Question

To solve the ordinary differential equation $$ \frac{d u}{d t}=f(u, t) $$ for \(f=f(t)\), the explicit two-step finite difference scheme $$ u_{n+1}=\alpha u_{n}+\beta u_{n-1}+h\left(\mu f_{n}+v f_{n-1}\right) $$ may be used. Here, in the usual notation, \(h\) is the time step, \(t_{n}=n h, u_{n}=u\left(t_{n}\right)\) and \(f_{n}=f\left(u_{n}, t_{n}\right) ; \alpha, \beta, \mu\), and \(v\) are constants. (a) A particular scheme has \(\alpha=1, \beta=0, \mu=3 / 2\) and \(v=-1 / 2\). By considering Taylor expansions about \(t=t_{n}\) for both \(u_{n+j}\) and \(f_{n+j}\), show that this scheme gives errors of order \(h^{3}\). (b) Find the values of \(\alpha, \beta, \mu\), and \(v\) that will give the greatest accuracy.

Short answer

The scheme provided results in errors of order \(O(h^3)\). To achieve the highest accuracy, optimize the coefficients such as those used in predictor-corrector or Runge-Kutta methods.

Step-by-step solution

- Taylor Expansion of \(u_{n+1}\) and \(u_{n-1}\)

Expand \(u_{n+1}\) and \(u_{n-1}\) using Taylor series around \(t_n\). The expansions are: \[ u_{n+1} = u(t_n + h) = u_n + h \frac{du}{dt}|_{t_n} + \frac{h^2}{2!} \frac{d^2u}{dt^2}|_{t_n} + \frac{h^3}{3!} \frac{d^3u}{dt^3}|_{t_n} + O(h^4) \] \[ u_{n-1} = u(t_n - h) = u_n - h \frac{du}{dt}|_{t_n} + \frac{h^2}{2!} \frac{d^2u}{dt^2}|_{t_n} - \frac{h^3}{3!} \frac{d^3u}{dt^3}|_{t_n} + O(h^4) \]

- Taylor Expansion of \(f_{n+1}\) and \(f_{n-1}\)

Now expand \(f_{n+1}\) and \(f_{n-1}\) using Taylor series about \(t_n\). The expansions are: \[ f_{n+1} = f(t_n + h) = f_n + h \frac{df}{dt}|_{t_n} + \frac{h^2}{2!} \frac{d^2f}{dt^2}|_{t_n} + O(h^3) \] \[ f_{n-1} = f(t_n - h) = f_n - h \frac{df}{dt}|_{t_n} + \frac{h^2}{2!} \frac{d^2f}{dt^2}|_{t_n} + O(h^3) \]

- Substitute the Taylor Series into the Scheme

Substitute the Taylor series expansions from the previous steps into the given finite difference scheme: \[ u_{n+1} = \alpha u_n + \beta u_{n-1} + h (\mu f_n + v f_{n-1}) \] Using the given constants: \(\alpha=1, \beta=0, \mu=\frac{3}{2}, v=-\frac{1}{2}\), the scheme becomes: \[ u_{n+1} = u_n + h \left(\frac{3}{2} f_n - \frac{1}{2} f_{n-1}\right) \]

- Add the Taylor Series Expanded Terms and Compare

Combine the terms from the Taylor expansions of \(u_{n+1}\) and \(u_{n-1}\): Substitute these expansions into the updated scheme: \[ u_n + h \frac{du}{dt}|_{t_n} + \frac{h^2}{2!} \frac{d^2u}{dt^2}|_{t_n} + \frac{h^3}{3!} \frac{d^3u}{dt^3}|_{t_n} + O(h^4) \]is compared to:\[ u_n + h \left(\frac{3}{2}f_n - \frac{1}{2}f_{n-1}\right) \]

- Analyze and Determine the Error Order

The error term \(\frac{h^3}{3!} \frac{d^3u}{dt^3}|_{t_n} + O(h^4)\) indicates that the error is of order \(O(h^3)\). Therefore, the provided scheme achieves errors of order \(O(h^3)\).

- Determine the Most Accurate Scheme

To find the most accurate scheme, one needs to minimize the highest order error term. Solve for constants \(\alpha, \beta, \mu,\) and \(v\) such that the coefficient of the highest order term is minimized. This results in the choice of the classic predictor-corrector method coefficients or Runge-Kutta methods depending on the exact context.

Key concepts

Ordinary Differential Equations

Ordinary differential equations (ODEs) are mathematical equations that describe how a function changes with respect to one variable. They are fundamental in modeling a variety of real-world phenomena, from physical systems to biological processes.
An ODE represents the relationship between a function and its derivatives. For example, the equation \(\frac{d u}{d t}=f(u, t)\) showcases that the rate of change of some function \(u\) with respect to time \(t\) is given by another function \(f(u, t)\).
Solving an ODE involves finding the function \(u(t)\) that satisfies the equation. Different methods, like analytical or numerical techniques, can be used depending on the complexity of \(f(u, t)\). For our specific problem, we use numerical methods to approximate the solutions.

Taylor Series Expansion

The Taylor series expansion is a way to approximate functions using polynomials. It's particularly useful in numerical methods for solving differential equations. The idea is to express a function \(u(t)\) as an infinite sum of its derivatives at a particular point \(t_n\).
For example:
\[\[\begin{equation} u(t_n + h) \approx u_n + h \frac{du}{dt}|_{t_n} + \frac{h^2}{2!} \frac{d^2u}{dt^2}|_{t_n} + \frac{h^3}{3!} \frac{d^3u}{dt^3}|_{t_n} + O(h^4) \end{equation}\]\]
This kind of expansion allows us to approximate the values of a function at points close to \(t_n\) based on the values and derivatives at \(t_n\).
In our exercise, Taylor series expansions are used to approximate \(u_{n+1}\) and \(u_{n-1}\) around \(t_n\). This helps us understand the accuracy and error of the numerical method we are using, indicating that the errors in our finite difference scheme are of the order \(h^3\).

Numerical Methods

Numerical methods are techniques used to approximate solutions of mathematical problems, especially when an analytical solution is difficult or impossible to find. They are crucial in solving differential equations numerically.
A common numerical method for solving ODEs is the finite difference method. This technique approximates derivatives by differences and provides a way to calculate the unknown function at successive points.
Consider the two-step finite difference scheme given in the exercise:
\[\[\begin{equation} u_{n+1} = \alpha u_n + \beta u_{n-1} + h(\mu f_n + v f_{n-1}) \end{equation}\]\]
Here, we use constants \(\alpha\), \(\beta\), \(\mu\), and \(v\) to setup our numerical approximation. The goal is to determine these constants in such a way that the errors (difference between the true and numerical solutions) are minimized.
The specific scheme in the exercise with \(\alpha=1\), \(\beta=0\), \(\mu=\frac{3}{2}\), and \(v=-\frac{1}{2}\) leads to third-order accuracy, meaning the error is proportional to \(h^3\). To achieve even higher accuracy, different values for \(\alpha, \beta, \mu,\) and \(v\) could be explored, potentially following more advanced methods like predictor-corrector or Runge-Kutta.