5. Numerical Solutions of Partial Differential Equations

Partial Differential Equations (PDEs) are equations that involve functions of several variables and their partial derivatives. PDEs are used to model many physical phenomena, such as heat conduction, fluid dynamics, and electromagnetic fields. Solving PDEs analytically is often impossible for real-world problems, especially when boundary conditions or complex geometries are involved. Numerical methods provide approximate solutions to PDEs by discretizing the problem and solving it using computational techniques. In this chapter, we will focus on two popular numerical methods for solving PDEs: Finite Difference Methods (FDM) and Finite Element Methods (FEM).

Partial Differential Equations (PDEs) are important in mathematics and physics because they help describe how physical quantities change with respect to each other. For instance, they can represent the distribution of heat in an object or the motion of fluid. However, these equations can become highly complicated, making them difficult to solve precisely, especially when the systems involved have complex shapes or specific boundaries. This is why we turn to numerical methods, which allow us to get approximate solutions by breaking down these complex problems into smaller, manageable pieces. In this section, we will explore two well-known techniques: Finite Difference Methods (FDM) and Finite Element Methods (FEM).

Imagine baking a cake. While there are many precise ingredients and steps needed to make the perfect cake (analogous to solving PDEs analytically), often we have to adjust the recipe based on what we have at hand (like numerical methods). Sometimes you might decide to estimate the baking temperature or time, which can be viewed as numerical approximations to the perfect cake recipe — and similarly, numerical methods help us approximate solutions where exact answers are hard to get.

Finite Difference Methods (FDM) are among the simplest and most widely used numerical methods for solving PDEs. They involve discretizing the domain and approximating the derivatives in the PDE using finite differences. This transforms the PDE into a system of algebraic equations that can be solved using standard numerical techniques.

Finite Difference Methods provide a framework for approximating solutions to PDEs. The fundamental idea is to replace the continuous variables with discrete points, creating a grid or mesh that represents the domain under consideration. We then use finite differences to approximate the derivatives specified in the PDE, replacing infinitesimal changes with finite steps. This process allows us to reformulate the PDE into linear algebraic equations, which can be solved using computational techniques that are widely available and easy to implement.

Think of someone trying to climb a mountain by taking steps rather than walking continuously. If they take steps of set height (like finite differences), they can still reach the top but might miss nuances and smaller details of the mountain's path. Similarly, FDM takes finite steps to approach solutions to PDEs, making it manageable even when the exact slope or path is complex.

1. Discretizing the Domain:

• The domain is discretized into a grid or mesh, where the continuous variables are approximated by values at specific grid points. For example, for a 1D problem, the domain [a,b] is discretized into points x0,x1,x2,…,xn.

2. Approximating Derivatives:

• The partial derivatives in the PDE are approximated by finite differences. For example:
  • Forward Difference: df/dx ≈ (f(x+h)−f(x))/h
  • Backward Difference: df/dx ≈ (f(x)−f(x−h))/h
  • Central Difference: df/dx ≈ (f(x+h)−f(x−h))/(2h)

The first step in applying FDM is to discretize the problem’s domain. This means dividing the domain into a series of discrete points, allowing for the approximation of continuous functions. In a simple 1D case, you might choose specific locations along a line to evaluate the function. Once you have these points, the next step is to approximate derivatives at these points using finite differences. This can take the form of forward, backward, or central differencing, each of which estimates the derivative based on function values from neighboring points. This process converts the PDE into a form that can be solved as a system of equations, making it feasible with numerical techniques.

Imagine you're reporting the temperature along a road at certain points instead of measuring it continuously. By using a thermometer at specific locations (discretizing), you can get readings (approximations) for each point along the road. You might then calculate the change in temperature using nearby readings. This method helps ensure you get a comprehensive picture of temperature changes without having to measure continuously.

Consider the 1D heat equation: ∂u/∂t=α∂²u/∂x² with initial condition u(x,0)=u0(x) and boundary conditions u(0,t)=uL and u(L,t)=uR.

1. Discretizing the spatial domain: Divide the spatial interval [0,L] into N equally spaced points with spacing Δx=L/N.
2. Discretizing the time domain: Divide the time interval into steps Δt, so the solution is approximated at time points tn=nΔt.
3. Applying the finite difference approximation:

• The spatial second derivative is approximated using the central difference: ∂²u/∂x² ≈ (ui+1n−2uin+ui−1n)/(Δx)²
• The time derivative is approximated using the forward difference: ∂u/∂t ≈ (uin+1−uin)/Δt

4. Substituting these into the heat equation, we get: uin+1−uin/Δt=α(ui+1n−2uin+ui−1n)/(Δx)²
5. Rearranging: uin+1=uin+αΔt/Δx²(ui+1n−2uin+ui−1n). This equation can be solved iteratively for each time step.

Let's apply FDM to solve the heat equation, which describes how heat evolves over time in a one-dimensional rod. First, we identify our spatial and temporal domains. The rod is divided into N equally spaced points, allowing us to measure temperature at each point at different times. Next, we break the time into intervals, at which we want to evaluate the temperature. By applying finite difference rules for both space and time, we can formulate a new equation that expresses how temperature changes over time based on spatial temperatures. This iterative equation allows us to calculate the temperature at the next time step using known values from the current step.

Think of cooking a pot of soup. You want to know how the temperature changes as you cook it. Instead of checking the temperature at every millisecond, you can check it at specific intervals (e.g., every minute). You can then use the temperature from adjacent minutes to estimate its future temperature. Each checking allows you to predict the next state of your soup's temperature, much like how we predict heat changes using the iterative method in FDM.

Advantages:

• Simple to implement and widely used.
• Effective for problems with uniform grids and simple geometries.

Disadvantages:

• Less accurate for complex geometries or problems with irregular boundaries.
• Requires grid refinement for higher accuracy, leading to higher computational costs.

Finite Difference Methods are praised for their simplicity and ease of implementation, making them a go-to choice for many problems involving PDEs. They work exceptionally well on uniform grids and for straightforward geometrical setups, where the approximation can be reasonably accurate without extensive adjustments. However, the effectiveness can diminish when faced with irregular boundaries or complex geometries. In such cases, obtaining high accuracy may necessitate refining the mesh, which increases computational demands, both in terms of processing power and time.

Consider drawing a straight line versus a wavy line. The straight line is easy to draw and requires a simple, straightforward approach (like FDM with uniform grids). However, when you want to draw the wavy line, you might need a more detailed strategy and additional time to capture each peak and trough (similar to refining your grid for accuracy). What is simple becomes complicated very quickly!

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