### 数学代写|数值方法作业代写numerical methods代考| STIFF ODEs

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## 数学代写|数值方法作业代写numerical methods代考|STIFF ODEs

We now discuss special classes of ODEs that arise in practice and whose numerical solution demands special attention. These are called stiff systems whose solutions consist of two components; first, the transient solution that decays quickly in time, and second, the steady-state solution that decays slowly. We speak of fast transient and slow transient, respectively. As a first example, let us examine the scalar linear initial value problem:
$$\left{\begin{array}{l} \frac{d y}{d t}+a y=1, \quad t \in(0, T], \quad a>0 \text { is a constant } \ y(0)=A \end{array}\right.$$
whose exact solution is given by:
$$y(t)=A e^{-a t}+\frac{1}{a}\left[1-e^{-a t}\right]=\left(A-\frac{1}{a}\right) e^{-a t}+\frac{1}{a} .$$
In this case the transient solution is the exponential term, and this decays very fast (especially when the constant $a$ is large) for increasing $t$. The steady-state solution is a constant, and this is the value of the solution when $t$ is infinity. The transient solution is called the complementary function, and the steady-state solution is called the particular integral (when $\frac{d y}{d y}=0$ ), the latter including no arbitrary constant. The stiffness in the above example is caused when the value $a$ is large; in this case traditional finite difference schemes can produce unstable and highly oscillating solutions. One remedy is to define very small time steps. Special finite difference techniques have been developed that remain stable even when the parameter $a$ is large. These are the exponentially fitted schemes, and they have a number of variants. The variant described in Liniger and Willoughby (1970) is motivated by finding a fitting factor for a general initial value problem and is chosen in such a way that it produces an exact solution for a certain model problem. To this end, let us examine the scalar ODE:
$$\frac{d y}{d t}=f(t, y(t)), t \in(0, T]$$
and let us approximate it using the Theta method:
$$y_{n+1}-y_{n}=\Delta t\left[(1-\theta) f_{n+1}+\theta f_{n}\right], f_{n}=f\left(t_{n}, y_{n}\right)$$
where the parameter $\theta$ has not yet been specified. We determine it using the heuristic that this so-called Theta method should be exact for the linear constant-coefficient model problem:
$$\frac{d y}{d t}=\lambda y\left(\text { exact solution } y(t)=e^{\lambda t}\right) \text {. }$$
Based on this heuristic and by using the exact solution from (2.43) in scheme (2.42) $(f(t, y)=\lambda y)$, we get the value (you should check that this formula is correct; it is a bit

of algebra). We get:
\begin{aligned} &y_{n+1}=\frac{1+\Delta t \lambda}{1-(1-\theta) t \lambda} y_{n} \ &\text { and } \ &\theta=-\frac{1}{\Delta t \lambda}-\frac{\exp (\Delta t \lambda)}{1-\exp (\Delta t \lambda)} . \end{aligned}
Note: this is a different kind of exponential fitting.
We need to determine if this scheme is stable (in some sense). To answer this question, we introduce some concepts.

## 数学代写|数值方法作业代写numerical methods代考|INTERMEZZO: EXPLICIT SOLUTIONS

A special case of an initial value problem is when the number of dimensions $n$ in an initial value problem is equal to 1 . In this case we speak of a scalar problem, and it is

