Partial Differential Equation Toolbox

Wave Equation on a Square Domain

This example shows how to solve the wave equation using the hyperbolic function in the Partial Differential Equation Toolbox™.

Problem Definition

The standard second-order wave equation is

$$ \frac{\partial^2 u}{\partial t^2} - \nabla\cdot\nabla u = 0.$$

To express this in toolbox form, note that the hyperbolic function solves problems of the form

$$ d\frac{\partial^2 u}{\partial t^2} - \nabla\cdot(c\nabla u) + au =

So the standard wave equation has coefficients $d = 1$, $c = 1$, $a = 0$, and $f = 0$.

c = 1;
a = 0;
f = 0;
d = 1;


Solve the problem on a square domain. The squareg function describes this geometry. Create a model object and include the geometry. Plot the geometry and view the edge labels.

numberOfPDE = 1;
model = createpde(numberOfPDE);
pdegplot(model, 'edgeLabels', 'on');
ylim([-1.1 1.1]);
axis equal
title 'Geometry With Edge Labels Displayed';
xlabel x
ylabel y

Boundary Conditions

Set zero Dirichlet boundary conditions on left (edge 4) and right (edge 2) and zero Neumann boundary conditions on the top (edge 1) and bottom (edge 3). The zero Neumann boundary condition is the default, so it is not necessary to include this setting.

applyBoundaryCondition(model,'Edge',([1 3]),'g',0); % Not strictly necessary

Generate Mesh

Create and view a finite element mesh for the problem.

ylim([-1.1 1.1]);
axis equal
xlabel x
ylabel y

Create Initial Conditions

The initial conditions:

  • $u(x,0) = \arctan\left(\cos\left(\frac{\pi x}{2}\right)\right)$.

  • $\left.\frac{\partial u}{\partial t}\right|_{t = 0} = 3\sin(\pi x) \exp\left(\sin\left(\frac{\pi y}{2}\right)\right)$.

This choice avoids putting energy into the higher vibration modes and permits a reasonable time step size.

u0 = 'atan(cos(pi/2*x))';
ut0 = '3*sin(pi*x).*exp(sin(pi/2*y))';

Define Solution Times

Find the solution at 31 equally-spaced points in time from 0 to 5.

n = 31;
tlist = linspace(0,5,n);

Calculate the Solution

u = hyperbolic(u0,ut0,tlist,model,c,a,f,d);
428 successful steps
62 failed attempts
982 function evaluations
1 partial derivatives
142 LU decompositions
981 solutions of linear systems

Animate the Solution

Plot the solution for all times. Keep a fixed vertical scale by first calculating the maximum and minimum values of u over all times, and scale all plots to use those $z$-axis limits.

umax = max(max(u));
umin = min(min(u));
for i = 1:n
    axis([-1 1 -1 1 umin umax]);
    caxis([umin umax]);
    xlabel x
    ylabel y
    zlabel u
    M(i) = getframe;