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数学代写|偏微分方程代写partial difference equations代考|HIGHER-ORDER APPROXIMATIONS

Thus far, our discussion of the finite difference method has been restricted to secondorder accurate schemes. In many applications, such as computation of turbulent flow [4] or acoustics calculations [5], second-order accuracy is often considered inadequate. This is because with a second-order scheme, the number of nodes required to make the computed solution either grid independent or able to capture the physics at all relevant length scales is often too large, thereby making the computations too resource and time intensive. In such a scenario, a higher-order scheme is warranted so that the same accuracy can be attained with a coarser grid.

Following our basic procedure, higher-order difference approximations can clearly be derived by using additional Taylor series expansions and extending the stencil beyond the three points (for 1D) that we have been using thus far. Such a derivation procedure, although conceptually straightforward, can quickly become tedious. Instead, we use a procedure here that recursively makes use of expressions that we have already derived in preceding sections. The procedure used here is adapted from the general procedure used to derive so-called compact difference schemes. For additional details on compact difference schemes beyond what is discussed here, the reader is referred to the popular article by Lele [6] and the references cited therein.
Our objective is to derive a difference approximation for the second derivative that shall have accuracy better than second order on a uniform grid. To begin the derivation, we first write Eq. (2.8) in an alternative form, as follows
$$
\phi_j^{i i}=\frac{\phi_{j+1}-2 \phi_j+\phi_{j-1}}{\delta^2}-\frac{\delta^2}{12} \phi_j^{i v}-\frac{\delta^4}{360} \phi_j^{v i}+\ldots,
$$
where the subscripts $j, j+1$, and $j-1$ have been used to denote nodal locations, and the superscript $i i$ has been used to denote the order of the derivative. The last two terms in Eq. (2.60) represent the truncation error. Thus far, we have been concerning ourselves only with the leading order (fourth derivative containing) truncation error term. Here, since our objective is to eliminate that term, it is important to also consider the next term in the truncation error expression, as has been done in Eq. (2.60). The grid spacing, $\Delta x$, has been denoted by $\delta$ for simplicity of notation. By mathematical induction, from Eq. (2.60) it follows that
$$
\phi_j^{i v}=\frac{\phi_{j+1}^{i i}-2 \phi_j^{i i}+\phi_{j-1}^{i i}}{\delta^2}-\frac{\delta^2}{12} \phi_j^{v i}-\frac{\delta^4}{360} \phi_j^{v i i i}+\ldots
$$
Substituting Eq. (2.61) inte Eq. (2.60) for the fourth derivative, and simplifying, we obtain
$$
\phi_j^{i i}=\frac{\phi_{j+1}-2 \phi_j+\phi_{j-1}}{\delta^2}-\frac{1}{12}\left(\phi_{j+1}^{i i}-2 \phi_j^{i i}+\phi_{j-1}^{i i}\right)+\frac{\delta^4}{240} \phi_j^{v i}+\frac{\delta^6}{4320} \phi_j^{v i i i}+\ldots
$$

数学代写|偏微分方程代写partial difference equations代考|DIFFERENCE APPROXIMATIONS IN THE CYLINDRICAL

Many scientific and engineering computations involve cylindrical shapes. In fact, outside of a rectangular prism (brick) shape, the cylindrical shape is probably the second most commonly encountered regular shape in engineering applications. Furthermore, in many applications, angular symmetry is valid, making the computations so-called 2D axisymmetric. In this section, we will develop finite difference approximations to the Poisson equation in cylindrical coordinates using the same principles discussed earlier but with an emphasis on issues that are unique to the cylindrical coordinate system. The cylindrical conrdinate system has three independent variahles: $r, \theta$, and $z$, as shown in Fig. 2.9. The three coordinates are generally referred to as the radial, azimuthal (or simply, angular), and axial coordinates, respectively.

