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## 物理代写|空气动力学代写Aerodynamics代考|Fundamental Stability Theory

This chapter examines the stability of difference schemes for initial value problems defined by ordinary or partial differential equations. Three simple examples are examined first: an ordinary differential equation, the linear advection equation, which is the prototype for hyperbolic equations, and the diffusion equations. These serve to illustrate that in all three cases the numerical scheme can become unstable if the time step is too large, or in the case of a hyperbolic equation, if the difference scheme does not contain the proper region of dependence. The use of implicit schemes can remove the limit on the time step, at the expense of greater solution complexity.

Next, the general definitions of consistency, convergence, and stability are introduced in terms of an arbitrary norm, leading to the Lax equivalence theorem that consistent and stable schemes must converge to the true solution in the limit as the mesh interval and time step are reduced toward zero. Then stability in the Euclidean norm is examined, and the von Neumann stability test is introduced as a convenient way to deduce the stability of any linear scheme.

Nonlinear conservation laws generally admit solutions containing discontinuities, such as shock waves in a fluid flow. This motivates the need for difference schemes in conservation form. Moreover, the linear stability theory is no longer adequate, and schemes that pass a von Neumann test can easily admit highly oscillatory solutions in the neighborhood of shock waves. Thus alternative measures of stability are needed, leading to the introduction of concepts such as total variation diminishing (TVD) and local extremum diminishing (LED) schemes. Generally it proves necessary to use upwind-biased schemes to meet these criteria, either by direct construction or through the introduction of a controlled amount of artificial diffusion.

The simplest example of an initial value problem for a hyperbolic equation is provided by the linear advection equation
$$\frac{\partial u}{\partial t}+a \frac{\partial u}{\partial x}=0, \quad t>0, \quad-\infty \leq x \leq \infty,$$
which defines the solution $u(x, t)$ for a given initial condition
$$u(x, 0)=f(x) .$$
For convenience assume that a is positive. Then this equation describes a simple wave motion in which the solution is transported to the right at a speed $a$. Along a line $x-a t=\xi$, as illustrated in Figure 4.4, we have
$$\frac{d u}{d t}=\frac{d}{d t} u(a t+\xi, t)=a \frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0 .$$
Thus $u$ is constant along this line, which is a characteristic – that is a line along which $\frac{\partial}{\partial t}=a \frac{\partial}{\partial x}$ is an internal operator – giving no information about derivatives normal to the line. This means that

1. $u$ cannot be arbitrarily specified along such a line, because $u$ is constant according to the equation
2. data given along the line is not sufficient to continue the solution into a larger region.
The general solution of (4.5) has the form
$$u(x, t)=f(\xi)=f(x-a t) .$$
This represents the solution uniquely in terms of the initial values. Conversely every $u$ of the form (4.6) is a solution of (4.5) provided it is differentiable, as can be verified by substituting $f(x-a t)$ for $u(x, t)$. Note that $u(x, t)$ depends only on previous values along the characteristic passing through the point $(x, t)$ and hence only on $f(x)$ at a single point on the axis.
Consider now a difference approximation with interval $\Delta x, \Delta t$ of the form
$$\frac{v(x, t+\Delta t)-v(x, t)}{\Delta t}+a \frac{v(x+\Delta x, t)-v(x, t)}{\Delta x}=0 .$$

## 物理代写|空气动力学代写空气动力学代考|线性平流方程

$$\frac{\partial u}{\partial t}+a \frac{\partial u}{\partial x}=0, \quad t>0, \quad-\infty \leq x \leq \infty,$$
，它定义了给定初始条件的解$u(x, t)$
$$u(x, 0)=f(x) .$$

$$\frac{d u}{d t}=\frac{d}{d t} u(a t+\xi, t)=a \frac{\partial u}{\partial x}+\frac{\partial u}{\partial t}=0 .$$

1. $u$不能沿着这条直线任意指定，因为根据
2. 公式，$u$是常数，沿着这条直线给出的数据不足以将解延续到更大的区域。
(4.5)的通解形式为
$$u(x, t)=f(\xi)=f(x-a t) .$$
这表示了根据初值唯一的解。相反，形式(4.6)的每个$u$都是(4.5)的解，只要它是可微的，这可以通过将$f(x-a t)$替换为$u(x, t)$来验证。注意，$u(x, t)$只依赖于经过$(x, t)$点的特征的先前值，因此只依赖于轴上单个点上的$f(x)$。现在考虑一个具有区间$\Delta x, \Delta t$的差分逼近形式
$$\frac{v(x, t+\Delta t)-v(x, t)}{\Delta t}+a \frac{v(x+\Delta x, t)-v(x, t)}{\Delta x}=0 .$$

