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数学代写|数值分析代写numerical analysis代考|DIFFERENCE METHODS FOR THE DIFFUSION EQUATION
Perhaps the simplest PDE is the diffusion equation, so-called because it can be used to model a number of processes that are driven by diffusion, such as heat transfer and some types of slow mass transfer.
We seek the unknown function $u(x, t)$ such that
$$
\begin{aligned}
u_{t} &=a u_{x x}+f(x, t), \quad t>0,0<x<1 \
u(0, t) &=g_{0}(t) \
u(1, t) &=g_{1}(t) \
u(x, 0) &=u_{0}(x)
\end{aligned}
$$
Here $g_{0}$ and $g_{1}$ are the (known) boundary data, $f$ is a known source term, and $u_{0}$ is the (known) initial data. More general and involved boundary conditions are possible, and we can generalize the equation somewhat, but this is the usual standard form. For simplicity we assume homogeneous boundary data , meaning that $g_{0}=g_{1}=0$ for all time $t$. In the most commonplace examples, the unknown $u$ gives the temperature at each point $x$ along a thin rod, for each time $t$. The boundary conditions (9.3) and (9.4) give the temperature at which each end of the rod is maintained, and the initial condition (9.4) gives the temperature distribution at the start of the process. See a text, such as Humi and Miller [8], for background material on the diffusion equation.
The parameter $a$, which must be positive, is the thermal diffusivity of the material being studied. If $a$ is large, it means that diffusion is rapid; if $a$ is small, then diffusion is slow.
We will study two basic methods for the diffusion equation in this section: the explicit method, and the Crank-Nicolson method. In the next section we discuss the finite element method for problems of this type.
Before proceeding with the derivations and analysis, we need to deal with some notational issues. Our approximations will be constructed, as was the case for ODE problems in Chapter 6, only at discrete points on a grid defined over space and time. There will be times when we will want to consider the approximation as a discretely defined function, and there will be times when we want to consider it as a vector in a standard Euclidean space. At times we will also want or need to consider the collection of exact solution values defined at the same grid points. To accomplish all this with a minimum of excessive notation, we will use the following conventions.
If the dependent variable in a problem is $u$, then $u_{h, \Delta t}$ will denote the approximation, defined on a grid using $h$ as the spatial step and $\Delta t$ as the time step. Specific values can be denoted using the standard functional notation, $u_{h, \Delta t}\left(x_{i}, t_{n}\right)$, which we occasionally abbreviate as $u_{i}^{n}$. If we wish to consider the Euclidean vector of $u_{h, \Delta t}$ values at a specific point in time, $t_{n}$, then we will write $u_{h, \Delta t^{n}}^{n}$. The similar notation $u^{n}$ (note the lack of subscript) will be used for the vector of exact solution values at time $t_{n}$, using the same grid points. Some of this notation is abusive of standard conventions, but it enables us to work with a minimum of confusing new symbols. The need for most of this will not occur until $\S 9.1 .3$, where we begin to discuss implicit methods and the Crank-Nicolson method.
数学代写|数值分析代写numerical analysis代考|FINITE ELEMENT METHODS FOR THE DIFFUSION EQUATION
In $\S 6.10 .3$ we outlined the finite element method for two-point boundary value problems. Can we do something like that for the diffusion equation? Of course we can. Consider the following simple example:
$$
\begin{aligned}
u_{t} &=a u_{x x}+f(x, t) \quad 0 \leq x \leq 1, \quad t>0, \
u(0, t) &=0 \
u(1, t) &=0 \
u(x, 0) &=u_{0}(x) .