In the cylindrical coordinate system, the Poisson equation, which has been our governing equation thus far, is written as
$$
\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2}+\frac{\partial^2 \phi}{\partial z^2}=S_\phi .
$$
As in the case of Cartesian coordinates, one can formulate 1D, 2D, and 3D problems in the cylindrical coordinate system. Formulating a $1 \mathrm{D}$ problem in the $z$ direction is no different than formulating a 1D problem in the Cartesian coordinate system, i.e., Eq. (2.71) would reduce to Eq. (2.1). Such a scenario has been discussed at length already and requires no further discussion. A 1D problem in the $\theta$ direction is rarely encountered for practical applications. The most common scenario for which a 1D problem may be formulated is one where the solution in the radial direction is sought. For example, trying to determine the temperature distribution inside a long nuclear fuel rod would require solution of the 1D heat conduction equation in radial coordinates. Therefore, we start our discussion with an ordinary differential equation in radial coordinates, written as
$$
\frac{1}{r} \frac{d}{d r}\left(r \frac{d \phi}{d r}\right)=\frac{d^2 \phi}{d r^2}+\frac{1}{r} \frac{d \phi}{d r}=S_\phi .
$$
The second part of Eq. (2.72) was obtained by expanding the derivative of the product of two terms. The nodal system in this particular case is shown in Fig. 2.10.
In order to derive finite difference approximations for the first and second derivatives, we perform two Taylor series expansions as before – one forward and one backward – as follows:
$$
\begin{aligned}
&\phi_{i+1}=\phi_i+\left.(\Delta r) \frac{d \phi}{d r}\right|i+\left.\frac{(\Delta r)^2}{2 !} \frac{d^2 \phi}{d r^2}\right|_i+\left.\frac{(\Delta r)^3}{3 !} \frac{d^3 \phi}{d r^3}\right|_i+\left.\frac{(\Delta r)^4}{4 !} \frac{d^4 \phi}{d r^4}\right|_i+\ldots, \quad \text { (2.73a) } \ &\phi{i-1}=\phi_i-\left.(\Delta r) \frac{d \phi}{d r}\right|_i+\left.\frac{(\Delta r)^2}{2 !} \frac{d^2 \phi}{d r^2}\right|_i-\left.\frac{(\Delta r)^3}{3 !} \frac{d^3 \phi}{d r^3}\right|_i+\left.\frac{(\Delta r)^4}{4 !} \frac{d^4 \phi}{d r^4}\right|_i+\ldots, \quad \text { (2.73h) }
\end{aligned}
$$

数学代写|偏微分方程代写partial difference equations代考|Math442

偏微分方程代考

数学代写|偏微分方程代写partial difference equations代考|HIGHER-ORDER APPROXIMATIONS

到目前为止,我们对有限差分法的讨论仅限于二阶精确方案。在许多应用中,例如湍流计算 [4] 或声学计 算 [5],二阶精度通常被认为是不够的。这是因为对于二阶方案,使计算解决方案独立于网格或能够捕获所 有相关长度尺度的物理所需的节点数量通常太大,从而使计算过于耗费资源和时间。在这种情况下,需要 使用更高阶的方案,以便使用较粗的网格可以获得相同的精度。
按照我们的基本程序,通过使用额外的泰勒级数展开并将模板扩展到我们迄今为止一直使用的三个点(对 于一维) 之外,可以清楚地得出高阶差分近似。这样的推导过程虽然在概念上很简单,但很快就会变得乏 味。相反,我们在这里使用了一个过程,它递归地使用我们在前面几节中已经导出的表达式。这里使用的 过程改编自用于导出所谓紧凑差分方案的一般过程。有关此处讨论之外的紧凑差分方案的更多详细信息, 请参阅 Lele [6] 的热门文章及其引用的参考文献。
我们的目标是推导出二阶导数的差分近似,其精度应优于均匀网格上的二阶。为了开始推导,我们首先写 方程。(2.8) 以另一种形式,如下
$$
\phi_j^{i i}=\frac{\phi_{j+1}-2 \phi_j+\phi_{j-1}}{\delta^2}-\frac{\delta^2}{12} \phi_j^{i v}-\frac{\delta^4}{360} \phi_j^{v i}+\ldots,
$$
下标在哪里 $j, j+1$ ,和 $j-1$ 已用于表示节点位置,上标 $i i$ 已用于表示导数的阶数。等式中的最后两项。 (2.60) 表示截断误差。到目前为止,我们只关注前导 (包含四阶导数) 截断误差项。在这里,由于我们的 目标是消除该项,因此还必须考虑截断误差表达式中的下一项,正如在方程式中所做的那样。(2.60)。网格 间距, $\Delta x$ ,表示为 $\delta$ 为了符号的简单。通过数学归纳,从方程式。(2.60) 由此可见
$$
\phi_j^{i v}=\frac{\phi_{j+1}^{i i}-2 \phi_j^{i i}+\phi_{j-1}^{i i}}{\delta^2}-\frac{\delta^2}{12} \phi_j^{v i}-\frac{\delta^4}{360} \phi_j^{v i i i}+\ldots
$$
代入方程式。(2.61) 积分方程。(2.60) 对于四阶导数,并简化,我们得到
$$
\phi_j^{i i}=\frac{\phi_{j+1}-2 \phi_j+\phi_{j-1}}{\delta^2}-\frac{1}{12}\left(\phi_{j+1}^{i i}-2 \phi_j^{i i}+\phi_{j-1}^{i i}\right)+\frac{\delta^4}{240} \phi_j^{v i}+\frac{\delta^6}{4320} \phi_j^{v i i i}+\ldots
$$

