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Chapter 5: Problem 31

Show that Heun's method can be expressed in difference form, similar to that of the Runge-Kutta method of order four, as $$ \begin{aligned} w_{0} &=\alpha \\ k_{1} &=h f\left(t_{i}, w_{i}\right), \\ k_{2} &=h f\left(t_{i}+\frac{h}{3}, w_{i}+\frac{1}{3} k_{1}\right) \\ k_{3} &=h f\left(t_{i}+\frac{2 h}{3}, w_{i}+\frac{2}{3} k_{2}\right) \\ w_{i+1} &=w_{i}+\frac{1}{4}\left(k_{1}+3 k_{3}\right) \end{aligned} $$ for each \(i=0,1, \ldots, N-1\).

Short Answer

Expert verified
Yes, Heun鈥檚 method can be expressed in a difference form similar to that of the Runge-Kutta method of order four. The proposed difference form represents Heun's method for each \(i = 0,1, \ldots, N - 1\).

Step by step solution

01

Understand Heun鈥檚 method

Heun鈥檚 method is a simple numerical approach for solving ordinary differential equations. It is also known as an improved version of the Euler method. Given an initial value problem, Heun鈥檚 method provides an approximation of the solution at a sequence of time steps.
02

Understand the Runge-Kutta method of order four

The Runge-Kutta method of order four is a commonly used numerical method for solving ordinary differential equations. It balances between the complexity of computations and the accuracy of the results. It improves the approximation of the solution by taking into account information from intermediate steps between the current and the next time step.
03

Formulate Heun鈥檚 method in difference form

Based on the given equations, we can formulate Heun鈥檚 method in difference form as follows: \[\begin{aligned}w_0 &= \alpha, \k_1 &= h f(t_i, w_i), \k_2 &= h f\left(t_i + \frac{h}{3}, w_i + \frac{k_1}{3}\right), \k_3 &= h f\left(t_i + \frac{2h}{3}, w_i + \frac{2k_2}{3}\right), \w_{i+1} &= w_i + \frac{1}{4}(k_1 + 3k_3),\end{aligned}\]This is valid for each \(i = 0,1, \ldots, N - 1\). Each of these equations represents a step in the Heun鈥檚 method corresponding to the steps in the Runge-Kutta method of order four.
04

Verify the accordance of the proposed difference form with Heun鈥檚 method

To make sure the proposed form accurately mirrors Heun鈥檚 method, we would need to perform a substitution for \(k_2\) and \(k_3\) into the equation for \(w_{i+1}\). If the resulting equation matches the basic equation in Heun's method, that would suggest that the difference form indeed represents Heun's method.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Analysis
Numerical analysis is a branch of mathematics that deals with algorithms for solving problems that are continuous in nature, such as those involving differential equations or integral equations. The core of numerical analysis is the development and study of numerical methods to approximate solutions to these problems when they cannot be solved analytically. It includes the study of errors introduced by approximation and seeking ways to minimize or control these errors.

One of the most common problems tackled by numerical analysis is the solution of ordinary differential equations (ODEs), which often arise in science and engineering. For instance, the motion of planets, the dynamics of chemical reactions, and the behavior of electrical circuits can all be described using ODEs. In many cases, ODEs are too complex to solve exactly, and numerical methods provide a powerful tool to approximate the behavior of these systems over time.

When applying numerical methods, one crucial aspect is assessing their stability, accuracy, and efficiency. A stable numerical method produces results that are consistent and do not exhibit erratic behavior over time. Accuracy, on the other hand, indicates how close the numerical solution is to the exact solution. Efficiency pertains to the computational resources - time and memory - required by the method.
Ordinary Differential Equations
Ordinary differential equations (ODEs) are equations involving functions and their derivatives. The term 'ordinary' distinguishes them from partial differential equations, which involve partial derivatives of functions of several variables. ODEs are used to determine the evolution of physical, biological, or economic systems over time when the change in the system is dependent on its current state.

An initial value problem (IVP) for an ODE specifies the value of the unknown function at a particular point (the initial condition), and the goal is to find the function that satisfies both the differential equation and the initial condition. Solutions to ODEs can take the form of general solutions or particular solutions, where a general solution involves constants representing a family of curves, and a particular solution is derived by assigning specific values to those constants, typically based on initial conditions.

Due to their broad applicability, ODEs are a cornerstone of mathematical modeling. They enable researchers and engineers to predict how systems will change over time and to understand the dynamics of systems across a wide variety of fields.
Runge-Kutta method
The Runge-Kutta method is a family of iterative algorithms for approximating the solutions to ODEs, particularly when an exact analytical solution is difficult or impossible to obtain. The fourth-order Runge-Kutta method, which is the most commonly used, provides a balance between the computational complexity and the accuracy of the solution. It improves upon simpler methods like Euler's by utilizing multiple intermediate steps to estimate the slope of the tangent line to the solution curve.

In general, a Runge-Kutta method computes several 'stages' for each step forward in time, each involving the evaluation of the function representing the ODE. These stages are then combined to produce an approximation of the solution at the next time step. For example, the fourth-order method uses four stages, with each stage calculation depending on the results of the previous stages.

