2024_06_22_deafddc186e6d5a1e984g

Chapter 5 Initial-Value Problems for Ordinary Differential Equations

The Elementary Theory

Euler's Method

Higher-Order Taylor Methods

Runge-Kutta Methods

*Error Control and the RKF Method

Multistep Methods
*Variable Step-Size Multistep Methods

*Extrapolation Methods

*Higher-Order Equations

Stability

Stiff Differential Equations

Initial-Value Problems for ODE

Stability

One-Step Methods

Definition 5.18

A one-step difference-equation method with local truncation error

at the

th step is said to be consistent with the differential equation it approximates if
一步差分方程方法在第

步的局部截断误差

被认为与其近似的微分方程一致

Initial-Value Problems for ODE Stability

One-Step Methods

This method is much practical than previous one.

Definition 5.19

A one-step difference-equation method is said to be convergent with respect to the differential equation it approximates if

where

denotes the exact value of the solution of the differential equation and

is the approximation obtained from the difference method at the

th step.
其中

表示微分方程解的精确值，

是在第

步通过差分方法得到的近似值。

Initial-Value Problems for ODE

Example 1 Show that Euler's method is convergent.

Solution Examining Inequality (5.10) on page 271, in the error-bound formula for Euler's method, we see that under the hypotheses of Theorem 5.9,
在第 271 页上检查不等式解（5.10），在欧拉方法的误差界公式中，我们看到在定理 5.9 的假设下，

However,

, and

are all constants and

So Euler's method is convergent with respect to a differential equation satisfying the conditions of this definition. The rate of convergence is

.
因此，欧拉方法在满足此定义条件的微分方程上是收敛的。收敛速度为

。

Initial-Value Problems for ODE Stability

Multistep Methods

For multistep methods, the problems involved with consistency, convergence, and stability are compounded because of the number of approximations involved at each step. In the onestep methods, the approximation

depends directly only on the previous approximation

, whereas the multistep methods use at least two of the previous approximations, and the usual methods that are employed involve more.
对于多步方法，由于每一步涉及的近似次数增加，一致性、收敛性和稳定性问题变得更加复杂。在单步方法中，近似

仅直接依赖于上一次的近似

，而多步方法至少使用两个先前的近似值，通常采用的方法涉及更多的近似值。

Initial-Value Problems for ODE

Stability

The general multistep method for approximating the solution to the initial-value problem

has the form

for each

, where

are constants and, as usual,

and

Initial-Value Problems for ODE Stability

The local truncation error for a multistep method expressed in this form is

for each

. As in the one-step methods, the local truncation error measures how the solution

to the differential equation fails to satisfy the difference equation.
对于每个

。与一步方法一样，局部截断误差衡量了解微分方程的解

未能满足差分方程。

The convergence and stability can be obtained.

Chapter 5 Initial-Value Problems for Ordinary Differential Equations

The Elementary Theory

Euler's Method

Higher-Order Taylor Methods

Runge-Kutta Methods

*Error Control and the RKF Method

Multistep Methods
*Variable Step-Size Multistep Methods

*Extrapolation Methods

*Higher-Order Equations

Stability

Stiff Differential Equations

Initial-Value Problems for ODE Stiff Differential Equations

All the methods for approximating the solution to initial-value problems have error terms that involve a higher derivative of the solution of the equation. If the derivative can be reasonably bounded, then the method will have a predictable error bound that can be used to estimate the accuracy of the approximation. Even if the derivative grows as the steps increase, the error can be kept in relative control, provided that the solution also grows in magnitude. Problems frequently arise, however, when the magnitude of the derivative increases but the solution does not. In this situation, the error can grow so large that it dominates the calculations. Initial-value problems for which this is likely to occur are called stiff equations and are quite common, particularly in the study of vibrations, chemical reactions, and electrical circuits.
所有逼近初值问题解的方法都包含涉及方程解的高阶导数的误差项。如果导数可以合理地被限制，那么该方法将具有可预测的误差界限，可用于估计逼近的准确性。即使导数随着步数增加而增长，只要解的幅度也增长，误差仍然可以相对控制。然而，当导数的幅度增加而解却没有增长时，问题经常出现。在这种情况下，误差可能会增长到足以主导计算。可能发生这种情况的初值问题被称为刚性方程，特别常见，尤其在振动、化学反应和电路研究中。

