Question

The first order ODE system introduced in the previous exercise for \(\mathbf{q}\) and \(\mathbf{v}\) is in partitioned form. It is also a Hamiltonian system with a separable Hamiltonian; i.e., the ODE for \(\mathbf{q}\) depends only on \(\mathbf{v}\) and the \(\mathrm{ODE}\) for \(\mathbf{v}\) depends only on \(\mathbf{q}\). This can be used to design special discretizations. Consider a constant step size \(h\). (a) The symplectic Euler method applies backward Euler to the \(\mathrm{ODE} \mathbf{q}^{\prime}=\mathbf{v}\) and forward Euler to the other ODE. Show that the resulting method is explicit and first order accurate. (b) The leapfrog, or Verlet, method can be viewed as a staggered midpoint discretization and is given by $$ \begin{aligned} \mathbf{q}_{i+1 / 2}-\mathbf{q}_{i-1 / 2} &=h \mathbf{v}_{i} \\ M\left(\mathbf{q}_{i+1 / 2}\right)\left(\mathbf{v}_{i+1}-\mathbf{v}_{i}\right) &=h \mathbf{f}\left(\mathbf{q}_{i+1 / 2}\right) \end{aligned} $$ Thus, the mesh on which the q-approximations "live" is staggered by half a step compared to the \(\mathbf{v}\) -mesh. The method can be kick-started by $$ \mathbf{q}_{1 / 2}=\mathbf{q}_{0}+h / 2 \mathbf{v}_{0} $$ To evaluate \(\mathbf{q}_{i}\) at any mesh point, the expression $$ \mathbf{q}_{i}=\frac{1}{2}\left(\mathbf{q}_{i-1 / 2}+\mathbf{q}_{i+1 / 2}\right) $$ can be used. Show that this method is explicit and second order accurate.

Short answer

Question: Show that the symplectic Euler method is explicit and first order accurate, and the leapfrog method is explicit and second order accurate for solving a first order ODE system. Answer: The symplectic Euler method is explicit and first order accurate, as it employs the backward Euler for one ODE and forward Euler for the other ODE, both of which are first-order schemes. The leapfrog method, on the other hand, is explicit and second order accurate, as it is based on the second-order central difference scheme, using known values at the previous time step to compute the solution at the next time step.

Step-by-step solution

Backward Euler for \(\mathbf{q}' = \mathbf{v}\)

As we are applying backward Euler for the ODE \(\mathbf{q}' = \mathbf{v}\), we have: $$ \frac{\mathbf{q}_{i+1} - \mathbf{q}_i}{h} = \mathbf{v}_{i+1} $$ Rearrange to find \(\mathbf{q}_{i+1}\): $$ \mathbf{q}_{i+1} = \mathbf{q}_i + h\mathbf{v}_{i+1} $$

Forward Euler for the other ODE

We apply the forward Euler to the other ODE: $$ M(\mathbf{q}_{i+1})\frac{\mathbf{v}_{i+1} - \mathbf{v}_i}{h} = - \mathbf{f}(\mathbf{q}_i) $$ Rearrange to find \(\mathbf{v}_{i+1}\): $$ \mathbf{v}_{i+1} = \mathbf{v}_i - \frac{h}{M(\mathbf{q}_{i+1})} \mathbf{f}(\mathbf{q}_i) $$

Show the method is explicit and first order accurate

We have found that: $$ \mathbf{q}_{i+1} = \mathbf{q}_i + h\mathbf{v}_{i+1} $$ and $$ \mathbf{v}_{i+1} = \mathbf{v}_i - \frac{h}{M(\mathbf{q}_{i+1})} \mathbf{f}(\mathbf{q}_i) $$ These equations are explicit as they only use known values at the previous time step to compute the solution at the next time step. Since we used the backward Euler for \(\mathbf{q}'\) and forward Euler for the other ODE, both schemes are first-order accurate. So, the symplectic Euler method is explicit and first order accurate. (b) Leapfrog Method:

Calculate \(\mathbf{q}_{i + 1/2}\) and \(\mathbf{q}_i\)

Using the given formulae, $$ \mathbf{q}_{i + 1/2} = \mathbf{q}_{i - 1/2} + h\mathbf{v}_i $$ and $$ \mathbf{q}_i = \frac{1}{2}(\mathbf{q}_{i - 1/2} + \mathbf{q}_{i + 1/2}) $$