## 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。

Posted on Categories:Aerodynamics, 物理代写, 空气动力学

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## 物理代写|空气动力学代写Aerodynamics代考|Multi-dimensional Finite Element Schemes with Linear Elements

Finite element methods are very easy to define for triangular and tetrahedral meshes using piecewise linear trial solutions. Then we introduce basis functions $\phi_j$ that have the value unity at the $j$ th node and zero at all other nodes, as illustrated in Figure $3.8$ for a triangular mesh. We can visualize the basis function of each node as a tent surrounding the node. Consider now Laplace’s equation in a triangulated domain $\mathcal{D}$ with Dirichlet boundary conditions on the boundary $\mathcal{B}$,
\begin{aligned} &u_{x x}+u_{y y}=0 \text { in } \mathcal{D} \ &u \text { specified on } \mathcal{B} . \end{aligned}

The corresponding weak form is obtained by multiplying $(3.45)$ by a test function $\psi(x, y)$ and integrating by parts to obtain
$$\int_{\mathcal{B}} \psi \nabla u_h \cdot \mathbf{n} d l-\int_{\mathcal{D}} \nabla u_h \cdot \nabla \psi d \mathcal{S}=0,$$
where $\mathbf{n}$ is the unit normal to the boundary. The trial solution is
$$u_h=\sum_{j=1}^n u_j \phi_j(x, y)$$

## 物理代写|空气动力学代写Aerodynamics代考|Further Analysis of the Discrete Laplacian

If we consider the stencil illustrated in Figure $3.8$, evaluation of formulas (3.47), (3.48), and (3.49) reduces the equation for node 0 to the form
$$r_0=\sum s_{k 0}\left(u_k-u_0\right),$$
where $s_{k 0}$ are the entries of the stiffness matrix between node $k$ and 0 . If we consider edge 20 , say, the total contribution of $u_2$, applying the formulas (3.47) and (3.48) to the triangles 012 and 023 with areas $\mathcal{S}1$ and $\mathcal{S}_2$ respectively, is \begin{aligned} s{20}=& \frac{1}{4 \mathcal{S}1}\left(\left(y_0-y_1\right)\left(y_2-y_1\right)+\left(x_0-x_1\right)\left(x_2-x_1\right)\right) \ &+\frac{1}{4 \mathcal{S}_2}\left(\left(y_0-y_3\right)\left(y_2-y_3\right)+\left(x_0-x_3\right)\left(x_2-x_3\right)\right) . \end{aligned} Let $l{k 0}$ be the length of the edge $k 0$. Then, we find that
\begin{aligned} s_{20} &=\frac{l_{01} l_{21} \cos \theta_{012}}{2 l_{01} l_{21} \sin \theta_{012}}+\frac{l_{03} l_{23} \cos \theta_{032}}{2 l_{03} l_{23} \sin \theta_{032}} \ &=\frac{1}{2}\left(\cot \theta_{012}+\cot \theta_{032}\right), \end{aligned}
where $\theta_{012}$ and $\theta_{032}$ are the angles between edges $l_{01}$ and $l_{21}$ and $l_{03}$ and $l_{23}$, as illustrated in Figure 3.9.
In the case of the Poisson equation
$$u_{x x}+u_{y y}=f,$$

we must also evaluate
$$\int f \phi_0 d \mathcal{S}$$
assuming $f$ is represented by a piecewise linear approximation. Carrying out the integrations, we find that the equation for node 0 is
$$\sum_{k=1}^n s_{k 0}\left(u_k-u_0\right)=M_{00} f_0+M_{0 k} f_k,$$
where $n$ is the number of neighbors, and the coefficients
$$M_{00}=\frac{1}{6} \sum_{k=1}^n s_k, \quad M_{0 k}=\frac{s_k+s_{k+1}}{12}$$
are entries of the stiffness matrix.

