple numbering. We apply our method to some models and compare its results with the exact one and other classical numerical methods. We can conclude that our results are very striking. The ziti's δ-method that we obtained is faster and more efficient and robust.
机译：在数值分析中众所周知，大多数数值方案都有不希望的振荡，尤其是在区域边界附近或物理现象（空区域，塌陷，边界层等）附近（在数学上不可见），例如：Burgers方程（解决方案会在有限时间内失去规律性）。在微分问题解决方案呈现出奇异性的情况下（无法轻易通过数值检测到的震动，爆炸），经典方案通常无法正常运行，并且在最佳情况下，我们面临着非常困难的算法，尤其是在多个维度。与经典方法相比，我们的目标是通过保留传统方法的优势来构建一种较简单的方案，并在最困难的情况下获得可接受的解决方案，而不会因所选网格划分而产生复杂性。在本文中，我们应用了称为ziti的δ-方案的新方法，该方法能够抵抗奇点附近的此类振荡，并使我们能够检测到许多物理现象（例如：冲击波，稀疏波，物质量守恒）。 ..）。我们在任何具有简单编号的网格上描绘了多维偏微分方程和系统的zitiδ-方案。我们将我们的方法应用于某些模型，并将其结果与精确的一种和其他经典数值方法进行比较。我们可以得出结论，我们的结果非常惊人。我们获得的ziti的δ方法更快，更有效，更健壮。

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