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## 数学代写|偏微分方程代写partial difference equations代考|OTHER DETERMINISTIC METHODS

Outside of the three volume discretization based methods discussed in the preceding three subsections, a myriad of other methods for solving PDEs exist and continue to be used. It is probably fair to state, however, that these “other” methods are either still in their infancy – relatively speaking – or have been used only for a special class of problems for which they were designed. In terms of usage, there is no doubt that the three methods discussed in subsections 1.3.1 through 1.3.3 far outweigh any other method for solving PDEs. While construction of an exhaustive list of other methods for solving PDEs is quite daunting and, perhaps, of little use in the opinion of this author, at least a few of these other methods deserve a mention. These include the so-called meshless or meshfree method, the boundary element method, and the spectral method.

The name meshless or meshfree method is quite broad, and is applicable to any method that does not rely on a mesh. These may also include statistical or spectral methods. In the specific case of solution of PDEs, however, the name is reserved for methods where a cloud of points is used to develop an algebraic approximation to the PDE. These points have domains of influence, but do not have rigid interconnections, as in a traditional mesh. One of the notable features of meshless methods is that the points (or nodes) can be allowed to move freely, which, in the context of a traditional mesh, may end up twisting the mesh to a state where it cannot be used, and may require remeshing – a step that brings a new set of difficulties to the table. In light of this feature, meshless methods have grown in popularity for the solution of problems that involve moving boundaries, such as fluid-structure interaction [39.40]. multiphase flow $[41,42]$, moving body problems, and any problem where nodes may be created or destroyed due to material addition or removal [42]. The meshless method is a close relative of the smooth particle hydrodynamics method $[43,44]$ that was developed for fluid flow. Some researchers contend that, fundamentally, the method is closely related to either variational formulations using radial basis functions or the moving least squares method [45]. The general concept of not using a mesh has been used in a variety of methods that include the element-free Galerkin method [46], the meshless local Petrov-Galerkin method [47], the diffuse element method [48],and the immersed boundary method [49], among others. A good review of meshless methods, as it applies to the solution of advection-diffusion type PDEs, has been provided by Pepper [50].

The boundary element method (BEM) is a derivative of the finite element method. Its attractive feature is that it does not require a volumetric mesh, but only a surface mesh. This is particularly convenient for $2 \mathrm{D}$ geometries, wherein the only geometric input required is a set of coordinates of points. BEM is particularly suited to problems for which the interior solution is not of practical interest, but the quantities at the boundary are, for example, a heat transfer problem in which one is primarily interested in the heat flux and temperature at the boundaries. The method is significantly more efficient than the conventional finite element method. Unfortunately, the method has several serious limitations. BEM can only be used for PDEs with constant coefficients or in which the variation of the coefficients obey a certain format. This is because it is only applicable to problems for which Green’s functions can be computed easily. Nonlinear PDEs, PDEs with strong inhomogeneities, and 3D problems are generally outside the realm of pure BEM. The application of BEM to such scenarios usually requires a volumetric mesh and combination of BEM with a volume discretization method, which undermines its advantage of being an efficient method. On account of these limitations BEM has found limited use except for some specific problems in mechanics $[51,52]$, heat conduction $[53,54]$, and electrostatics [51].

## 数学代写|偏微分方程代写partial difference equations代考|OVERVIEW OF MESH TYPES

Since the focus of this text is the finite difference and finite volume methods, and both of those methods require a mesh for discretization of the governing PDE, a discussion of the mesh – both its type and attributes – is a prerequisite for understanding the discretization procedures presented in later chapters. The word, grid, usually refers to the interconnected nodes and lines (see Fig. 1.4), while the word, mesh, usually refers to the spaces enclosed by the lines. Throughout this text, this subtle distinction will be ignored since it only creates confusion in the mind of the reader. Henceforth, the words mesh and grid will be used interchangeably.

The history of discretization-based procedures for solving boundary value problems can be traced back to the works of Courant around 1928. In those days, when computers were almost nonexistent, solving an elliptic PDE on a square mesh with a few handful of node points was already considered ground breaking. The need for a complex procedure or algorithm to generate a mesh was never felt. As a matter of fact, this trend continued into the early 1960s, during which period, most reported computations were performed on rectangular domains with a perfectly orthogonal (Cartesian) mesh, and mostly for 2D geometries, as shown in Fig. 1.6(a). The seminal paper by Patankar and Spalding, published in 1972, which introduced the world to the famous SIMPLE algorithm [61], reports solution of the Navier-Stokes equation performed in a square domain with a $16 \times 16$ Cartesian mesh. With rapidly advancing computer technology, by the late 1970 s, the feasibility of performing slightly larger scale computations was realized.

Along with the desire to perform larger computations came the desire to perform computations in irregular geometries. In the earliest computations of this type, angled or curved boundaries were addressed using a stair-step grid [5,26], as shown in Fig. 1.6(b). With the growing emphasis on satisfying conservation laws, and the resulting proliferation of the finite volume method, the stair-step approach was soon replaced by the cut cell approach, which produced polygonal cells (control volumes) at curved boundaries, as depicted in Fig. 1.7. This was considered both a reasonable as well as convenient approximation – reasonable because with a fine mesh one can always approximate a curve with piecewise linear segments (it is a practice followed even today!), and convenient because the core finite volume procedure required little modification to accommodate one slanted edge.

# 偏微分方程代写

## 有限元方法代写

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。