## 物理代写|空气动力学代写空气动力学代考|具有线性元素的多维有限元格式

\begin{aligned} &u_{x x}+u_{y y}=0 \text { in } \mathcal{D} \ &u \text { specified on } \mathcal{B} . \end{aligned}

$$\int_{\mathcal{B}} \psi \nabla u_h \cdot \mathbf{n} d l-\int_{\mathcal{D}} \nabla u_h \cdot \nabla \psi d \mathcal{S}=0,$$
，其中$\mathbf{n}$是边界的法线单位。试解是
$$u_h=\sum_{j=1}^n u_j \phi_j(x, y)$$

## 物理代写|空气动力学代写空气动力学代考|离散拉普拉斯算子的进一步分析

$$r_0=\sum s_{k 0}\left(u_k-u_0\right),$$
，其中$s_{k 0}$是节点$k$和0之间的刚度矩阵项。如果我们考虑边20，假设$u_2$的总贡献，分别对面积为$\mathcal{S}1$和$\mathcal{S}2$的三角形012和023应用公式(3.47)和(3.48)，为\begin{aligned} s{20}=& \frac{1}{4 \mathcal{S}1}\left(\left(y_0-y_1\right)\left(y_2-y_1\right)+\left(x_0-x_1\right)\left(x_2-x_1\right)\right) \ &+\frac{1}{4 \mathcal{S}_2}\left(\left(y_0-y_3\right)\left(y_2-y_3\right)+\left(x_0-x_3\right)\left(x_2-x_3\right)\right) . \end{aligned}设$l{k 0}$为边$k 0$的长度。然后，我们发现
\begin{aligned} s{20} &=\frac{l_{01} l_{21} \cos \theta_{012}}{2 l_{01} l_{21} \sin \theta_{012}}+\frac{l_{03} l_{23} \cos \theta_{032}}{2 l_{03} l_{23} \sin \theta_{032}} \ &=\frac{1}{2}\left(\cot \theta_{012}+\cot \theta_{032}\right), \end{aligned}
，其中$\theta_{012}$和$\theta_{032}$是边$l_{01}$和$l_{21}$以及$l_{03}$和$l_{23}$之间的角度，如图3.9所示。在泊松方程
$$u_{x x}+u_{y y}=f,$$ 的情况下

$$\int f \phi_0 d \mathcal{S}$$
，假设$f$由分段线性逼近表示。进行积分，我们发现节点0的方程是
$$\sum_{k=1}^n s_{k 0}\left(u_k-u_0\right)=M_{00} f_0+M_{0 k} f_k,$$
，其中$n$是邻居数，系数
$$M_{00}=\frac{1}{6} \sum_{k=1}^n s_k, \quad M_{0 k}=\frac{s_k+s_{k+1}}{12}$$

## 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。

Posted on Categories:Aerodynamics, 物理代写, 空气动力学

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## 物理代写|空气动力学代写Aerodynamics代考|General Formulation

In contrast to finite difference methods, which approximate the differential operators appearing in a PDE, finite element methods approximate the solution. The approximate trial solution is represented as an expansion in a set of basis functions, which is then inserted in the exact differential equation. Suppose that we wish to solve an equation of the form
$$L u=f .$$
Let $\phi_j$ be a set of independent basis functions. Then we insert a trial solution
$$u_h=\sum_{j=1}^n u_j \phi_j$$
into (3.28) to obtain
$$L u_h-f=r_h,$$
where $r_h$ is the residual error.
The coefficients $u_j$ are now chosen to make the residual error as small as possible. In general there are not enough degrees of freedom to reduce the error to zero, so we try to minimize the error by choosing $u_j$ such that the residual is orthogonal to each of $n$ independent test functions $\psi_i$, yielding the weighted residual equations
$$\left(r_h, \psi_i\right)=\left(L u_h-f, \psi_i\right)=0, \quad i=1, \ldots, n,$$

where $(u, v)$ is an inner product in an appropriate space. Inserting the trial solution, we obtain a linear system of equations for the $n$ unknowns $u_j$. Important properties of the method, such as accuracy, computational cost, and complexity of the formulation, all depend on the choice of basis and test functions.

