How to solve the wave equation in 2D using the Runge-Kutta Method (Matlab Code Explained)

First, we express the wave equation as a system of 2 first order PDEs.

Then we discretize the spatial part of the second equation

And then we program it in Matlab

We modify Runge-Kutta integrator for Velocity and Displacement

We program the Runge-Kutta integrator in Matlab

Then we add a function that helps us update the plot as the simulation progress

From here we proceed with the following

1. Set the domain.
2. Compute dt to ensure a stable solution.
3. Prepare the initial condition.
4. Set function handle for the right-hand side of the PDE.
5. Initialize the plot.
6. Set function handle for plot update.
7. Start video writer.
8. Run the simulation.

1. Set the domain

Here we set the bounds of the domain, the number of discrete points in x and y direction, the size of a grid sell, and the mesh.

2. Compute dt to ensure stability of the solution

We use the Courant–Friedrichs–Lewy (CFL) Number, the velocity of propagation and the grid size to determine the time step size.

3. Prepare the initial condition.

The displacement is initialized to be a stationary “blob” somewhere in the domain. We define the blob to be centered at xc, yc

The code in matlab

4. Set function handle for the right-hand side of the pde.

We set the function handle that calls the wave equation we created earlier. This function computes the rate of change of velocity and displacement over time.

5. Initialize the plot

We plot the initial condition. Since this is a 2D problem the plot would be a 3D plot. the displacement would be along the third dimension.

6. Set function handle for plot update

We set the function that would help us update the plot as the simulation progresses. This function is pointing to the function we created earlier.

7. Start video writer.

Here we use the start the video writer (setting the name, format and frame rate of the video) that would be used to capture the sequence of event as the simulation progresses.

8. Run the simulation.

Finally, we run the simulation by solving for u and v at the next time step, then using these values as the initial, we move to the next time step until we get to the stated time for the simulation.

Below is the video created.