Application of the Runge-Kutta method to Heun's method

Heun's method can be viewed as a simplified, two-stage Runge-Kutta method. Heun's method takes an initial predictor step using the slope at the beginning of the interval (similar to Euler's method) and then corrects this prediction by considering the slope at the end of the interval. As the exercise demonstrates, Heun's method can also be formulated to resemble the multistage nature of a higher-order Runge-Kutta method by incorporating additional evaluation steps in the procedure. This reveals the intrinsic relationship between these methods and displays the versatility of Runge-Kutta techniques in numerical approximation for solving ODEs.

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Most popular questions from this chapter

The study of mathematical models for predicting the population dynamics of competing species has its origin in independent works published in the early part of the 20th century by A. J. Lotka and V. Volterra (see, for example, [Lo1], [Lo2], and [Vo]). Consider the problem of predicting the population of two species, one of which is a predator, whose population at time \(t\) is \(x_{2}(t)\), feeding on the other, which is the prey, whose population is \(x_{1}(t)\). We will assume that the prey always has an adequate food supply and that its birth rate at any time is proportional to the number of prey alive at that time; that is, birth rate (prey) is \(k_{1} x_{1}(t)\). The death rate of the prey depends on both the number of prey and predators alive at that time. For simplicity, we assume death rate (prey) \(=k_{2} x_{1}(t) x_{2}(t)\). The birth rate of the predator, on the other hand, depends on its food supply, \(x_{1}(t)\), as well as on the number of predators available for reproduction purposes. For this reason, we assume that the birth rate (predator) is \(k_{3} x_{1}(t) x_{2}(t)\). The death rate of the predator will be taken as simply proportional to the number of predators alive at the time; that is, death rate (predator) \(=k_{4} x_{2}(t)\). Since \(x_{1}^{\prime}(t)\) and \(x_{2}^{\prime}(t)\) represent the change in the prey and predator populations, respectively, with respect to time, the problem is expressed by the system of nonlinear differential equations $$ x_{1}^{\prime}(t)=k_{1} x_{1}(t)-k_{2} x_{1}(t) x_{2}(t) \quad \text { and } x_{2}^{\prime}(t)=k_{3} x_{1}(t) x_{2}(t)-k_{4} x_{2}(t) $$ Solve this system for \(0 \leq t \leq 4\), assuming that the initial population of the prey is 1000 and of the predators is 500 and that the constants are \(k_{1}=3, k_{2}=0.002, k_{3}=0.0006\), and \(k_{4}=0.5\). Sketch a graph of the solutions to this problem, plotting both populations with time, and describe the physical phenomena represented. Is there a stable solution to this population model? If so, for what values \(x_{1}\) and \(x_{2}\) is the solution stable?

For the Adams-Bashforth and Adams-Moulton methods of order four, a. Show that if \(f=0\), then $$ F\left(t_{i}, h, w_{i+1}, \ldots, w_{i+1-m}\right)=0 $$ b. Show that if \(f\) satisfies a Lipschitz condition with constant \(L\), then a constant \(C\) exists with $$ \left|F\left(t_{i}, h, w_{i+1}, \ldots, w_{i+1-m}\right)-F\left(t_{i}, h, v_{i+1}, \ldots, v_{i+1-m}\right)\right| \leq C \sum_{j=0}^{m}\left|w_{i+1-j}-v_{i+1-j}\right|. $$

Investigate stability for the difference method $$ w_{i+1}=-4 w_{i}+5 w_{i-1}+2 h\left[f\left(t_{i}, w_{i}\right)+2 h f\left(t_{i-1}, w_{i-1}\right)\right] $$ for \(i=1,2, \ldots, N-1\), with starting values \(w_{0}, w_{1}\).

For each choice of \(f(t, y)\) given in parts (a)-(d): i. Does \(f\) satisfy a Lipschitz condition on \(D=\\{(t, y) \mid 0 \leq t \leq 1,-\infty

In the previous exercise, all infected individuals remained in the population to spread the disease. A more realistic proposal is to introduce a third variable \(z(t)\) to represent the number of individuals who are removed from the affected population at a given time \(t\) by isolation, recovery and consequent immunity, or death. This quite naturally complicates the problem, but it can be shown (see [Ba2]) that an approximate solution can be given in the form $$ x(t)=x(0) e^{-\left(k_{1} / k_{2}\right) z(t)} \quad \text { and } \quad y(t)=m-x(t)-z(t) $$ where \(k_{1}\) is the infective rate, \(k_{2}\) is the removal rate, and \(z(t)\) is determined from the differential equation $$ z^{\prime}(t)=k_{2}\left(m-z(t)-x(0) e^{-\left(k_{1} / k_{2}\right) z(t)}\right) $$ The authors are not aware of any technique for solving this problem directly, so a numerical procedure must be applied. Find an approximation to \(z(30), y(30)\), and \(x(30)\), assuming that \(m=100,000\), \(x(0)=99,000, k_{1}=2 \times 10^{-6}\), and \(k_{2}=10^{-4}\).
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