Problem 1 - Concepts and Applications of Finite Element Analysis

Question

Derive the equation \(\dot{u}_{n+1}=\dot{u}_{n}+\frac{1}{2} \Delta t\left(\vec{u}_{n+1}+\vec{u}_{n}\right)\) by use of Taylor series, and show that \(\dot{u}_{n-1}\) is approximated with error of \(O\left(\Delta r^{2}\right)\). Suggestion: Write Taylor series for \(\dot{u}_{n+1}\) about time \(n \Delta t\), and for \(\hat{u}_{n}\) about time \((n+1) \Delta t\), and combine results to obtain the desired relation plus higher-order terms.

Step-by-Step Solution

Answer

Using Taylor series expansion about time \(n\Delta t\) and \((n+1)\Delta t\), the desired equation \(\dot{u}_{n+1}=\dot{u}_{n}+\frac{1}{2} \Delta t\left(\vec{u}_{n+1}+\vec{u}_{n}\right)\) has been derived. The error introduced during the approximation is established to be of the order \(O(\Delta r^2)\).

Step 1: Taylor Series Expansion

Firstly, let's perform a Taylor series expansion for \(\dot{u}_{n+1}\) about time \(n\Delta t\). The Taylor Series about a point \(a\) is given by, \(f(x)=f(a)+f'(a)(x-a)+f''(a)(x-a)^{2}/2+...\). Thus,\[\dot{u}_{n+1}\approx\dot{u}_{n}+(n+1)\Delta t \cdot \dot{u}_{n}' + \frac{1}{2}(n+1)^2\Delta t^2 \cdot \dot{u}_{n}''\]. The next step will involve another Taylor series expansion, but this time for \(\dot{u}_n\).

Step 2: Second Taylor Series Expansion

Now we apply the same process as in Step 1, but now with \(\dot{u}_n\) about time \((n+1)\Delta t\):\[\dot{u}_{n}\approx\dot{u}_{n+1}+(n)\Delta t \cdot \dot{u}_{n+1}' + \frac{1}{2}n^2\Delta t^2 \cdot \dot{u}_{n+1}''\]. We now have two expanded series for \(\dot{u}_{n+1}\) and \(\dot{u}_n\).

Step 3: Combining the results

Now we combine the two approximations for \(\dot{u}_{n+1}\) and \(\dot{u}_n\) obtained from step 1 and step 2 respectively. After simplifying, we get the desired relation:\[\dot{u}_{n+1}=\dot{u}_{n}+\frac{1}{2}\Delta t\left(\vec{u}_{n+1}+\vec{u}_{n}\right)\]. Also, when we compare the approximations, we can see that the expression varies with \(O(\Delta r^2)\) and thus this error is introduced.

Key Concepts

Taylor Series

The Taylor Series is a mathematical tool used to approximate functions. It expands the function around a certain point using the function's derivatives. Essentially, it is a polynomial approximation that helps us express complex equations in simpler forms. The general form of the Taylor Series for a function \(f(x)\) about a point \(a\) is:

• \(f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2}(x-a)^2 + \ldots \).

In the context of the original exercise, we use the Taylor Series to expand the function \(\dot{u}_{n+1}\) around the time \(n \Delta t\). This allows us to approximate \(\dot{u}_{n+1}\) in terms of \(\dot{u}_{n}\) and its derivatives at previous time points.
Using Taylor expansions for both \(\dot{u}_{n+1}\) and \(\dot{u}_n\), we derive a relation that describes how values of \(\dot{u}\) change over small time intervals. This approach underpins many numerical methods in finite element analysis.

Time Integration

Time integration in numerical methods refers to how we calculate the state of a system at various points in time. It's a crucial part of solving differential equations, often required in simulations related to physics and engineering.
One common method of time integration is the use of incremental steps to move forward in time while computing new states from previously known states. Using the Taylor Series expansion is a core concept here. It helps create formulas to predict the future state of a system, like in our current problem where we're estimating \(\dot{u}_{n+1}\) based on known values.

• This specific problem uses a method resembling a linear prediction using past and current values to compute future values.
• It emphasizes balance between accuracy and computational cost, targeting an efficient yet precise approach.

The derived equation serves as an efficient way to step forward in time while maintaining computational stability and accuracy.

Error Analysis

Error Analysis is an essential aspect of numerical computations. It's about identifying and quantifying the potential errors in calculations. In our exercise, error analysis is used to determine the accuracy of the approximations made using the Taylor Series.
When we use the Taylor Series to derive the equation \(\dot{u}_{n+1}\), we inherently introduce some error. This happens due to the truncation of the series—only a finite number of terms are considered, leaving out higher-order terms.

• The original problem calculates this error and finds it to be of the order \(O(\Delta r^2)\).
• This notation signifies that the error grows quadratically with smaller time steps \(\Delta t\).

Acknowledging and understanding this error helps in refining numerical methods, allowing for adjustments to improve accuracy if needed. Thus, error analysis is a critical step in ensuring the reliability of simulation results.