Question

The leapfrog method for the ODE \(y^{\prime}=f(t, y)\) is derived like the forward and backward Euler methods, except that centered differencing is used. This yields the formula $$ \frac{y_{i+1}-y_{i-1}}{2 h}=f\left(t_{i}, y_{i}\right) $$ Show that this is an explicit linear two-step method that is second order accurate and does not belong to the Adams or BDF families.

Short answer

Question: Demonstrate that the leapfrog method is an explicit linear two-step method and prove its second-order accuracy. Show that it does not belong to the Adams or BDF families. Answer: The leapfrog method is an explicit linear two-step method because its update equation is defined by known values and is linear in terms of the next step. It requires two previous points to calculate the next point, making it a two-step method. The local truncation error for the leapfrog method is O(h^2), indicating that it is second-order accurate. The leapfrog method does not belong to the Adams or BDF families because it involves both previous and next time steps, unlike these families' methods, which are based on polynomial interpolations of f(t, y) or approximating the derivative using only previous time steps.

Step-by-step solution

Identify the method as explicit#linear#two-step

The given formula for the leapfrog method is: $$ \frac{y_{i+1}-y_{i-1}}{2h} = f\left(t_i, y_i\right) $$ We can rewrite this formula by solving for \(y_{i+1}\): $$ y_{i+1} = y_{i-1} + 2hf\left(t_i, y_i\right) $$ As we see, the method is explicit since \(y_{i+1}\) is defined by the known values \(y_{i-1}\), \(t_i\), and \(y_i\). The method is linear because the update equation is linear in terms of \(y_{i+1}\). It is also a two-step method because we require two previous points, \(y_{i-1}\) and \(y_i\), to calculate the next point \(y_{i+1}\).

Prove second-order accuracy of the method

The local truncation error for the leapfrog method can be found by substituting the Taylor series expansion of \(y(t)\) about \(t_i\) into the leapfrog formula. The Taylor series expansion of \(y(t)\) is given as: $$ y(t_{i+1}) = y(t_i) + hy'(t_i) + \frac{h^2}{2}y''(t_i) + O(h^3) $$ $$ y(t_{i-1}) = y(t_i) - hy'(t_i) + \frac{h^2}{2}y''(t_i) - O(h^3) $$ Substitute these expansions into the leapfrog formula: $$ \frac{y(t_{i+1}) - y(t_{i-1})}{2h} = f(t_i, y(t_i)) $$ $$ \frac{2hy'(t_i) + 2O(h^3)}{2h} = f(t_i, y(t_i)) $$ Comparing both sides of the equation, we can see that: $$ y'(t_i) = f(t_i, y(t_i)) + O(h^2) $$ This demonstrates that the leapfrog method is second-order accurate, as the local truncation error is \(O(h^2)\).

Show that leapfrog method does not belong to the Adams or BDF families

The Adams family methods are based on the polynomial interpolations of \(f(t, y)\) to approximate the integral of \(y'(t)\). The backward differentiation formula (BDF) methods are a set of linear multistep methods that approximate the derivative using only previous time steps. Neither of these methods involve both previous and next time steps, as the leapfrog method does. In the leapfrog method, we have the following update equation: $$ y_{i+1} = y_{i-1} + 2hf(t_i, y_i) $$ This equation uses both the previous step, \(y_{i-1}\), and next step, \(y_{i+1}\), which is a notable difference from both the Adams or BDF families. Therefore, the leapfrog method does not belong to the Adams or BDF families.

Key concepts

Second-order accuracy

Second-order accuracy in numerical methods relates to how closely the numerical solution approximates the true solution of a differential equation. This is measured by the local truncation error, which is the error made in one step of the numerical method. In general, the smaller the error, the more accurate the method is considered.

For the leapfrog method, we determine accuracy using Taylor series expansions. By comparing terms in the Taylor series for solutions at times \(t_{i+1}\) and \(t_{i-1}\), we see that the expression \[\frac{y(t_{i+1}) - y(t_{i-1})}{2h} = f(t_i, y(t_i))\] authorized by the method contains the second-order terms \(O(h^2)\).

This means the leading error term in the local truncation error is a multiple of \(h^2\), demonstrating second-order accuracy. In essence, the smaller the step size \(h\), the more the method captures the curve's behavior accurately over that small interval.

• Second-order accuracy benefits: More accurate approximations over finer intervals.
• Applicability: Helpful in simulations requiring precise solutions.

Understanding this concept is crucial because it allows us to appreciate how changing step sizes can influence the overall solution's accuracy. Additionally, being second-order accurate ensures that the leapfrog method strikes a balance between computational efficiency and exactness, making it a valuable method for solving differential equations.

Explicit linear two-step method

The leapfrog method is an example of an explicit linear two-step method, a classification under numerical solutions for ordinary differential equations (ODEs). This means the method involves explicit computation of the next step without solving any algebraic equations that might depend on future states.

In the leapfrog approach, the formula \[y_{i+1} = y_{i-1} + 2hf(t_i, y_i)\]is used to compute the next value \( y_{i+1} \) based solely on previously computed values \( y_{i-1} \) and \( y_i \). The linearity arises because the right-hand side is a linear expression in terms of \( f(t_i, y_i) \) and does not involve any additional nonlinear transformations of \( y_{i+1} \).

The method is considered a two-step method as it relies on two previous time points, \( y_{i-1} \) and \( y_i \), to determine \( y_{i+1} \). This differs from one-step methods like the Runge-Kutta, which only require the last evaluated point to compute the next one.

Key points about explicit linear two-step methods involve:

• Simplicity: They often have straightforward implementations.
• Efficiency: No solutions for additional equations are required, increasing speed.
• Memory requirement: Two previous states need to be stored, which is manageable in many applications.

Understanding these properties helps evaluate when and where to apply such methods effectively, especially in scenarios where quick calculations are essential and memory constraints are minimal.

Numerical methods for ODEs

Numerical methods for ordinary differential equations (ODEs) are a suite of tools and techniques used to find solutions for differential equations that cannot be solved analytically. These methods allow us to approximate solutions with a certain degree of accuracy, typically determined by the method's order and specific implementation details.

One such method is the leapfrog technique, which belongs to a class of linear multistep methods. Whereas some methods like Euler's or Runge-Kutta are one-step, relying on a single previous point, the leapfrog method calculates the next state using two preceding points.

Here are some important points about numerical methods for ODEs:

• Safety: Some methods provide better stability over others, important for long-time integration.
• Performance: Methods vary in computational speed and load, influencing performance in large simulations.
• Flexibility: Different problems require unique methods; flexibility helps achieve more tailored solutions.

These methods are indispensable in fields like physics, engineering, and finance, where real-world phenomena are modeled by differential equations. Understanding not just the available methods but their strengths and ideal use cases enable scientists and engineers to perform simulations accurately and efficiently, adapting their technique of choice to the right scenario including the use of second-order accurate, two-step processes like leapfrog for specific problem types.