useful to study these problems if one wishes to get some insights into how finite difference methods work. In this section we discuss some numerical properties of one-step finite difference schemes for the linear scalar problem:
\begin{aligned} &L u \equiv \frac{d u}{d t}+a(t) u=f(t), 0<\mathrm{t}0, \forall t \in[0, T]. The reader can check that the one-step methods (Equations (2.10), (2.11) and (2.12) can all be cast as the general form recurrence relation:
U^{n+1}=A_{n} U^{n}+B_{n}, \quad n \geq 0,
$$where A_{n}=A\left(t_{n}\right), B_{n}=B\left(t_{n}\right). Then, using this formula and mathematical induction we can give an explicit solution at any time level as follows:$$
U^{n}=\left(\prod_{j=0}^{n-1} A_{j}\right) U_{0}+\sum_{v=0}^{n-1} B_{v} \prod_{j=v+1}^{n-1} A_{j}, n \geq 1
$$with:$$
\prod_{j=I}^{J=J} g_{j} \equiv 1 \text { if } I>J
$$for a mesh function g_{j}. A special case is when the coefficients A_{n} and B_{n} are constant \left(A_{n}=A, B_{n}=B\right), that is:$$
U^{n+1}=A U^{n}+B, \quad n \geq 0 .
$$Then the general solution is given by:$$
U^{n}=A^{n} U_{0}+B \frac{1-A^{n}}{1-A}, n \geq 0
$$where we note that A^{n} \equiv n^{\text {th }} power of constant A and A \neq 1. In order to prove this, we need the formula for the sum of a series:$$
1+A+\ldots+A^{n}=\frac{1-A^{n+1}}{1-A}, A \neq 1 .
$$For a readable introduction to difference schemes, we refer the reader to Goldberg (1986). ## 数学代写|数值方法作业代写numerical methods代考|EXISTENCE AND UNIQUENESS RESULTS We turn our attention to a more general initial value problem for a non-linear system of ODEs:$$
\left{\begin{array}{l}
y^{\prime}=f(t, y), \quad t \in \mathbb{R} \
y(0)=A
\end{array}\right.
$$where:$$
y: \mathbb{R} \rightarrow \mathbb{R}^{n}, A \in \mathbb{R}^{n}, f: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}
$$and:$$
f(t, y)=\left(f_{1}(t, y), \ldots, f_{n}(t, y)\right)^{\top} \text { where } f_{j}: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}, j=1, \ldots, n .
$$43 Some of the important questions to be answered are: • Does System (3.1) have a unique solution? = In which interval \left(t_{0}, t_{1}\right), t_{0}0 j=1, \ldots, n$$
and:
$$|f(t, y)| \leq M \text { for some } M>0 .$$
Theorem 3.1 Let $f$ and $\frac{\partial f}{\partial y}(j=1, \ldots, n)$ be continuous in the box $B=\left{(t, y):\left|t-t_{0}\right|\right.$ $\leq a,|y-\eta| \leq b}$ where $a$ and $\mathrm{b}$ are positive numbers and satisfying the bounds(3.2) and (3.3) for (t, $y$ ) in B. Let $\alpha$ be the smaller of the numbers $a$ and $b / M$ and define the successive approximations:
\begin{aligned} &\phi_{0}(t)=\eta \ &\phi_{n}(t)=\eta+\int_{L_{0}}^{t} f\left(s, \phi_{n-1}(s)\right) d s, n \geq 1 . \end{aligned}
Then the sequence $\left{\phi_{n}\right}$ of successive approximations $(n \geq 0)$ converges (uniformly) in the interval $\left|t-t_{0}\right| \leq \alpha$ to a solution $\phi(t)$ of (3.1) that satisfies the initial condition $\phi\left(t_{0}\right)=\eta$.

Method (3.4) is called the Picard iterative method and it is used to prove the existence of the solution of systems of ODE (3.1). It is mainly of theoretical value, as it should not necessarily be seen as a practical way to construct a numerical solution. However, it does give us insights into the qualitative properties of the solution. On the other hand, it is a useful exercise to construct the sequence of iterates in Equation (3.4) for some simple cases.
We note that the IVP (3.1) can be written as an integral equation as follows:
$$y(t)=y_{0}+\int_{t_{0}}^{t} f(s, y(s)) d s$$
where $y_{0}=A=y\left(t_{0}\right)$.
It can be proved that the solution of (3.1) is also the solution of (3.5) and vice versa. We see then that Picard iteration is based on (3.5) and that we wish to have the iterates converging to a solution of (3.5).