\end{aligned}
$$
It is relatively straight-forward to apply a finite element discretization to this problem. We assume a grid on $[0,1]$ so that
$$
0=x_{0}<x_{1}<\cdots<x_{N-1}<x_{N}=1
$$
and then look for an approximate solution in the form
$$
u_{h}(x, t)=\sum_{j=1}^{N-1} v_{j}(t) \phi_{j}^{h}(x)
$$
where the $\phi_{j}^{h}$ are the tent functions from $\S 6.10 .3$. Note that the coefficients $\left{v_{j}\right}$ are now taken to be functions of time, instead of constants. We proceed generally as we do in $\S 6.10 .3$ : First, multiply the PDE (9.22) by a smooth function $v$ which also satisfies the boundary conditions, and integrate by parts over the interval. We get
$$
\int_{0}^{1} u_{t}(x, t) v(x) d x+a \int_{0}^{1} u_{x}(x, t) v_{x}(x) d x=\int_{0}^{1} f(x, t) v(x) d x .
$$
This is analogous to (6.105). We now replace $u$ with $u_{h}$ and $v$ with $\phi_{j}^{h}$ to get
$$
\int_{0}^{1}\left{\left(\phi_{i}^{h}(x)\right)\left(\sum_{j=1}^{N-1} v_{j}^{\prime}(t) \phi_{j}^{h}(x)\right)+a\left(\phi_{i}^{h}(x)\right)^{\prime}\left(\sum_{j=1}^{N-1} v_{j}(t)\left(\phi_{j}^{h}(x)\right)^{\prime}\right)\right} d x=\int_{0}^{1} \phi_{i}^{h}(x) f(x) d x
$$
数学代写|数值分析代写NUMERICAL ANALYSIS代考|DIFFERENCE METHODS FOR POISSON EQUATIONS
In this section we discuss methods for the approximate solution of PDEs of the form
$$
-\Delta u=f, \quad(x, y) \in D ; u=g, \quad(x, y) \in \Gamma=\partial D
$$
where $\Delta$ is the Laplace operator,
$$
\Delta u=\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}
$$
$D$ is an open, connected subset of $\mathbb{R}^{2}$; and $\Gamma$ is the boundary of $D$. The unknown function $u$ therefore depends on two independent variables $x$ and $y$. Equations of this type are known as Poisson equations, after the French mathematician Siméon-Denis Poisson. ${ }^{3}$
For simplicity’s sake we will assume that $D$ is the unit square $(0,1) \times(0,1)$, although much more general domains and, of course, much more general equations can be considered. It is also possible to consider problems posed in three dimensions.
In terms of the general theory, problems of this form have much in common with the boundary value problems studied in $\S 6.10 .1$. The approximation is constructed by replacing the derivatives in the PDE with difference quotients, resulting in a system of linear equations. While we can apply Gaussian elimination and similar ideas to the solution of these systems, the size of the problem often mandates that we look at different methods.
Since the difference methods are fairly straightforward to apply, we will spend most of our time discussing efficient means for solving the large systems of linear equations that result from discretizing such problems. Thus, although the chapter is about approximating PDEs, this section has more to do with linear algebra than anything else.
We first define grids in both the $x$ and $y$ directions, in the usual way:
$$
0=x_{0}<x_{1}<x_{2}<\cdots<x_{N-1}<x_{N}=1
$$
and
$$
0=y_{0}<y_{1}<y_{2}<\cdots<y_{M-1}<y_{M}=1 .
$$
For simplicity’s sake, we assume that $N=M$, which implies an equivalent grid, with the same spacing $h$ in each direction. We denote the approximate solution as the array of values $u_{i, j} \approx u\left(x_{i}, y_{j}\right)$, where $x_{i}$ and $y_{j}$ are points on a grid over $D$. Second, we use these in the usual difference approximation to the second derivative to write the approximate form of the PDE as
$$
-\left[\frac{u_{i+1, j}-2 u_{i, j}+u_{i-1, j}}{h^{2}}\right]-\left[\frac{u_{i, j+1}-2 u_{i, j}+u_{i, j-1}}{h^{2}}\right]=f\left(x_{i}, y_{j}\right)
$$
which can be simplified somewhat to yield
$$
-u_{i+1, j}-u_{i-1, j}+4 u_{i, j}-u_{i, j+1}-u_{i, j-1}=h^{2} f_{i, j}, \quad 1 \leq i, j \leq N-1
$$
where we have written $f_{i, j}=f\left(x_{i}, y_{j}\right)$. This is a system of $(N-1)^{2}$ equations in $(N-1)^{2}$ unknowns. To write it in a matrix-vector format, we define the vector $u_{h}$ as a one-dimensional arrangement of the $u_{i, j}$ values:
$$
u_{h}=\left(u_{1,1}, u_{1,2}, u_{1,3}, \ldots, u_{1, N-1}, u_{2,1}, \ldots, u_{N-1, N-1}\right)^{T}
$$
We could just as easily have ordered the $u_{i, j}$ in another way (by columns, instead of rows, for example). The ordering of the unknowns can make a difference, especially in more complicated settings.
