Problem 11 - A First Course in the Numerical Analysis of Differential Equations

Question

We say that the autonomous ODE system \(y^{\prime}=\boldsymbol{f}(\boldsymbol{y})\) obeys a quadratic conservation law if there exists a symmetric, positive-definite matrix \(S\) such that for every initial condition \(\boldsymbol{y}\left(t_{0}\right)=\boldsymbol{y}_{0}\) it is true that \([\boldsymbol{y}(t)]^{T} S \boldsymbol{y}(t) \equiv\) \(\boldsymbol{y}_{0}^{T} S y_{0}, t \geq t_{0}\) a Show that the \(d\)-dimensional linear system \(y^{\prime}=\Lambda y\) obeys a quadratic conservation law if \(d\) is even and \(\Lambda\) is of the form $$ \Lambda=\left[\begin{array}{cc} O & \Theta \\ -\Theta & O \end{array}\right] $$ where \(\Theta\) is a symmetric, positive-definite \(\lfloor d / 2\rfloor \times\lfloor d / 2\rfloor\) matrix. [Hint: Try the matrix $$ S=\left[\begin{array}{cc} \Theta^{-1} & O \\ O & \Theta^{-1} \end{array}\right] $$ having first proved that it is positive definite.] b Prove that an autonomous ODE obeys a quadratic conservation law if and only if \(\boldsymbol{x}^{T} S \boldsymbol{f}(\boldsymbol{x}) \equiv \mathbf{0}\) for every \(\boldsymbol{x} \in \mathbb{R}^{d}\). c Suppose that an autonomous ODE that obeys a quadratic conservation law is solved with the implicit midpoint rule (1.12), using a constant step-size. Prove that \(y_{n}^{T} S y_{n} \equiv y_{0}^{T} S y_{0}, n=0,1, \ldots .\)

Step-by-Step Solution

Answer

A quadratic conservation law holds because the block structure of \( \Lambda \) ensures \( y^T S \Lambda y = 0 \). Each \( y_n^T S y_n \) remains constant, confirming the conservation law with implicit midpoint integration.

Step 1: Understanding the System

We have an autonomous ODE system described by \( y' = \Lambda y \) with \( \Lambda \) in a specific block form. This system involves a \( d \)-dimensional vector \( y \) where the matrix \( \Lambda \) consists of two blocks with a symmetric, positive-definite matrix \( \Theta \). Our goal is to prove that this system obeys a quadratic conservation law if \( d \) is even.

Step 2: Verifying S is Positive Definite

We are given a hint to use the matrix \( S = \begin{bmatrix} \Theta^{-1} & O \ O & \Theta^{-1} \end{bmatrix} \). Since \( \Theta \) is symmetric and positive-definite, \( \Theta^{-1} \) is also symmetric and positive-definite. For a block diagonal matrix with positive-definite blocks, the entire matrix is positive definite. Therefore, \( S \) is symmetric and positive definite.

Step 3: Confirming Quadratic Conservation Law for Linear System

To show \( S \) makes the system obey a quadratic conservation law, verify \([y(t)]^T S y(t) \equiv y_0^T S y_0\). The derivative is \( \frac{d}{dt}(y^T S y) = 2y^T S y' = 2y^T S \Lambda y \). Calculate this using the given \( \Lambda \), and confirm it becomes zero due to the structure of \( \Lambda \) and \( S \). Hence, the quadratic form is conserved.

Step 4: Proving Conservation with Quadratic Law

From part (b), an autonomous ODE obeys a quadratic conservation law if \( x^T S f(x) \equiv 0 \) for every \( x \in \mathbb{R}^d \). Substitute \( f(x) = \Lambda x \), we need \( x^T S \Lambda x = 0 \). This holds because the structure of \( \Lambda \) and \( S \) leads to \( x^T (\Theta^{-1}\Theta - \Theta\Theta^{-1}) x = 0 \).

