### 数学代写|数值方法作业代写numerical methods代考|Scalar Non-Linear Problems and Predictor-Corrector Method

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• Statistical Inference 统计推断
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• 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|Scalar Non-Linear Problems and Predictor-Corrector Method

Real-life problems are very seldom linear. In general, we model applications using nonlinear IVPs:
$$\left{\begin{array}{l} u^{\prime} \equiv \frac{d u}{d t}=f(t, u), t \in(0, T] \ u(0)=A . \end{array}\right.$$
Here $f(t, u)$ is a non-linear function in $u$ in general. Of course, Equation (2.28) contains Equation (2.1) as a special case. However, it is not possible to come up with an exact solution for (2.28) in general, and we must resort to some numerical techniques. Approximating (2.28) poses challenges because the resulting difference schemes may also be non-linear, thus forcing us to solve the discrete system at each time level by Newton’s method or some other non-linear solver. For example, consider applying the trapezoidal method to (2.28):
$$u_{n+1}=u_{n}+\frac{k}{2}\left[f\left(t_{n}, u_{n}\right)+f\left(t_{n+1}, u_{n+1}\right)\right] n=0, \ldots, N-1$$
where $f(t, u)$ is non-linear. Here see that the unknown term $u$ is on both the left-and right-hand sides of the equation, and hence it is not possible to solve the problem explicitly in the way that we did for the linear case. However, not all is lost, and to this end we introduce the predictor-corrector method that consists of a set consisting of two difference schemes; the first equation uses the explicit Euler method to produce an intermediate solution called a predictor that is then used in what could be called a modified trapezoidal rule:
Predictor: $\bar{u}{n+1}=u{n}+k f\left(t_{n}, u_{n}\right)$
Corrector: $u_{n+1}=u_{n}+\frac{k}{2}\left[f\left(t_{n}, u_{n}\right)+f\left(t_{n+1}, \bar{u}{n+1}\right)\right]$ or: $$u{n+1}=u_{n}+\frac{k}{2}\left{f\left(t_{n}, u_{n}\right)+f\left(t_{n+1}, u_{n}+k f\left(t_{n}, u_{n}\right)\right)\right} .$$
The predictor-corrector is used in practice; it can be used with non-linear systems and stochastic differential equations (SDE). We discuss this topic in Chapter $13 .$

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

We give an introduction to a technique that allows us to improve the accuracy of finite difference schemes. This is called Richardson extrapolation in general. We take a specific case to show the essence of the method, namely the implicit Euler method (2.11).

We know that it is first-order accurate and that it has good stability properties. We now apply the method on meshes of size $k$ and $k / 2$, and we can show that the approximate solutions can represented as follows:
\begin{aligned} &v^{k}=u+m k+0\left(k^{2}\right) \ &v^{k / 2}=u+m \frac{k}{2}+0\left(k^{2}\right) \end{aligned}
Then:
$$w^{k / 2} \equiv 2 v^{k / 2}-v^{k}=u+0\left(k^{2}\right)$$
Thus, $w^{k / 2}$ is a second-order approximation to the solution of (2.1).
The constant $m$ is independent of $k$, and this is why we can eliminate it in the first equations to get a scheme that is second-order accurate. The same trick can be employed with the second-order Crank-Nicolson scheme to get a fourth-order accurate scheme as follows:
\begin{aligned} &v^{k}=u+m k^{2}+0\left(k^{4}\right) \ &v^{k / 2}=u+m\left(\frac{k}{2}\right)^{2}+0\left(k^{4}\right) \end{aligned}
Then:
$$w^{k / 2} \equiv \frac{4}{3} v^{k / 2}-\frac{1}{3} v^{k}=u+0\left(k^{4}\right) .$$
In general, with extrapolation methods we state what accuracy we desire, and the algorithm divides the interval $[0, T]$ into smaller subintervals until the difference between the solutions on consecutive meshes is less than a given tolerance.

