数学代写|数值方法作业代写numerical methods代考|Common Schemes

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

数学代写|数值方法作业代写numerical methods代考|Common Schemes

We now introduce a number of important and useful difference schemes that approximate the solution of Equation (2.1). These schemes will pop up all over the place in later chapters. Understanding how the schemes work in a simpler context will help you appreciate them when we tackle partial differential equations based on the Black-Scholes model. They also help in our understanding of notation, jargon, and syntax.
The main schemes are:

• Explicit Euler
• Implicit Euler
• Crank-Nicolson (or Box scheme)
• The trapezoidal method.
The explicit Euler method is given by:
\begin{aligned} &\frac{u^{n+1}-u^{n}}{k}+a^{n} u^{n}=f^{n}, n=0, \ldots, N-1 \ &u^{0}=A \end{aligned}

whereas the implicit Euler method is given by:
\begin{aligned} &\frac{u^{n+1}-u^{n}}{k}+a^{n+1} u^{n+1}=f^{n+1}, n=0, \ldots, N-1 \ &u^{0}=A \end{aligned}
Notice the difference: in Equation (2.10) the solution at level $n+1$ can be directly calculated in terms of the solution at level $n$, while in Equation (2.11) we must rearrange terms in order to calculate the solution at level $n+1$.

The next scheme is called the Crank-Nicolson or box scheme, and it can be seen as an average of explicit and implicit Euler schemes. It is given as (see notation in Equation (2.7)):
$\frac{u^{n+1}-u^{n}}{k}+a^{n, \frac{1}{2}} u^{n, \frac{1}{2}}=f^{n, \frac{1}{2}}, n=0, \ldots, N-1$ $u^{0}=A$ where $u^{n, \frac{1}{2}} \equiv \frac{1}{2}\left(u^{n}+u^{n+1}\right)$
It is useful to know that the three schemes can be merged into one generic scheme as it were by introducing a parameter $\theta$ (the scheme is sometimes called the Theta method):
\begin{aligned} &L(k) u^{n} \equiv \frac{u^{n+1}-u^{n}}{k}+a^{n, \theta} u^{n, \theta}=f^{n, \theta} \ &u^{n, \theta} \equiv \theta u^{n}+(1-\theta) u^{n+1}, 0 \leq \theta \leq 1 \ &f^{n, \theta} \equiv f\left(\theta t_{n}+(1-\theta) t_{n+1}\right) \end{aligned}
and the special cases are given by:
\begin{aligned} &\theta=1, \text { explicit Euler } \ &\theta=0, \text { implicit Euler } \ &\theta=\frac{1}{2}, \text { Crank-Nicolson. } \end{aligned}
The solution of Equation (2.13) is given by:
$$u^{n+1, \theta} \equiv u^{n+1,}=\frac{\left(1-k \theta a^{n, \theta}\right) u^{n}+k f^{n, \theta}}{1+k(1-\theta) a^{n, \theta}} .$$
This equation is useful because it can be mapped to $\mathrm{C}++$ code and will be used by other schemes by defining the appropriate value of the parameter $\theta$.

数学代写|数值方法作业代写numerical methods代考|Discrete Maximum Principle

Having developed some difference schemes, we would like to have a way of determining if the discrete solution is a good approximation to the exact solution in some sense. Although we do not deal with this issue in great detail, we do look at stability and convergence issues.

Definition 2.1 The one-step difference scheme $L(k)$ of the form (2.13) is said to be positive if:
$$L(k) w^{n} \geq 0, n=0, \ldots, N-1, w^{0} \geq 0$$
implies that $w^{n} \geq 0 \forall n=0, \ldots, N$. Here, $w^{n}$ is a mesh function defined at the mesh points $t_{n}$.

Based on this definition, we see that the implicit Euler scheme is always positive while the explicit Euler scheme is positive if the term:
$$1-k a^{n} \geq 0 \text { or } k \leq \frac{1}{a^{n}}, n \geq 0$$
is positive. Thus, if the function $a(t)$ achieves large values (and this happens in practice), we will have to make $k$ very small in order to produce good results. Even worse, if $k$ does not satisfy the constraint in (2.18) then the discrete solution looks nothing like the exact solution, and so-called spurious oscillations occur. This phenomenon occurs in other finite difference schemes, and we propose a number of remedies later in this book.
Definition $2.2$ A difference scheme is stable if its solution is based in much the same way as the solution of the continuous problem (2.1) (see Theorem 2.1), that is:
$$\left|u^{n}\right| \leq \frac{N}{\alpha}+|A|, \quad n \geq 0$$
where:
$$a\left(t_{n}\right) \geq \alpha, n \geq 0,\left|f\left(t_{n}\right)\right| \leq N, n \geq 0$$
and:
$$u^{0}=A$$

Based on the fact that a scheme is stable and consistent (see Dahlquist and Björck (1974)), we can state in general that the error between the exact and discrete solutions is bounded by some polynomial power of the step-size $k$ :
$$\left|u^{n}-u\left(t_{n}\right)\right| \leq M k^{p}, \quad p=1,2, \ldots, n \geq 0$$
where $M$ is a constant that is independent of $k$. For example, in the case of schemes $2.10$, $2.11$ and $2.12$ we have:
Implicit Euler: $\left|u^{n}-u\left(t_{n}\right)\right| \leq M k, n=0, \ldots, N$
Crank-Nicolson (Box): $\left|u^{n}-u\left(t_{n}\right)\right| \leq M k^{2}, n=0, \ldots, N$
Explicit Euler: $\left|u^{n}-u\left(t_{n}\right)\right| \leq M k, n=0, \ldots, N$ if $1-a^{n} k>0$.
Thus, we see that the Box method is second-order accurate and is better than the implicit Euler scheme, which is only first-order accurate.