## 物理代写|空气动力学代写Aerodynamics代考|Galerkin method for the one-dimensional Poisson equation

Consider the one-dimensional Poisson equation
$$u_{x x}=f$$
on a domain from $a$ to $b$ with Dirichlet boundary conditions. This could represent, for example, steady heat conduction with a constant conductivity $\kappa$. The distribution of the temperature $T$ is then governed by
$$\frac{d}{d x} \kappa \frac{d T}{d x}=q(x),$$
where $q$ is the rate of heat input. Then we recover (3.31) with $u=\kappa T, f=q$.
Using the superscript prime to denote differentiation with respect to $x$, the Galerkin method leads to the residual equation
$$\int_a^b u^{\prime \prime} \phi_i d x=\int_a^b f \phi_i d x$$
With a piecewise linear trial solution, the second derivative $u_h^{\prime \prime}$ is not defined at the nodes. However, we can circumvent this difficulty by integrating the residual equation by parts to obtain
$$-\int_a^b u_h^{\prime} \phi_i^{\prime} d x=\int_a^b f \phi_i d x$$

## 物理代写|空气动力学代写空气动力学代考|一般配方

$$L u=f .$$

$$u_h=\sum_{j=1}^n u_j \phi_j$$

$$L u_h-f=r_h,$$
，其中$r_h$是残差。现在选择系数$u_j$是为了使残余误差尽可能小。一般来说，没有足够的自由度来将误差降至零，因此我们试图通过选择$u_j$使残差与$n$独立测试函数$\psi_i$的每个正交来最小化误差，得到加权残差方程
$$\left(r_h, \psi_i\right)=\left(L u_h-f, \psi_i\right)=0, \quad i=1, \ldots, n,$$

，其中$(u, v)$是适当空间中的内积。插入试解，我们得到了$n$未知数$u_j$的线性方程组。该方法的重要性质，如精度、计算成本和公式的复杂性，都取决于基和测试函数的选择

## 物理代写|空气动力学代写空气动力学代考|一维泊松方程的伽略金方法

$$u_{x x}=f$$
。例如，这可以表示具有恒定导电性的稳定热传导$\kappa$。温度的分布$T$由
$$\frac{d}{d x} \kappa \frac{d T}{d x}=q(x),$$

$$\int_a^b u^{\prime \prime} \phi_i d x=\int_a^b f \phi_i d x$$

$$-\int_a^b u_h^{\prime} \phi_i^{\prime} d x=\int_a^b f \phi_i d x$$

## 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。

Posted on Categories:Aerodynamics, 物理代写, 空气动力学

## avatest™帮您通过考试

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## 物理代写|空气动力学代写Aerodynamics代考|Symmetric Form

Consider the equations of one-dimensional flow in primitive variables:
$$\begin{array}{r} \frac{\partial \rho}{\partial t}+u \frac{\partial \rho}{\partial x}+\rho \frac{\partial u}{\partial x}=0 \ \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+\frac{1}{\rho} \frac{\partial p}{\partial x}=0 \ \frac{\partial p}{\partial t}+\rho c^{2} \frac{\partial u}{\partial x}+u \frac{\partial p}{\partial x}=0 \end{array}$$

Then subtracting the first equation multiplied by $c^{2}$ from the third equation, we find that
$$\frac{\partial p}{\partial t}-c^{2} \frac{\partial \rho}{\partial t}+u\left(\frac{\partial p}{\partial x}-c^{2} \frac{\partial \rho}{\partial x}\right)=0$$
This is equivalent to a statement that the entropy
$$S=\log \left(\frac{p}{\rho^{\gamma}}\right)=\log p-\gamma \log \rho$$
is constant since
$$d S=\frac{d p}{p}-\gamma \frac{d \rho}{\rho}=\frac{1}{p}\left(d p-c^{2} d \rho\right)$$

## 物理代写|空气动力学代写Aerodynamics代考|Riemann Invariants

In the case of one-dimensional isentropic flow, these equations reduce to
$$\frac{2}{\gamma-1} \frac{\partial c}{\partial t}+\frac{2 u}{\gamma-1} \frac{\partial c}{\partial x}+c \frac{\partial u}{\partial x}=0$$

\begin{aligned} \frac{\partial u}{\partial t}+\frac{2 c}{\gamma-1} \frac{\partial c}{\partial x}+u \frac{\partial u}{\partial x} &=0 \ \frac{\partial S}{\partial t}+u \frac{\partial S}{\partial x} &=0 \end{aligned}
Now the first two equations can be added and subtracted to yield
$$\frac{\partial R^{+}}{\partial t}+(u+c) \frac{\partial R^{+}}{\partial x}=0$$
and
$$\frac{\partial R^{-}}{\partial t}+(u-c) \frac{\partial R^{-}}{\partial x}=0,$$
where $R^{+}$and $R^{-}$are the Riemann invariants
$$R^{+}=u+\frac{2 c}{\gamma-1}, R^{-}=u-\frac{2 c}{\gamma-1},$$
which remain constant as they are transported at the wave speeds $u+c$ and $u-c$. The Riemann invariants prove to be useful in the formulation of far field boundary conditions designed to minimize wave reflection.