数值分析代写
数学代写|数值分析代写NUMERICAL ANALYSIS代考|DIFFERENCE METHODS FOR THE DIFFUSION EQUATION
也许最简单的 PDE 是扩散方程，之所以这么称呼它，是因为它可以用来模拟许多由扩散驱动的过程，例如传热和某些类型的缓慢传质。
我们寻求未知的功能在(X,吨)这样
在吨=一种在XX+F(X,吨),吨>0,0<X<1 在(0,吨)=G0(吨) 在(1,吨)=G1(吨) 在(X,0)=在0(X)
这里G0和G1是ķn这在n边界数据，F是一个已知的源项，并且在0是个ķn这在n初始数据。更一般和更复杂的边界条件是可能的，我们可以对方程进行一些概括，但这是通常的标准形式。为简单起见，我们假设同质边界数据，这意味着G0=G1=0一直以来吨. 在最常见的例子中，未知在给出每个点的温度X沿着一根细棒，每次吨. 边界条件9.3和9.4给出杆的每一端保持的温度，以及初始条件9.4给出过程开始时的温度分布. 查看文本，例如 Humi 和 Miller8，对于扩散方程的背景材料。
参数一种，它必须是正的，是正在研究的材料的热扩散率。如果一种大，说明扩散快；如果一种小，则扩散慢。
在本节中，我们将研究扩散方程的两种基本方法：显式方法和 Crank-Nicolson 方法。在下一节中，我们将讨论此类问题的有限元方法。
在进行推导和分析之前，我们需要处理一些符号问题。我们的近似值将被构造，就像第 6 章中 ODE 问题的情况一样，仅在空间和时间上定义的网格上的离散点处。有时我们希望将近似视为离散定义的函数，有时我们希望将其视为标准欧几里得空间中的向量。有时我们还希望或需要考虑在相同网格点定义的精确解值的集合。为了以最少的过多符号完成所有这些，我们将使用以下约定。
如果问题中的因变量是在， 然后在H,Δ吨将表示近似值，使用网格定义H作为空间步骤和Δ吨作为时间步长。可以使用标准功能符号表示特定值，在H,Δ吨(X一世,吨n)，我们有时将其缩写为在一世n. 如果我们想考虑欧几里得向量在H,Δ吨特定时间点的值，吨n，那么我们会写在H,Δ吨nn. 类似的符号在n n这吨和吨H和l一种Cķ这Fs在bsCr一世p吨将用于时间精确解值的向量吨n，使用相同的网格点。其中一些符号滥用了标准约定，但它使我们能够使用最少的令人困惑的新符号。大部分的需求不会发生，直到§§9.1.3，我们开始讨论隐式方法和 Crank-Nicolson 方法。
数学代写|数值分析代写NUMERICAL ANALYSIS代考|FINITE ELEMENT METHODS FOR THE DIFFUSION EQUATION
在§§6.10.3我们概述了两点边值问题的有限元方法。我们可以为扩散方程做类似的事情吗？当然，我们可以。考虑以下简单示例：
在吨=一种在XX+F(X,吨)0≤X≤1,吨>0, 在(0,吨)=0 在(1,吨)=0 在(X,0)=在0(X).
对这个问题应用有限元离散化是相对简单的。我们假设一个网格[0,1]以便
0=X0<X1<⋯<Xñ−1<Xñ=1
然后在表格中寻找一个近似解
在H(X,吨)=∑j=1ñ−1在j(吨)φjH(X)
在哪里φjH是来自的帐篷函数§§6.10.3. 注意系数\左{v_{j}\右}\左{v_{j}\右}现在被认为是时间的函数，而不是常数。我们通常像我们在§§6.10.3：首先，乘以 PDE9.22通过平滑函数在也满足边界条件，在区间内按部分积分。我们得到
∫01在吨(X,吨)在(X)dX+一种∫01在X(X,吨)在X(X)dX=∫01F(X,吨)在(X)dX.
