MATLAB Examples

Wave Equation on a Square Domain

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

Contents

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 solvepde function solves problems of the form

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

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

c = 1;
a = 0;
f = 0;
m = 1;

Geometry

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);
geometryFromEdges(model,@squareg);
pdegplot(model,'EdgeLabels','on');
ylim([-1.1 1.1]);
axis equal
title 'Geometry With Edge Labels Displayed';
xlabel x
ylabel y

Specify PDE Coefficients

specifyCoefficients(model,'m',m,'d',0,'c',c,'a',a,'f',f);

Boundary Conditions

Set zero Dirichlet boundary conditions on the left (edge 4) and right (edge 2) and zero Neumann boundary conditions on the top (edge 1) and bottom (edge 3).

applyBoundaryCondition(model,'dirichlet','Edge',[2,4],'u',0);
applyBoundaryCondition(model,'neumann','Edge',([1 3]),'g',0);

Generate Mesh

Create and view a finite element mesh for the problem.

generateMesh(model);
figure
pdemesh(model);
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 = @(location) atan(cos(pi/2*location.x));
ut0 = @(location) 3*sin(pi*location.x).*exp(sin(pi/2*location.y));
setInitialConditions(model,u0,ut0);

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

Set the SolverOptions.ReportStatistics of model to 'on'.

model.SolverOptions.ReportStatistics ='on';
result = solvepde(model,tlist);
u = result.NodalSolution;
440 successful steps
35 failed attempts
952 function evaluations
1 partial derivatives
107 LU decompositions
951 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.

figure
umax = max(max(u));
umin = min(min(u));
for i = 1:n
    pdeplot(model,'XYData',u(:,i),'ZData',u(:,i),'ZStyle','continuous',...
                  'Mesh','off','XYGrid','on','ColorBar','off');
    axis([-1 1 -1 1 umin umax]);
    caxis([umin umax]);
    xlabel x
    ylabel y
    zlabel u
    M(i) = getframe;
end

To play the animation, use the movie(M) command.