## 物理代写|空气动力学代写Aerodynamics代考|Symmetric Form

$$\frac{\partial \rho}{\partial t}+u \frac{\partial \rho}{\partial x}+\rho \frac{\partial u}{\partial x}=0 \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}+\frac{1}{\rho} \frac{\partial p}{\partial x}=0 \frac{\partial p}{\partial t}+\rho c^{2} \frac{\partial u}{\partial x}+u \frac{\partial p}{\partial x}=0$$

$$\frac{\partial p}{\partial t}-c^{2} \frac{\partial \rho}{\partial t}+u\left(\frac{\partial p}{\partial x}-c^{2} \frac{\partial \rho}{\partial x}\right)=0$$

$$S=\log \left(\frac{p}{\rho^{\gamma}}\right)=\log p-\gamma \log \rho$$

$$d S=\frac{d p}{p}-\gamma \frac{d \rho}{\rho}=\frac{1}{p}\left(d p-c^{2} d \rho\right)$$

## 物理代与写|空气动力学代写Aerodynamics代考|Riemann Invariants

$$\begin{gathered} \frac{2}{\gamma-1} \frac{\partial c}{\partial t}+\frac{2 u}{\gamma-1} \frac{\partial c}{\partial x}+c \frac{\partial u}{\partial x}=0 \ \frac{\partial u}{\partial t}+\frac{2 c}{\gamma-1} \frac{\partial c}{\partial x}+u \frac{\partial u}{\partial x}=0 \frac{\partial S}{\partial t}+u \frac{\partial S}{\partial x} \quad=0 \end{gathered}$$

$$\frac{\partial R^{+}}{\partial t}+(u+c) \frac{\partial R^{+}}{\partial x}=0$$

$$\frac{\partial R^{-}}{\partial t}+(u-c) \frac{\partial R^{-}}{\partial x}=0,$$

$$R^{+}=u+\frac{2 c}{\gamma-1}, R^{-}=u-\frac{2 c}{\gamma-1},$$

## 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。

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## 物理代写|空气动力学代写Aerodynamics代考|Analysis of the Equations of gas Dynamics: The Jacobian Matrices

The Euler equations for the three-dimensional flow of an inviscid gas are obtained by eliminating the viscous stress and the heat conduction term from the Navier-Stokes equations. Following the notation of Section 2.1, the conservation laws for mass, momentum, and energy can be written in combined form as
$$\frac{\partial \boldsymbol{w}}{\partial t}+\frac{\partial}{\partial x_{i}} \boldsymbol{f}{i}(\boldsymbol{w})=0,$$ where the state and flux vectors are $$\boldsymbol{w}=\rho\left[\begin{array}{c} 1 \ u{1} \ u_{2} \ u_{3} \ E \end{array}\right], \quad \boldsymbol{f}{i}=\rho u{i}\left[\begin{array}{c} 1 \ u_{1} \ u_{2} \ u_{3} \ H \end{array}\right]+p\left[\begin{array}{c} 0 \ \delta_{i 1} \ \delta_{i 2} \ \delta_{i 3} \ 0 \end{array}\right] .$$
Also,
$$p=(\gamma-1) \rho\left(E-\frac{u^{2}}{2}\right), \quad H=E+\frac{p}{\rho}=\frac{c^{2}}{\gamma-1}+\frac{u^{2}}{2},$$
where $u$ is the speed, and $c$ is the speed of sound:
$$u^{2}=u_{i} u_{i}, \quad c^{2}=\frac{\gamma p}{\rho} .$$