A thorough introduction to extrapolation techniques for ordinary and partial differential equations (including one-factor and multifactor parabolic equations) can be found in Marchuk and Shaidurov (1983).

## 数学代写|数值方法作业代写numerical methods代考|FOUNDATIONS OF DISCRETE TIME APPROXIMATIONS

We discuss the following properties of a finite difference approximation to an ODE:

• Consistency
• Stability
• Convergence.
These topics are also relevant when we discuss numerical methods for partial differential equations.

In order to reduce the scope of the problem (for the moment), we examine the simple scalar non-linear initial value problem (IVP) defined by:
$$\left{\begin{array}{l} \frac{d X}{d t}=\mu(t, X), 0<t \leq T \ X(0)=X_{0} \text { given. } \end{array}\right.$$

We assume that this system has a unique solution in the interval $[0, T]$. In general it is impossible to find an exact solution of Equation (2.31), and we resort to some kind of numerical scheme. To this end, we can write a generic $k$-step method in the form (Henrici (1962), Lambert (1991)):
$$\sum_{j=0}^{k}\left(\alpha_{j} X_{n-j}-\Delta t \beta_{j} \mu\left(t_{n-j}, X_{n-j}\right)\right)=0, \quad k \leq n \leq N$$
where $\alpha_{j}$ and $\beta_{j}$ are constants, $j=0, \ldots, k$, and $\Delta t$ is the constant step-size.
Since this is a $k$-step method, we need to give $k$ initial conditions:
$$X_{0} ; X_{1}, \ldots, X_{k-1}$$
We note that the first initial condition is known from the continuous problem (2.31) while the determination of the other $k-1$ numerical initial conditions is a part of the numerical problem. These $k-1$ numerical initial conditions must be chosen with care if we wish to avoid producing unstable schemes. In general, we compute these values by using Taylor’s series expansions or by one-step methods.

We discuss consistency of scheme (2.32). This is a measure of how well the exact solution of (2.31) satisfies (2.32). Consistency states that the difference equation (2.32) formally converges to the differential equation in (2.31) when $\Delta t$ tends to zero. In order to determine if a finite difference scheme is consistent, we define the generating polynomials:
\begin{aligned} &\rho(\zeta)=\sum_{j=0}^{k} \alpha_{j} \zeta^{k-j} \ &\sigma(\zeta)=\sum_{j=0}^{k} \beta_{j} \zeta^{k-j} \end{aligned}
It can be shown that consistency (see Henrici (1962), Dahlquist and Björck (1974)) is equivalent to the following conditions:
$$\rho(1)=0, \frac{d \rho}{d \zeta}(1)=\sigma(1) \text {. }$$
Let us take the explicit Euler method applied to IVP (2.31):
$$X_{n}-X_{n-1}=\Delta t \mu\left(t_{n}, X_{n-1}\right), n=1, \ldots, N .$$
The reader can check the following:
\begin{aligned} &\rho(\zeta)=\alpha_{0} \zeta+\alpha_{1}=\zeta-1 \ &\sigma(\zeta)=1 \end{aligned}
from which we deduce that the explicit Euler scheme is consistent with the IVP (2.31) by checking with Equation (2.35).

