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## 数学代写|偏微分方程代写partial difference equations代考|DIFFERENCE APPROXIMATIONS AND TRUNCATION ERRORS

In order to understand the application of the finite difference method to the solution of differential equations, let us consider a simple second-order linear ordinary differential equation (ODE) of the following form:
$$\frac{d^2 \phi}{d x^2}=S_\phi,$$
where $\phi$ is the dependent variable or unknown that needs to be determined via solution of the differential equation. The independent variable, on the other hand, is denoted by $x$. The quantity, $S_\phi$, on the right-hand side of Eq. (2.1), could be, in general, a function of both the independent and dependent variables, i.e., $S_\phi=S_\phi(x, \phi)$. This term is often referred to as the source term or source. If the functional dependence of $S_\phi$ is nonlinear, then Eq. (2.1) would be nonlinear. Otherwise, Eq. (2.1) would be a linear ODE. For the discussion at hand, let us assume that $S_\phi$ is such that Eq. (2.1) is linear. Since Eq. (2.1) is a second-order ODE, it will also require two boundary conditions in $x$. Let us assume that the boundary conditions are given by
\begin{aligned} &\phi\left(x_{\mathrm{L}}\right)=\phi_{\mathrm{L}}, \ &\phi\left(x_{\mathrm{R}}\right)=\phi_{\mathrm{R}}, \end{aligned}
where $x_{\mathrm{L}}$ and $x_{\mathrm{R}}$ denote values of $x$ at the left and right ends of the domain of interest, respectively. For the discussion at hand, the type of boundary condition being applied is not relevant. Rather, the fact that these conditions are posed on the two ends is the only point to be noted.

One critical point to note is that Eq. (2.1) is valid in the open interval $\left(x_{\mathrm{L}}, x_{\mathrm{R}}\right)$, not in the closed interval $\left[x_{\mathrm{L}}, x_{\mathrm{R}}\right]$. At the end points of the domain, only the boundary conditions are valid. Let us now consider the steps that are undertaken to obtain a closedform analytical solution to Eq. (2.1) subject to the boundary conditions given by Eq. (2.2). For simplicity, we will assume that $S_\phi=0$. Integrating Eq. (2.1) twice, we obtain $\phi(x)=C_1 x+C_2$, where $C_1$ and $C_2$ are the undetermined constants of integration. To determine these two constants, we generally substitute the two boundary conditions [Eq. (2.2)] into $\phi(x)=C_1 x \perp C_2$ and solve for $C_1$ and $C_2$. This procedure has the underlying assumption that the second derivative of $\phi$ is continuous in the closed interval $\left[x_{\mathrm{L}}, x_{\mathrm{R}}\right]$, and therefore, the validity of the governing equation is implied to extend to the end points of the domain. In principle, this may not always be the case, and strictly speaking, the boundary conditions [Eq. (2.1)] should be interpreted as follows.

## 数学代写|偏微分方程代写partial difference equations代考|GENERAL PROCEDURE FOR DERIVING DIFFERENCE APPROXIMATIONS

In the preceding section, derivation of a central difference approximation to the second derivative was presented. The procedure was relatively straightforward to grasp because the nodal spacing used was uniform, and only three points in the stencil were used. In general, however, several important questions need to be addressed before one can concoct a foolproof recipe for deriving difference approximations. These questions include the following:

1. Is there a relationship between the order of the derivative to be derived and the number of Taylor series expansions that may be used?
2. What should be the pivot point in the Taylor series expansions? For example, in Eq. (2.6), why did we expand $\phi_{i+1}$ and $\phi_{i-1}$ about $\phi_i$ and not the other way around?
3. Given the derivative order and the number of Taylor series expansions to be used, can we predict $a$ priori what the order of the resulting error will be?

There are a minimum number of Taylor series expansions one must use to derive an expression for the derivative of a certain order. For example, with just two nodes of the stencil and one Taylor series expansion, it is impossible to derive an expression for the second derivative. A minimum of two Taylor series expansions and three nodes must be used. In general, to derive an expression for the $m$-th derivative, at least $m$ Taylor series expansions and $m+1$ nodes must be used. The order of the error, $n$, using the minimum number of Taylor series expansions would be greater than or equal to 1 . For example, the central difference scheme employed two Taylor series expansions, i.e., the bare minimum, but resulted in a second-order error. This is because in this particular case, with equal grid spacing the third derivative containing terms fortuitously cancelled out. If more than the bare minimum number of Taylor series expansions is used, it is possible to increase the order of the error by manipulating the expansions to the Taylor series expansion is performed, is dictated by the objective of the derivation. For example, in the derivation presented in Section 2.1, the pivot point used was the node $i$ because the second derivative was sought at that node.

Some of the rules of thumb just discussed are brought to light by the example that follows. It highlights a general procedure for deriving a difference approximation.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|DIFFERENCE APPROXIMATIONS AND TRUNCATION ERRORS

$$\frac{d^2 \phi}{d x^2}=S_\phi,$$

$$\phi\left(x_{\mathrm{L}}\right)=\phi_{\mathrm{L}}, \quad \phi\left(x_{\mathrm{R}}\right)=\phi_{\mathrm{R}},$$

## 数学代写|偏微分方程代写partial difference equations代考|GENERAL PROCEDURE FOR DERIVING DIFFERENCE APPROXIMATIONS

1. 要导出的导数的阶数和可以使用的泰勒级数展开的数量之间是否存在关系？
2. 泰勒级数展开中的支点应该是什么？例如，在方程式中。(2.6)，我们为什么要展开φ一世+1和φ一世−1关于φ一世而不是相反？
3. 给定导数阶和要使用的泰勒级数展开的数量，我们可以预测一个先验结果错误的顺序是什么？

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