## 物理代写|空气动力学代写Aerodynamics代考|Two-Dimensional Flow

The equations of two-dimensional flow have the simpler form
$$\frac{\partial \boldsymbol{w}}{\partial t}+\frac{\partial \boldsymbol{f}{1}(\boldsymbol{w})}{\partial x{1}}+\frac{\partial \boldsymbol{f}{2}(\boldsymbol{w})}{\partial x{2}}=0,$$
where the state and flux vectors are
$$\boldsymbol{w}=\left[\begin{array}{c} p \ m_{1} \ m_{2} \ e \end{array}\right], \quad \boldsymbol{f}{1}=u{1} \boldsymbol{w}+p\left[\begin{array}{c} 0 \ 1 \ 0 \ u_{1} \end{array}\right], \quad \boldsymbol{f}{2}=u{2} \boldsymbol{w}+p\left[\begin{array}{c} 0 \ 0 \ 1 \ u_{2} \end{array}\right]$$
Now the flux vector normal to an edge with normal components $n_{1}$ and $n_{2}$ is
$$f=n_{1} f_{1}+n_{2} f_{2} .$$

## 物理代写|空气动力学代写Aerodynamics代考|Analysis of the Equations of gas Dynamics: The Jacobian Matrices

$$\frac{\partial \boldsymbol{w}}{\partial t}+\frac{\partial}{\partial x_{i}} \boldsymbol{f} i(\boldsymbol{w})=0$$

$$\boldsymbol{w}=\rho\left[1 u 1 u_{2} u_{3} E\right], \quad \boldsymbol{f} i=\rho u i\left[1 u_{1} u_{2} u_{3} H\right]+p\left[0 \delta_{i 1} \delta_{i 2} \delta_{i 3} 0\right] .$$

$$p=(\gamma-1) \rho\left(E-\frac{u^{2}}{2}\right), \quad H=E+\frac{p}{\rho}=\frac{c^{2}}{\gamma-1}+\frac{u^{2}}{2},$$

$$u^{2}=u_{i} u_{i}, \quad c^{2}=\frac{\gamma p}{\rho}$$

## 物理代写|空气动力学代写Aerodynamics代考|Two-Dimensional Flow

$$\frac{\partial \boldsymbol{w}}{\partial t}+\frac{\partial \boldsymbol{f} 1(\boldsymbol{w})}{\partial x 1}+\frac{\partial \boldsymbol{f} 2(\boldsymbol{w})}{\partial x 2}=0$$

$$f=n_{1} f_{1}+n_{2} f_{2}$$

## 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。

Posted on Categories:Aerodynamics, 物理代写, 空气动力学

## avatest™帮您通过考试

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## 物理代写|空气动力学代写Aerodynamics代考|The Development of Methods for the Euler and Navier–Stokes Equations

By the 1980s, advances in computer hardware had made it feasible to solve the full Euler equations using software that could be cost-effective in industrial use. The idea of directly discretizing the conservation laws to produce a finite volume scheme had been introduced by MacCormack (MacCormack \& Paullay 1972). Most of the early flow solvers tended to exhibit strong pre- or post-shock oscillations. Also, in a workshop held in Stockholm in 1979 (Rizzi \& Viviand 1981), it was apparent that none of the existing schemes converged to a steady state. These difficulties were resolved during the following decade.

The Jameson-Schmidt-Turkel scheme (Jameson, Schmidt, \& Turkel 1981), which used Runge-Kutta time stepping and a blend of second- and fourth-differences (both to control oscillations and to provide background dissipation), consistently demonstrated convergence to a steady state, with the consequence that it has remained one of the most widely used methods to the present day.

A fairly complete understanding of shock capturing algorithms was achieved, stemming from the ideas of Godunov, Van Leer, Harten, and Roe. The issue of oscillation control and positivity had already been addressed by Godunov (1959) in his pioneering work in the 1950s (translated into English in 1959). He had introduced the concept of representing the flow as piecewise constant in each computational cell and solving a Riemann problem at each interface, thus obtaining a first order accurate solution that avoids nonphysical features such as expansion shocks. When this work was eventually recognized in the West, it became very influential. It was also widely recognized that numerical schemes might benefit from distinguishing the various wave speeds, and this motivated the development of characteristics-based schemes.

