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

## 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}

# 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

$$\phi_j^{i i}=\frac{\phi_{j+1}-2 \phi_j+\phi_{j-1}}{\delta^2}-\frac{\delta^2}{12} \phi_j^{i v}-\frac{\delta^4}{360} \phi_j^{v i}+\ldots,$$

$$\phi_j^{i v}=\frac{\phi_{j+1}^{i i}-2 \phi_j^{i i}+\phi_{j-1}^{i i}}{\delta^2}-\frac{\delta^2}{12} \phi_j^{v i}-\frac{\delta^4}{360} \phi_j^{v i i i}+\ldots$$

$$\phi_j^{i i}=\frac{\phi_{j+1}-2 \phi_j+\phi_{j-1}}{\delta^2}-\frac{1}{12}\left(\phi_{j+1}^{i i}-2 \phi_j^{i i}+\phi_{j-1}^{i i}\right)+\frac{\delta^4}{240} \phi_j^{v i}+\frac{\delta^6}{4320} \phi_j^{v i i i}+\ldots$$

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

$$\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial \phi}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 \phi}{\partial \theta^2}+\frac{\partial^2 \phi}{\partial z^2}=S_\phi .$$

$$\frac{1}{r} \frac{d}{d r}\left(r \frac{d \phi}{d r}\right)=\frac{d^2 \phi}{d r^2}+\frac{1}{r} \frac{d \phi}{d r}=S_\phi .$$

$$\phi_{i+1}=\phi_i+\left.(\Delta r) \frac{d \phi}{d r}\right|_i+\left.\frac{(\Delta r)^2}{2 !} \frac{d^2 \phi}{d r^2}\right|_i+\left.\frac{(\Delta r)^3}{3 !} \frac{d^3 \phi}{d r^3}\right|_i+\left.\frac{(\Delta r)^4}{4 !} \frac{d^4 \phi}{d r^4}\right|_i+\ldots, \quad(2.73 \mathrm{a}) \quad \phi i-1=\phi_i$$

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