数学代写|数值方法作业代写numerical methods代考|Exponential Fitting

We now introduce a special class of schemes with desirable properties. These are schemes that are suitable for problems with rapidly increasing or decreasing solutions. In the literature these are called stiff or singular perturbation problems (see Duffy (1980)). We can motivate these schemes in the present context. Let us take the problem (2.1) when $a(t)$ is constant and $f(t)$ is zero. The solution $u(t)$ is given by a special case of (2.2), namely:
$$u(t)=A e^{-a t} .$$
If $a$ is large then the derivatives of $u(t)$ tend to increase; in fact, at $t=0$, the derivatives are given by:
$$\frac{d^{k} u(0)}{d t^{k}}=A(-a)^{k}, \quad k=0,1,2, \ldots$$
The physical interpretation of this fact is that a boundary layer exits near $t=0$ where $u$ is changing rapidly, and it has been shown that classical finite difference schemes fail to give acceptable answers when $a$ is large (typically values between 1000 and 10000). We get so-called spurious oscillations, and this problem is also encountered when solving one-factor and multifactor Black-Scholes equations using finite difference methods. We have resolved this problem using so-called exponentially fitted schemes. We motivate the scheme in the present context, and later chapters describe how to apply it to more complicated cases.

In order to motivate the fitted scheme, consider the case of constant $a(t)$ and $f(t)=0$. We wish to produce a difference scheme in such a way that the discrete solution is equal to the exact solution at the mesh points for this constant-coefficient case. We introduce a so-called fitting factor $\sigma$ in the new scheme:
$$\left{\begin{array}{l} \sigma\left(\frac{u^{n+1}-u^{n}}{k}\right)+a^{n, \theta} u^{n, \theta}=f^{n, \theta}, n=0, \ldots, N-1,0 \leq \theta \leq 1 \ u^{0}=A . \end{array}\right.$$
The motivation for finding the fitting factor is to demand that the exact solution of (2.1) (which is known) has the same values as the discrete solution of (2.24) at the mesh points.

Plugging the exact solution (2.22) into $(2.24)$ and doing some simple arithmetic, we get the following representation for the fitting factor $\sigma$ :
$$\sigma=\frac{a k\left(\theta+(1-\theta) e^{-a k}\right)}{1-e^{-a k}}$$
Having found the fitting factor for the constant coefficient case, we generalise to a scheme for the case (2.1) as follows:
\begin{aligned} &\sigma^{n, \theta} \frac{u^{n+1}-u^{n}}{k}+a^{n, \theta} u^{n, \theta}=f^{n, \theta}, n=0, \ldots, N-1,0 \leq \theta \leq 1 \ &u^{0}=A \ &\sigma^{n, \theta}=\frac{a^{n, \theta}\left(\theta+(1-\theta) e^{-a^{n, \theta} k}\right)}{1-e^{-a^{n}, \theta_{k}}} k . \end{aligned}
In practice we work with a number of special cases:
In the final case coth $(x)$ is the hyperbolic cotangent function.

数学代写|数值方法作业代写numerical methods代考|Common Schemes

• 显式欧拉
• 隐式欧拉
• Crank-Nicolson（或 Box 方案）
• 梯形法。
显式欧拉方法由下式给出：
在n+1−在nķ+一种n在n=Fn,n=0,…,ñ−1 在0=一种

θ=1, 显式欧拉  θ=0, 隐式欧拉  θ=12, 曲柄-尼科尔森。

数学代写|数值方法作业代写numerical methods代考|Discrete Maximum Principle

1−ķ一种n≥0 或者 ķ≤1一种n,n≥0

|在n|≤ñ一种+|一种|,n≥0

|在n−在(吨n)|≤米ķp,p=1,2,…,n≥0

数学代写|数值方法作业代写numerical methods代考|Exponential Fitting

dķ在(0)d吨ķ=一种(−一种)ķ,ķ=0,1,2,…

$$\left{σ(在n+1−在nķ)+一种n,θ在n,θ=Fn,θ,n=0,…,ñ−1,0≤θ≤1 在0=一种.\对。$$

σ=一种ķ(θ+(1−θ)和−一种ķ)1−和−一种ķ

σn,θ在n+1−在nķ+一种n,θ在n,θ=Fn,θ,n=0,…,ñ−1,0≤θ≤1 在0=一种 σn,θ=一种n,θ(θ+(1−θ)和−一种n,θķ)1−和−一种n,θķķ.

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