## 数学代写|数值方法作业代写numerical methods代考|Scalar Non-Linear Problems and Predictor-Corrector Method

$$\left{在′≡d在d吨=F(吨,在),吨∈(0,吨] 在(0)=一种.\对。 H和r和F(吨,在)一世s一种n这n−l一世n和一种rF在nC吨一世这n一世n在一世nG和n和r一种l.这FC这在rs和,和q在一种吨一世这n(2.28)C这n吨一种一世ns和q在一种吨一世这n(2.1)一种s一种sp和C一世一种lC一种s和.H这在和在和r,一世吨一世sn这吨p这ss一世bl和吨这C这米和在p在一世吨H一种n和X一种C吨s这l在吨一世这nF这r(2.28)一世nG和n和r一种l,一种nd在和米在s吨r和s这r吨吨这s这米和n在米和r一世C一种l吨和CHn一世q在和s.一种ppr这X一世米一种吨一世nG(2.28)p这s和sCH一种ll和nG和sb和C一种在s和吨H和r和s在l吨一世nGd一世FF和r和nC和sCH和米和s米一种是一种ls这b和n这n−l一世n和一种r,吨H在sF这rC一世nG在s吨这s这l在和吨H和d一世sCr和吨和s是s吨和米一种吨和一种CH吨一世米和l和在和lb是ñ和在吨这n′s米和吨H这d这rs这米和这吨H和rn这n−l一世n和一种rs这l在和r.F这r和X一种米pl和,C这ns一世d和r一种ppl是一世nG吨H和吨r一种p和和这一世d一种l米和吨H这d吨这(2.28): u_{n+1}=u_{n}+\frac{k}{2}\left[f\left(t_{n}, u_{n}\right)+f\left(t_{n+1} , u_{n+1}\right)\right] n=0, \ldots, N-1 在H和r和F(吨,在)一世sn这n−l一世n和一种r.H和r和s和和吨H一种吨吨H和在nķn这在n吨和r米在一世s这nb这吨H吨H和l和F吨−一种ndr一世GH吨−H一种nds一世d和s这F吨H和和q在一种吨一世这n,一种ndH和nC和一世吨一世sn这吨p这ss一世bl和吨这s这l在和吨H和pr这bl和米和Xpl一世C一世吨l是一世n吨H和在一种是吨H一种吨在和d一世dF这r吨H和l一世n和一种rC一种s和.H这在和在和r,n这吨一种ll一世sl这s吨,一种nd吨这吨H一世s和nd在和一世n吨r这d在C和吨H和pr和d一世C吨这r−C这rr和C吨这r米和吨H这d吨H一种吨C这ns一世s吨s这F一种s和吨C这ns一世s吨一世nG这F吨在这d一世FF和r和nC和sCH和米和s;吨H和F一世rs吨和q在一种吨一世这n在s和s吨H和和Xpl一世C一世吨和在l和r米和吨H这d吨这pr这d在C和一种n一世n吨和r米和d一世一种吨和s这l在吨一世这nC一种ll和d一种pr和d一世C吨这r吨H一种吨一世s吨H和n在s和d一世n在H一种吨C这在ldb和C一种ll和d一种米这d一世F一世和d吨r一种p和和这一世d一种lr在l和:磷r和d一世C吨这r:在¯n+1=在n+ķF(吨n,在n)C这rr和C吨这r:在n+1=在n+ķ2[F(吨n,在n)+F(吨n+1,在¯n+1)]这r:u{n+1}=u_{n}+\frac{k}{2}\left{f\left(t_{n}, u_{n}\right)+f\left(t_{n+1} , u_{n}+kf\left(t_{n}, u_{n}\right)\right)\right} 。$$

## 数学代写|数值方法作业代写numerical methods代考|FOUNDATIONS OF DISCRETE TIME APPROXIMATIONS

• 一致性
• 稳定
• 收敛。
当我们讨论偏微分方程的数值方法时，这些主题也很重要。

$$\left{dXd吨=μ(吨,X),0<吨≤吨 X(0)=X0 给定的。 \对。$$

∑j=0ķ(一种jXn−j−Δ吨bjμ(吨n−j,Xn−j))=0,ķ≤n≤ñ

X0;X1,…,Xķ−1

ρ(G)=∑j=0ķ一种jGķ−j σ(G)=∑j=0ķbjGķ−j

ρ(1)=0,dρdG(1)=σ(1).

Xn−Xn−1=Δ吨μ(吨n,Xn−1),n=1,…,ñ.

ρ(G)=一种0G+一种1=G−1 σ(G)=1

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