这类似于6.105. 我们现在替换在和在H和在和φjH要得到
\int_{0}^{1}\left{\left(\phi_{i}^{h}(x)\right)\left(\sum_{j=1}^{N-1} v_{j} ^{\prime}(t) \phi_{j}^{h}(x)\right)+a\left(\phi_{i}^{h}(x)\right)^{\prime}\left (\sum_{j=1}^{N-1} v_{j}(t)\left(\phi_{j}^{h}(x)\right)^{\prime}\right)\right} d x=\int_{0}^{1} \phi_{i}^{h}(x) f(x) d x\int_{0}^{1}\left{\left(\phi_{i}^{h}(x)\right)\left(\sum_{j=1}^{N-1} v_{j} ^{\prime}(t) \phi_{j}^{h}(x)\right)+a\left(\phi_{i}^{h}(x)\right)^{\prime}\left (\sum_{j=1}^{N-1} v_{j}(t)\left(\phi_{j}^{h}(x)\right)^{\prime}\right)\right} d x=\int_{0}^{1} \phi_{i}^{h}(x) f(x) d x
数学代写|数值分析代写NUMERICAL ANALYSIS代考|DIFFERENCE METHODS FOR POISSON EQUATIONS
在本节中，我们讨论形式为 PDE 的近似解的方法
−Δ在=F,(X,是)∈D;在=G,(X,是)∈Γ=∂D
在哪里Δ是拉普拉斯算子，
Δ在=∂2在∂X2+∂2在∂是2
D是一个开放的、连通的子集R2; 和Γ是的边界D. 未知功能在因此取决于两个自变量X和是. 这种类型的方程被称为泊松方程，以法国数学家 Siméon-Denis Poisson 命名。3
为简单起见，我们假设D是单位平方(0,1)×(0,1)，尽管可以考虑更通用的域，当然也可以考虑更通用的方程。也可以考虑三个维度提出的问题。
在一般理论方面，这种形式的问题与研究的边值问题有很多共同之处。§§6.10.1. 通过用差商替换 PDE 中的导数来构造近似值，从而得到一个线性方程组。虽然我们可以将高斯消除和类似的想法应用于这些系统的解决方案，但问题的规模通常要求我们采用不同的方法。
由于差分方法的应用相当简单，因此我们将花费大部分时间讨论解决由离散此类问题产生的大型线性方程组的有效方法。因此，尽管本章是关于逼近 PDE，但本节更多地与线性代数有关。
我们首先在两者中定义网格X和是方向，以通常的方式：$$
-\left[\frac{u_{i+1, j}-2 u_{i, j}+u_{i-1, j}}{h^{2}}\right]-\left[\frac{u_{i, j+1}-2 u_{i, j}+u_{i, j-1}}{h^{2}}\right]=f\left(x_{i}, y_{j}\right)
$$
which can be simplified somewhat to yield
$$
-u_{i+1, j}-u_{i-1, j}+4 u_{i, j}-u_{i, j+1}-u_{i, j-1}=h^{2} f_{i, j}, \quad 1 \leq i, j \leq N-1
$$
where we have written $f_{i, j}=f\left(x_{i}, y_{j}\right)$. This is a system of $(N-1)^{2}$ equations in $(N-1)^{2}$ unknowns. To write it in a matrix-vector format, we define the vector $u_{h}$ as a one-dimensional arrangement of the $u_{i, j}$ values:
$$
u_{h}=\left(u_{1,1}, u_{1,2}, u_{1,3}, \ldots, u_{1, N-1}, u_{2,1}, \ldots, u_{N-1, N-1}\right)^{T}
$$
We could just as easily have ordered the $u_{i, j}$ 未知数的排序可能会有所不同，尤其是在更复杂的设置中。
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