## 物理代写|空气动力学代写Aerodynamics代考|Overview of the Simulation Process

The essential steps of developing a numerical simulation of a physical problem can be outlined as follows:

1. Formulate a mathematical model of the physical problem that captures the important aspects for the purpose in hand and can provide the desired accuracy. Here it should be noted that models of widely varying complexity and levels of fidelity can be useful. For example, potential flow models based on Laplace’s equation can provide reasonably accurate predictions of low speed aerodynamic flows over streamlined shapes at a low computational cost, and this is very useful at an early stage in the design when rapid turnaround is crucial. At the final stage of the design, one would wish to confirm the expected performance using a model with the highest possible fidelity, typically the Reynolds averaged Navier-Stokes equations in actual practice.
2. Analyze the mathematical properties of the model, such as proper formulation of boundary conditions that ensure the existence and convergence of a solution.
3. Formulate a discrete numerical scheme to approximate the mathematical model that has been selected. Analyze the stability, accuracy, and convergence of the scheme. Can we prove, for example, that the error in the numerical approximation decreases as some power of the mesh spacing when the spacing is progressively reduced?
4. Implement the discrete scheme in software that makes efficient use of the available hardware. This is becoming harder with the emergence of parallel systems with multiple levels of parallelism down to multiple threads within each core of multi-core processing chips, which are in turn arranged in parallel clusters. This stage also requires the use of every possible procedure to assure that the software is actually correct.
5. Validate the software by showing that it produces trustworthy results in practice. Here we should distinguish between the questions of whether the software is correct and whether the selected mathematical model adequately represents the physics. To address the first question, we may test whether the results are correct for some limiting situations for which the true answer is known. For example, an arbitrary body has zero drag in inviscid flow. Or is the numerical solution symmetric for flow over a symmetric profile at zero angle of attack? We should also test the convergence of the numerical solution as the grid is refined. Does it exhibit the expected order of accuracy? We may also compare the results with those obtained by other software developed to solve the same problem. Workshops such as the AIAA Drag Prediction Workshops can play a useful role in this process. Finally, once a sufficiently high confidence level has been established for the software, comparisons with experimental data can be used to address the question of whether the mathematical model adequately represents the physical problem of interest.

## 物理代写|空气动力学代写Aerodynamics代考|The Development of Methods for the Euler and Navier–Stokes Equations

Jameson-Schmidt-Turkel 方案 (Jameson, Schmidt, \& Turkel 1981) 使用 Runge-Kutta 时间步长以及二阶和四阶差分的混合（用于控制振荡和提供背景耗散），始终显示收敛到一个稳定的状态，其结果是它至今仍是最广泛使用的方法之一。

## 物理代写|空气动力学代写Aerodynamics代考|Overview of the Simulation Process

1. 制定物理问题的数学模型，该模型捕捉手头目的的重要方面，并可以提供所需的准确性。这里应该注意的是，具有广泛不同的复杂性和保真度级别的模型可能是有用的。例如，基于拉普拉斯方程的势流模型可以以较低的计算成本对流线型形状上的低速气动流动提供合理准确的预测，这在设计的早期阶段非常有用，因为快速周转至关重要。在设计的最后阶段，人们希望使用具有最高保真度的模型来确认预期性能，通常是实际实践中的雷诺平均 Navier-Stokes 方程。
2. 分析模型的数学属性，例如确保解决方案存在和收敛的边界条件的正确公式化。
3. 制定离散数值方案来近似已选择的数学模型。分析方案的稳定性、准确性和收敛性。例如，我们能否证明当网格间距逐渐减小时，数值近似中的误差会随着网格间距的某些幂次而减小？
4. 在软件中实施离散方案，以有效利用可用硬件。随着具有多级并行性的并行系统的出现，这变得越来越困难，多核处理芯片的每个核心内都有多个线程，这些处理器又排列在并行集群中。这个阶段还需要使用所有可能的程序来确保软件实际上是正确的。
5. 通过证明软件在实践中产生可信赖的结果来验证软件。在这里，我们应该区分软件是否正确以及选择的数学模型是否充分代表物理的问题。为了解决第一个问题，我们可以测试结果对于已知真实答案的某些限制情况是否正确。例如，任意物体在无粘性流动中的阻力为零。或者在零攻角的对称剖面上流动的数值解是对称的吗？随着网格的细化，我们还应该测试数值解的收敛性。它是否表现出预期的准确性顺序？我们也可以将结果与为解决相同问题而开发的其他软件获得的结果进行比较。AIAA 阻力预测研讨会等研讨会可以在此过程中发挥有用的作用。最后，一旦为软件建立了足够高的置信水平，就可以使用与实验数据的比较来解决数学模型是否充分代表感兴趣的物理问题的问题。

## 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。