3. Discretization Technique

There are significant benefits in obtaining a theoretical prediction of a physical phenomenon. The phenomena of interest here are governed by differential equations, which involve replacing continuous information contained in the exact solution of these equations with discrete values.

Discretization techniques are used to convert continuous models defined by differential equations into a form that can be solved with numerical methods. Continuous models assume smooth changes across the entire domain; discretization breaks the domain into small chunks (or points). This enables us to approximate the solution at those specific points rather than needing a complete continuous solution.

Think of a smooth piece of string representing a continuous function. If you want to find out how long the string is, you could choose points along the string to measure, rather than trying to measure its entire length at once. Each measurement gives you a discrete point that helps estimate the overall length.

There is something called the Taylor Series Formulation which we generally use. Usually, finite difference equations consist of approximating the derivatives in differential equations via a truncated Taylor series.

The Taylor Series is a mathematical formula that expresses a function as an infinite sum of terms calculated from the function's derivatives at a single point. In the context of discretization, we truncate this series to a finite number of terms, which allows us to approximate derivatives. By doing this, we can replace the continuous derivatives in our equations with finite difference approximations that can be solved numerically.

Imagine you're trying to describe the path of a car using its speed and acceleration at a specific moment. The Taylor Series helps us estimate the car's position at various future times using only its current speed and acceleration. Just as you might only need a few measurements (like speed at that moment) to predict where the car will be, discretization uses a few terms from the Taylor series.

Truncating the series just after the third term and adding and subtracting the two equations, we can obtain expressions for first and second derivatives that lead to finite difference equations.

When we truncate the Taylor series after three terms and manipulate it by adding and subtracting, we derive formulas for the first and second derivatives. These formulas are essential for forming finite difference equations, which approximate the original differential equations. This is crucial for transitioning from the theoretical model to a numerical method that we can implement in simulations or calculations.

Consider baking a cake. When following a recipe, you might measure out ingredients using rough approximations (like 'a handful of flour'). Truncating the Taylor series is like deciding that, for simplicity, you'll only exactly measure the first few necessary ingredients instead of weighing out everything perfectly. This simplified approach still lets you create a great cake without needing to be exact in every detail.

Analytical solution of partial differential equations provides closed-form expressions, while numerical solutions based on finite differences provide values at discrete points in the domain.

Analytical solutions yield exact results for the entire domain and provide a comprehensive understanding of the behavior of the system. However, they are not always feasible for complex systems. In contrast, numerical solutions give approximate values at specific points (grid points) in the domain, which allows us to simulate scenarios that are difficult to solve analytically.

Imagine you’re trying to predict the weather. An analytical model might give you a perfect formula for how temperature changes based on time, but if it's too complex, we use numerical weather forecasting, which provides temperature predictions for specific locations at specific times (like your city today) even if it isn't perfect everywhere.

In the next lecture, we will start with elementary finite difference quotients.

This section sets the stage for the upcoming lectures, where students will learn about the basic building blocks of finite difference methods. These quotients will form the foundation for solving various differential equations using discretization techniques.

Like learning to walk before you run, understanding elementary finite difference quotients will prepare students for more complex applications in numerical methods, ensuring they have a solid footing as they progress in their studies.

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