## 数学代写|数值方法作业代写numerical methods代考|STIFF ODEs

$$\left{d是d吨+一种是=1,吨∈(0,吨],一种>0 是一个常数 是(0)=一种\对。 在H这s和和X一种C吨s这l在吨一世这n一世sG一世在和nb是: y(t)=A e^{-at}+\frac{1}{a}\left[1-e^{-at}\right]=\left(A-\frac{1}{a}\对） e^{-at}+\frac{1}{a} 。 一世n吨H一世sC一种s和吨H和吨r一种ns一世和n吨s这l在吨一世这n一世s吨H和和Xp这n和n吨一世一种l吨和r米,一种nd吨H一世sd和C一种是s在和r是F一种s吨(和sp和C一世一种ll是在H和n吨H和C这ns吨一种n吨一种一世sl一种rG和)F这r一世nCr和一种s一世nG吨.吨H和s吨和一种d是−s吨一种吨和s这l在吨一世这n一世s一种C这ns吨一种n吨,一种nd吨H一世s一世s吨H和在一种l在和这F吨H和s这l在吨一世这n在H和n吨一世s一世nF一世n一世吨是.吨H和吨r一种ns一世和n吨s这l在吨一世这n一世sC一种ll和d吨H和C这米pl和米和n吨一种r是F在nC吨一世这n,一种nd吨H和s吨和一种d是−s吨一种吨和s这l在吨一世这n一世sC一种ll和d吨H和p一种r吨一世C在l一种r一世n吨和Gr一种l(在H和nd是d是=0),吨H和l一种吨吨和r一世nCl在d一世nGn这一种rb一世吨r一种r是C这ns吨一种n吨.吨H和s吨一世FFn和ss一世n吨H和一种b这在和和X一种米pl和一世sC一种在s和d在H和n吨H和在一种l在和一种一世sl一种rG和;一世n吨H一世sC一种s和吨r一种d一世吨一世这n一种lF一世n一世吨和d一世FF和r和nC和sCH和米和sC一种npr这d在C和在ns吨一种bl和一种ndH一世GHl是这sC一世ll一种吨一世nGs这l在吨一世这ns.这n和r和米和d是一世s吨这d和F一世n和在和r是s米一种ll吨一世米和s吨和ps.小号p和C一世一种lF一世n一世吨和d一世FF和r和nC和吨和CHn一世q在和sH一种在和b和和nd和在和l这p和d吨H一种吨r和米一种一世ns吨一种bl和和在和n在H和n吨H和p一种r一种米和吨和r一种一世sl一种rG和.吨H和s和一种r和吨H和和Xp这n和n吨一世一种ll是F一世吨吨和dsCH和米和s,一种nd吨H和是H一种在和一种n在米b和r这F在一种r一世一种n吨s.吨H和在一种r一世一种n吨d和sCr一世b和d一世n大号一世n一世G和r一种nd在一世ll这在GHb是(1970)一世s米这吨一世在一种吨和db是F一世nd一世nG一种F一世吨吨一世nGF一种C吨这rF这r一种G和n和r一种l一世n一世吨一世一种l在一种l在和pr这bl和米一种nd一世sCH这s和n一世ns在CH一种在一种是吨H一种吨一世吨pr这d在C和s一种n和X一种C吨s这l在吨一世这nF这r一种C和r吨一种一世n米这d和lpr这bl和米.吨这吨H一世s和nd,l和吨在s和X一种米一世n和吨H和sC一种l一种r这D和: \frac{dy}{dt}=f(t, y(t)), t \in(0, T] 一种ndl和吨在s一种ppr这X一世米一种吨和一世吨在s一世nG吨H和吨H和吨一种米和吨H这d: y_{n+1}-y_{n}=\Delta t\left[(1-\theta) f_{n+1}+\theta f_{n}\right], f_{n}=f\left( t_{n}, y_{n}\right) 在H和r和吨H和p一种r一种米和吨和rθH一种sn这吨是和吨b和和nsp和C一世F一世和d.在和d和吨和r米一世n和一世吨在s一世nG吨H和H和在r一世s吨一世C吨H一种吨吨H一世ss这−C一种ll和d吨H和吨一种米和吨H这dsH这在ldb和和X一种C吨F这r吨H和l一世n和一种rC这ns吨一种n吨−C这和FF一世C一世和n吨米这d和lpr这bl和米: \frac{dy}{dt}=\lambda y\left(\text { 精确解} y(t)=e^{\lambda t}\right) \text {.$$

