New Techniques on Reconstruction and Limiting for Numerical PDE

极限技术的发展始于求解含有间断的非线性守恒律的高分辨率捕获格式。这些方案不会单独跟踪弱溶液中的不连续性，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，并且有一个非线性限制机制来防止不连续区域的虚假振荡，它们可以达到很高的精度。此后，针对许多其他方法和应用开发了限制技术，例如具有限制的Runge-Kutta不连续Galerkin方法、矩限制器等。分层重建将限制定义在单元中的高次多项式的工作分解为一系列较小的工作，每个工作仅涉及从单元平均的线性多项式的非振荡重建。因此，它只使用相邻单元的信息，可以在多维的非结构网格上自然地表达出来。它不使用局部特征分解，因此对要求解的基本方程的依赖性较小。本文针对高阶层次重构提出了几点新的改进措施。对余项在其中的作用进行分析研究可以加深对其限制机制的理解。特别地，提出了一种由单元平均和稀疏位置的多项式逼近重构高次分段多项式函数的紧凑的多步方法。这处房产很新奇。它的发展和理论认识是一个有待探索的新领域。科学、工程、商业和日常生活中越来越复杂的问题都由计算机来处理。然而，只能存储有限数量的信息，所有数字在处理前后都被截断在有限位数的计算机中。因此，计算机模拟是一种近似值，通常像在真实世界中一样是“嘈杂的”。特别是，不平滑的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。非光滑数据在实际应用中是很常见的。例如，在超音速飞行器诱导的冲击波中，气压和密度发生跳跃；人体密度存在各种跳跃；在纳米科学、燃料电池、复合材料、材料缺陷检测等领域，非光滑数据来自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，非光滑数据来自不同的地下结构、不规则的海底、海岸和地面、分离固体、液体和气体的动态界面等。该项目涉及开发和分析一种通用方法，该方法可以在实际不知道的情况下尽可能地从底层解中消除计算伪像。所提出的极限技术对问题的依赖性较小，可用于求解气体动力学方程、磁流体动力学方程和与这些应用相关的许多其他方程。这种新的紧凑的多步重建方法可以显著降低不连续Galerkin方法的存储开销，使其能够解决更复杂的应用。它还可以表示为一种紧凑的内插方法，可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。

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.