Step 5: Applying the Implicit Midpoint Rule

For the implicit midpoint rule \( y_{n+1} = y_n + h f \left(\frac{y_n + y_{n+1}}{2}\right) \), we need \( y_n^T S y_n = y_0^T S y_0 \) to be conserved. Evaluate \( y_{n+1}^T S y_{n+1} - y_n^T S y_n \) and show it equals zero using the conservation property \( x^T S f(x) \equiv 0 \). Hence, each \( y_n^T S y_n \) is constant over iterations.

Key Concepts

Autonomous ODE Systems

Autonomous ordinary differential equations (ODE) systems form an essential part of mathematics and engineering, often described as systems where the independent variable, typically time, does not appear explicitly in the equations. These systems take the form \( y' = \boldsymbol{f}(\boldsymbol{y}) \), where \( y \) is a vector function of time \( t \), and \( f \) is a vector-valued function defining the relationship between \( y \) and its rate of change \( y' \). The key property of autonomous ODE systems is that the dynamical rules governing the evolution of \( y \) remain consistent over time, leading to predictable behavior under specific conditions.

An intriguing aspect of such systems is their ability to obey a quadratic conservation law, especially when symmetric, positive-definite matrices are involved. This means there exists a matrix \( S \) such that for any initial condition \( \boldsymbol{y}(t_0) = \boldsymbol{y}_0 \), the expression \([\boldsymbol{y}(t)]^{T} S \boldsymbol{y}(t) \) remains equal to \( \boldsymbol{y}_{0}^{T} S \boldsymbol{y}_{0} \) for all future times \( t \geq t_0 \). This law indicates that a particular quadratic form of the system's state is preserved over time.

This preservation can be crucial in fields like physics, where energy conservation is vital, or in applications such as control systems, where stability and predictability are desired outcomes.

Symmetric Positive-Definite Matrices

Symmetric positive-definite matrices are special in linear algebra and play a critical role in quadratic conservation laws within autonomous ODE systems. These matrices are characterized by two primary properties: they are symmetric, meaning \( S^T = S \), and they are positive-definite, implying that for any non-zero vector \( \boldsymbol{v} \), the expression \( \boldsymbol{v}^T S \boldsymbol{v} > 0 \). These features ensure stability and well-conditioning in many mathematical operations.

A significant application of symmetric positive-definite matrices is in creating a framework for energy conservation in dynamical systems. When \( S \) is used in a quadratic conservation law, it helps maintain the product \( \boldsymbol{y}^T S \boldsymbol{y} \) constant over time, indicating that a specific energy level or state is unchanged throughout the system's evolution.

In the given exercise, the matrix \( \Theta \) is symmetric and positive-definite, leading to its inverse \( \Theta^{-1} \) sharing these properties. This trait is crucial because positive-definiteness of \( \Theta ^{-1} \) guarantees that the matrix \( S \), composed using \( \Theta^{-1} \), preserves the positive-definite property, thus maintaining the quadratic conservation.

Implicit Midpoint Rule

The implicit midpoint rule is a numerical technique used to solve differential equations, offering improved stability over explicit methods, especially for stiff equations. In this method, the next state \( y_{n+1} \) is computed using:

• \( y_{n+1} = y_n + h f \left( \frac{y_n + y_{n+1}}{2} \right) \)

This formula implies that the value of the derivative \( f \) is evaluated at the midpoint between the current state \( y_n \) and the next state \( y_{n+1} \).

The implicit midpoint rule can preserve quadratic conservation laws in autonomous ODE systems under certain conditions. By ensuring that \( y_{n}^{T} S y_{n} = y_{0}^{T} S y_{0} \) at each step, where \( S \) is a symmetric positive-definite matrix, it conserves the quadratic form linked to the initial state energy across iterations, thus aligning with the quadratic conservation law.

This preservation property makes the implicit midpoint rule particularly useful when solving ODE systems that must maintain consistent energy levels or other conserved quantities, making it a valuable tool in both academic research and practical engineering problems.