数学代写|偏微分方程代写partial difference equations代考|DIFFERENCE APPROXIMATIONS IN THE CYLINDRICAL

许多科学和工程计算涉及圆柱形状。事实上,在矩形棱柱 (砖) 形状之外,圆柱形状可能是工程应用中第 二常见的规则形状。此外,在许多应用中,角对称是有效的,使得计算成为所谓的 $2 \mathrm{D}$ 轴对称。在本节 中,我们将使用前面讨论的相同原理开发柱坐标中泊松方程的有限差分近似,但重点是柱坐标系特有的问 题。圆柱坐标系具有三个独立变量: $r, \theta$ ,和 $z$ ,如图 $2.9$ 所示。这三个坐标通常分别称为径向坐标、方位 角坐标 (或简称角坐标) 和轴向坐标。
在圆柱坐标系中,泊松方程 (迄今为止一直是我们的控制方程) 写为
$$
\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2}+\frac{\partial^2 \phi}{\partial z^2}=S_\phi .
$$
与笛卡尔坐标的情况一样,可以在圆柱坐标系中制定 1D、2D 和 3D 问题。制定一个1D中的问题 $z$ 方向与 在笛卡尔坐标系中制定一维问题没有什么不同,即方程。(2.71) 将减少到等式。(2.1)。这种情况已经详 细讨论过,无需进一步讨论。中的一维问题 $\theta$ 在实际应用中很少遇到方向。可以制定一维问题的最常见情况 是寻求径向解决方案的情况。例如,试图确定长核燃料棒内部的温度分布需要求解径向坐标中的一维热传 导方程。因此,我们从径向坐标中的常微分方程开始讨论,写为
$$
\frac{1}{r} \frac{d}{d r}\left(r \frac{d \phi}{d r}\right)=\frac{d^2 \phi}{d r^2}+\frac{1}{r} \frac{d \phi}{d r}=S_\phi .
$$
等式的第二部分。(2.72) 是通过扩展两项乘积的导数获得的。这种特殊情况下的节点系统如图 2.10所示。 为了推导一阶和二阶导数的有限差分近似,我们像以前一样执行两个泰勒级数展开一一一个向前展开, 个向后展开一一如下:
$$
\phi_{i+1}=\phi_i+\left.(\Delta r) \frac{d \phi}{d r}\right|_i+\left.\frac{(\Delta r)^2}{2 !} \frac{d^2 \phi}{d r^2}\right|_i+\left.\frac{(\Delta r)^3}{3 !} \frac{d^3 \phi}{d r^3}\right|_i+\left.\frac{(\Delta r)^4}{4 !} \frac{d^4 \phi}{d r^4}\right|_i+\ldots, \quad(2.73 \mathrm{a}) \quad \phi i-1=\phi_i
$$

数学代写|偏微分方程代写partial difference equations代考

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