New Techniques on Reconstruction and Limiting for Numerical PDE

数值偏微分方程重构与限制新技术

• 批准号: 1115671
• 负责人: Yingjie Liu
• 金额: $ 17.1万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2014-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1115671&HistoricalAwards=false
• 关键词: New; Techniques; Reconstruction; Limiting; Numerical

The development of limiting techniques starts from high resolution capturing schemes for solving nonlinear conservation laws whose weak solutions contain discontinuities. These schemes do not trace discontinuities in a weak solution individually and automatically smear them into transition layers within a few mesh cells. They can achieve high order of accuracy if the solution is smooth and there is a nonlinear limiting mechanism to prevent spurious oscillations in the vicinities of discontinuities. The limiting techniques have since been developed for many other methods and applications, e.g., the Runge-Kutta discontinuous Galerkin methods with limiting, the moment limiter etc. Hierarchical reconstruction decomposes the job of limiting a high degree polynomial defined in a cell into a series of smaller jobs, each of which only involves the non-oscillatory reconstruction of a linear polynomial from cell averages. Therefore it only uses information from adjacent cells and can be naturally formulated on unstructured meshes in multi dimensions. It does not use local characteristic decomposition and thus is less dependent on the underlying equation to be solved. The principle investigator proposes several new improvements related to the hierarchical reconstruction in higher orders. The analytical study of the role of the remainder term in it could provide deeper understanding of the limiting mechanism. In particular, a compact, multi-step method is proposed to reconstruct a piecewise polynomial function of high degree from cell averages and sparsely located polynomial approximations. This property is novel. Its development and theoretical understanding is a new area to be explored.More and more complex problems from science, engineering, business and daily life are handled by computers. However, only a finite amount of information can be stored and all numbers are truncated in a computer with a finite number of digits before and after being processed. Therefore a computer simulation is an approximation and is usually "noisy" as in the real world. In particular, non-smooth data tends to induce artifacts in computational solutions, making them less useful or completely useless. Non-smooth data is common in real applications. For example, the air pressure and density have jumps across a shockwave induced by a supersonic aircraft; the human body contains various jumps in density; in nanoscience, fuel cells, composite materials, material defect detection etc, non-smooth data originates from interfaces between different materials, irregular boundaries and cracks; in simulations in environmental science, ocean and atmosphere, non-smooth data comes from heterogeneous underground structures, irregular seafloor, seashore and ground surface, dynamic interfaces separating solid, liquid and gas etc. The project involves the development and analysis of a general method which eliminates as much computational artifacts as possible from the underlying solution without actually knowing it. The proposed limiting techniques are less problem dependent and can be useful in solving gas dynamics equations, magnetohydrodynamics equations and many other equations related to these applications. The new compact, multi-step reconstruction method could significantly reduce the memory cost of the discontinuous Galerkin methods enabling them to solve more complicated applications. It can also be formulated as a compact interpolation method and can be broadly used in computer graphics, image processing and many other scientific and engineering computations.

极限技术的发展始于求解弱解包含不连续点的非线性守恒律的高分辨率捕获方案。这些方案不单独跟踪弱溶液中的不连续点，并自动将它们涂抹到几个网格单元内的过渡层中。如果解是光滑的，并且在不连续点附近有一个非线性限制机制来防止伪振荡，它们可以达到高阶精度。极限技术已经发展为许多其他方法和应用，如龙格-库塔不连续伽辽金方法的极限，矩限制器等。分层重构将限制在单元中定义的高次多项式的工作分解为一系列较小的工作，每个工作只涉及从单元平均值对线性多项式的非振荡重构。因此，它只使用相邻单元的信息，并且可以在多维的非结构化网格上自然地形成。它不使用局部特征分解，因此较少依赖于待解的底层方程。主要研究者提出了一些新的改进，涉及到高阶的层次重建。对剩余项在其中的作用进行分析研究，可以对限制机制有更深入的认识。特别地，提出了一种紧凑的、多步的方法，从单元平均和稀疏定位的多项式近似中重构高阶分段多项式函数。这种特性是新颖的。它的发展和理论认识是一个有待探索的新领域。计算机处理越来越多的科学、工程、商业和日常生活中的复杂问题。然而，只能存储有限的信息量，所有的数字在计算机中被截断，处理前后的数字数量是有限的。因此，计算机模拟是一种近似值，通常与现实世界一样是“有噪声的”。特别是，非光滑数据往往会在计算解决方案中产生伪影，使它们变得不那么有用或完全无用。非平滑数据在实际应用中很常见。例如，空气压力和密度在超音速飞机引起的冲击波中有跳跃；人体包含各种密度跳跃；在纳米科学、燃料电池、复合材料、材料缺陷检测等领域，非光滑数据来源于不同材料之间的界面、不规则的边界和裂纹；在环境科学、海洋和大气模拟中，非光滑数据来自非均匀地下结构、不规则海底、海岸和地表、固、液、气分离的动态界面等。该项目涉及到一个通用方法的开发和分析，该方法可以在不实际了解的情况下从底层解决方案中消除尽可能多的计算工件。所提出的限制技术较少依赖于问题，并且可以用于求解气体动力学方程，磁流体动力学方程以及与这些应用相关的许多其他方程。这种紧凑的多步重构方法可以显著降低不连续伽辽金方法的存储成本，使其能够解决更复杂的应用。它也可以被表述为一种紧凑的插值方法，可以广泛应用于计算机图形学、图像处理和许多其他科学和工程计算中。