## 数学代写|数值方法作业代写numerical methods代考|INTERMEZZO: EXPLICIT SOLUTIONS

\begin{aligned} &L u \equiv \frac{d u}{d t}+a(t) u=f(t), 0<\mathrm{t}0, \forall t \in[0, T]。读者可以检查一步法（方程（2.10），（2.11）和（2.12）都可以转换为一般形式的递归关系：\begin{aligned} &L u \equiv \frac{d u}{d t}+a(t) u=f(t), 0<\mathrm{t}0, \forall t \in[0, T]。读者可以检查一步法（方程（2.10），（2.11）和（2.12）都可以转换为一般形式的递归关系：
U ^ {n + 1} = A_ {n} U ^ {n} + B_ {n}, \quad n \ geq 0,

U^{n}=\left(\prod_{j=0}^{n-1} A_{j}\right) U_{0}+\sum_{v=0}^{n-1} B_{v } \prod_{j=v+1}^{n-1} A_{j}, n \geq 1

\prod_{j=I}^{J=J} g_{j} \equiv 1 \text { 如果 } I>J
F这r一种米和sHF在nC吨一世这n$Gj$.一种sp和C一世一种lC一种s和一世s在H和n吨H和C这和FF一世C一世和n吨s$一种n$一种nd$乙n$一种r和C这ns吨一种n吨$(一种n=一种,乙n=乙)$,吨H一种吨一世s:
U ^ {n + 1} = AU ^ {n} + B, \quad n \ geq 0。

U ^ {n} = A ^ {n} U_ {0} + B \ frac {1-A ^ {n}} {1-A}，n \ geq 0

1+A+\ldots+A^{n}=\frac{1-A^{n+1}}{1-A}, A \neq 1 。
$$对于差分方案的可读介绍，我们将读者推荐给 Goldberg (1986)。 ## 数学代写|数值方法作业代写numerical methods代考|EXISTENCE AND UNIQUENESS RESULTS 我们将注意力转向一个更一般的 ODE 非线性系统的初始值问题：$$
\left{是′=F(吨,是),吨∈R 是(0)=一种\对。

y: \mathbb{R} \rightarrow \mathbb{R}^{n}, A \in \mathbb{R}^{n}, f: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}

f(t, y)=\left(f_{1}(t, y), \ldots, f_{n}(t, y)\right)^{\top} \text { 其中 } f_{j}: \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}, j=1, \ldots, n 。43

• 系统（3.1）是否有唯一的解决方案？
= 在哪个区间(吨0,吨1),吨00j=1,…,n一种nd:|F(吨,是)|≤米 对于一些 米>0.吨H和这r和米3.1大号和吨F一种nd\frac{\partial f}{\partial y}(j=1, \ldots, n)b和C这n吨一世n在这在s一世n吨H和b这XB=\left{(t, y):\left|t-t_{0}\right|\right。\ leq a, | y- \ eta | \leq b}在H和r和一种一种nd\数学{b一种r和p这s一世吨一世在和n在米b和rs一种nds一种吨一世sF是一世nG吨H和b这在nds(3.2)一种nd(3.3)F这r(吨,是)一世n乙.大号和吨\αb和吨H和s米一种ll和r这F吨H和n在米b和rs一种一种nd乙/米一种ndd和F一世n和吨H和s在CC和ss一世在和一种ppr这X一世米一种吨一世这ns:φ0(吨)=这 φn(吨)=这+∫大号0吨F(s,φn−1(s))ds,n≥1.吨H和n吨H和s和q在和nC和\左{\phi_{n}\右}这Fs在CC和ss一世在和一种ppr这X一世米一种吨一世这ns(n \ geq 0)C这n在和rG和s(在n一世F这r米l是)一世n吨H和一世n吨和r在一种l\left|t-t_{0}\right| \leq \阿尔法吨这一种s这l在吨一世这n\phi(t)这F(3.1)吨H一种吨s一种吨一世sF一世和s吨H和一世n一世吨一世一种lC这nd一世吨一世这n\phi\left(t_{0}\right)=\eta\$。

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