Calculate \(\mathbf{v}_{i + 1}\)

By using the given formula, $$ M(\mathbf{q}_{i + 1/2})(\mathbf{v}_{i + 1} - \mathbf{v}_i) = h \mathbf{f}(\mathbf{q}_{i + 1/2}) $$ Rearrange to find \(\mathbf{v}_{i + 1}\): $$ \mathbf{v}_{i + 1} = \mathbf{v}_i + \frac{h}{M(\mathbf{q}_{i + 1/2})} \mathbf{f}(\mathbf{q}_{i + 1/2}) $$

Show the method is explicit and second order accurate

We have found that: $$ \mathbf{q}_{i + 1/2} = \mathbf{q}_{i - 1/2} + h\mathbf{v}_i $$ and $$ \mathbf{v}_{i + 1} = \mathbf{v}_i + \frac{h}{M(\mathbf{q}_{i + 1/2})} \mathbf{f}(\mathbf{q}_{i + 1/2}) $$ These equations are explicit as they use known values at the previous time step to compute the solution at the next time step. As the leapfrog method is a second-order central difference scheme, it is second order accurate. So, the leapfrog method is explicit and second order accurate.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are mathematical equations that involve functions and their derivatives. They express the relationship between a variable and its rate of change. Specifically, an ODE involves a single independent variable alongside one or more dependent variables.

Understanding ODEs is important because they are widely used across different scientific fields such as physics, engineering, and biology, allowing us to model real-world phenomena:

• Population growth, which can be modeled with simple first-order ODEs.
• Motion, which can be described using Newton's laws, results in systems of second-order ODEs.
• Chemical reactions that change over time.

In mathematical terms, an ODE for a function \(y(t)\) can be written as \(F(t, y, y') = 0\), where \(y'\) denotes the derivative of \(y\) with respect to \(t\). Many numerical methods exist to solve ODEs, providing approximations to complex problems that do not have simple analytical solutions.

Symplectic Euler Method

The Symplectic Euler Method is a numerical technique used to solve differential equations with a special structure known as Hamiltonian systems. These systems have a particular form that allows for the conservation of energy, which is crucial in simulating physical systems accurately over time.
There are two variations of Euler's method used in the Symplectic Euler Method: backward and forward.

• **Backward Euler**: In this variation, one calculates the \(\mathbf{q}' = \mathbf{v}\) part of the system using a backward step. The formula is \(\mathbf{q}_{i+1} = \mathbf{q}_{i} + h \mathbf{v}_{i+1}\), where \(h\) is the time step size.
• **Forward Euler**: This is used for the remaining part of the system \(\mathbf{v}\), using a forward step: \(\mathbf{v}_{i+1} = \mathbf{v}_i - \frac{h}{M(\mathbf{q}_{i+1})} \mathbf{f}(\mathbf{q}_i)\).

Both of these approaches make the Symplectic Euler Method a first-order accurate method, meaning it is most accurate for problems requiring simple solutions to evolve over large time steps. The method is also explicit, meaning that each step can be calculated directly from the previous one without iteration, making it computationally straightforward.

Leapfrog Method

The Leapfrog Method, sometimes called the Verlet method, is a second-order accurate numerical technique. This means it provides more precise results with each step compared to first-order methods like Symplectic Euler.

In the Leapfrog Method, the calculations for position (\(\mathbf{q}\)) and velocity (\(\mathbf{v}\)) are staggered. You calculate halfway between time steps, hence the name "leapfrog."

• **Position Update**: Calculate the intermediate position \(\mathbf{q}_{i+1/2} = \mathbf{q}_{i-1/2} + h \mathbf{v}_i\).
• **Velocity Update**: Once you have the intermediate position, you calculate the velocity for the next time step: \(\mathbf{v}_{i+1} = \mathbf{v}_i + \frac{h}{M(\mathbf{q}_{i+1/2})} \mathbf{f}(\mathbf{q}_{i+1/2})\).
• **Mesh Offset**: This method requires an offset beginning with \(\mathbf{q}_{1/2} = \mathbf{q}_0 + \frac{h}{2} \mathbf{v}_0\).

This staggered approach avoids calculating everything at once and improves accuracy without increasing computational burden significantly. The Leapfrog Method is widely used in computational physics and dynamics because it maintains the geometric properties of Hamiltonian systems, ensuring stability over long simulations.