Initial-Value Problems for ODE

Stiff Differential Equations

The system of initial-value problems

has the unique solution

Initial-Value Problems for ODE Stiff Differential Equations

The transient term

in the solution causes this system to be stiff. Applying Algorithm 5.7, the Runge-Kutta Fourth-Order Method for Systems, gives results listed in Table 5.22. When

, stability results and the approximations are accurate. Increasing the step size to

, however, leads to the disastrous results shown in the table.
解决方案中的瞬态项

导致系统变得僵硬。应用算法 5.7，系统的龙格-库塔四阶方法，得到表 5.22 中列出的结果。当

时，稳定性结果和近似值准确。然而，将步长增加到

会导致表中显示的灾难性结果。

Initial-Value Problems for ODE

Stiff Differential Equations

Table 5.22

0.1

1.793061

1.712219

-2.645169

-1.032001

-0.8703152

7.844527

0.2

1.423901

1.414070

-18.45158

-0.8746809

-0.8550148

38.87631

0.3

1.131575

1.130523

-87.47221

-0.7249984

-0.7228910

176.4828

0.4

0.9094086

0.9092763

-934.0722

-0.6082141

-0.6079475

789.3540

0.5

0.7387877

9.7387506

-1760.016

-0.5156575

-0.5155810

3520.00

0.6

0.6057094

0.6056833

-7848.550

-0.4404108

-0.4403558

15697.84

0.7

0.4998603

0.4998361

-34989.63

-0.3774038

-0.3773540

69979.87

0.8

0.4136714

0.4136490

- 155979.4

-0.3229535

-0.3229078

311959.5

0.9

0.3416143

0.3415939

-695332.0

-0.2744088

-0.2743673

1390664.

1.0

0.2796748

0.2796568

-3099671

-0.2298877

-0.2298511

6199352

Initial-Value Problems for ODE Stiff Differential Equations

Although stiffness is usually associated with systems of differential equations, the approximation characteristics of a particular numerical method applied to a stiff system can be predicted by examining the error produced when the method is applied to a simple test equation,
尽管刚度通常与微分方程系统相关联，但应用于刚性系统的特定数值方法的近似特性可以通过检查该方法应用于简单测试方程时产生的误差来预测

The solution to this equation is

, which contains the transient solution

. The steady-state solution is zero, so the approximation characteristics of a method are easy to determine.
该方程的解为

，其中包含瞬态解

。稳态解为零，因此方法的近似特性易于确定。

Initial-Value Problems for ODE

Stiff Differential Equations

First consider Euler's method applied to the test equation. Letting

and

, for

, Eq. (5.8) on page 266 implies that

Initial-Value Problems for ODE

Since the exact solution is

, the absolute error is

and the accuracy is determined by how well the term

approximates

. When

, the exact solution

decays to zero as

increases, but by Eq.(5.65), the approximation will have this property only if

, which implies that

. This effectively restricts the step size

for Euler's method to satisfy

.
准确性取决于术语

近似

的程度。当

时，精确解

随着

的增加而衰减为零，但根据方程（5.65），近似解只有在

时才具有这种特性，这意味着

。这有效地限制了欧拉方法的步长

以满足

。

Initial-Value Problems for ODE Stiff Differential Equations

Suppose now that a round-off error

is introduced in the initial condition for Euler's method,

At the

th step the round-off error is

Initial-Value Problems for ODE Stiff Differential Equations

Since

, the condition for the control of the growth of round-off error is the same as the condition for controlling the absolute error,

, which implies that

. So
自

以来，控制舍入误差增长的条件与控制绝对误差的条件相同，

，这意味着

。因此

Euler's method is expected| to be stable for

only if the step size

is less than

Initial-Value Problems for ODE Stiff Differential Equations

The situation is similar for other one-step methods. In general, a function

exists with the property that the difference method, when applied to the test equation, gives
其他一步法的情况类似。一般来说，存在一个函数

，具有这样的性质，即当应用于测试方程时，差分法会产生

The accuracy of the method depends upon how well

approximates

, and the error will grow without bound if

. An

th-order Taylor method, for example, will have stability with regard to both the growth of round-off error and absolute error, provided

is chosen to satisfy
该方法的准确性取决于

近似

的程度，如果

，误差将无限增长。例如，

阶泰勒方法在处理舍入误差和绝对误差增长方面具有稳定性，只要选择

以满足。

Chapter 5

Initial-Value Problems for

Ordinary Differential Equations

END