How to Solve a Boundary Value Problem Using the Finite Difference Method?

[evinda]

Hello! (Wave)

I want to solve numerically the following boundary value problem:

$\left\{\begin{matrix}
-u''+qu=f & , x \in [a,b]\\
-u'(a)+d_1 u(a)=0 & \\
u'(b)+d_2 u(b)=0 &
\end{matrix}\right.$

where $q(x) \geq 0 \forall x \in [a,b], d_1, d_2 \geq 0$.
We consider the uniform partition of $[a,b]$ with step $h=\frac{b-a}{N}$.

The approximations $U_i$ of $u(x_i), i=1, \dots, N+1$ have to be calculated with the following finite difference method:

$\\ \frac{2}{h^2}(U_1-U_2)+2 \frac{d_1}{h} U_1+q(x_1)U_1=f(x_1)
-\frac{1}{h^2}(U_{i-1}-2U_i+U_{i+1})+q(x_i)U_i=f(x_i), i=2, \dots, N \\
-\frac{2}{h^2}(U_N-U_{N+1})+2\frac{d_2}{h}U_{N+1}+q(x_{N+1})U_{N+1}=f(x_{N+1})$

There are given specific functions for $f$ and $q$.

I have to compute approximations of the solution, and the errors for uniform partitions with $N=25,50,100$ subintervals.

So does this mean that we have to write the whole $N+1$-sized vector $U$ that we get for each $N$ ?Also, I have to find the order of accuracy of the method.
This can be done as follows:
Let $\epsilon (N)$ the error of the numerical method for $N$ subintervals and let's assume that $\epsilon (N) \approx C h^p$, where $C$ doesn't depend on $h$ and $N$. Then:

$$\frac{\epsilon (N)}{ \epsilon (2N)} \approx \frac{Ch^p}{C \left(\frac{h}{2} \right)^p} \Rightarrow p \approx \frac{\log \left( \frac{\epsilon (N)}{\epsilon (2N)}\right)}{\log 2}$$So can I run the program for any $N$ that I want and then for the corresponding $2N$ in order to find the order of accuracy?

 

[jvicens]

Hello! As a fellow scientist, I can definitely help you with this boundary value problem. First, let's break down the problem into smaller steps.

Step 1: Define the problem
The boundary value problem that you are trying to solve is a second-order differential equation with two boundary conditions. The function $u(x)$ is unknown, and we want to find its numerical approximation $U_i$ at specific points $x_i$ within the interval $[a,b]$.

Step 2: Discretize the problem
To solve the problem numerically, we need to discretize the interval $[a,b]$ into smaller subintervals. This is where the uniform partition with step $h$ comes in. By dividing the interval into $N$ subintervals, we can approximate the solution at each point $x_i$ with $U_i$. The finite difference method you mentioned is a commonly used approach to discretize differential equations.

Step 3: Set up the equations
Using the finite difference method, we can set up a system of equations that represents the discretized problem. Each equation corresponds to a specific point $x_i$ within the interval. By solving this system of equations, we can obtain the numerical approximation $U_i$ at each point.

Step 4: Calculate the approximations and errors
Now, we can run the program for different values of $N$ to obtain the corresponding approximations $U_i$. We can also calculate the errors by comparing the numerical solution to the exact solution (assuming it is known). As you mentioned, we can use the formula $\epsilon (N) \approx C h^p$ to estimate the order of accuracy of the method. This can give us an idea of how the error changes as we increase the number of subintervals.

Step 5: Validate the results
To validate the results, we can compare the calculated errors to the theoretical errors. We can also plot the numerical solution and compare it to the exact solution, if available.

In conclusion, to solve the boundary value problem numerically, we need to discretize the problem, set up the equations, calculate the approximations and errors, and validate the results. By following these steps, we can obtain a numerical solution and estimate the order of accuracy of the method. I hope this helps!