Wave Equation on Square Domain
This example shows how to solve the wave equation using the
The standard second-order wave equation is
To express this in toolbox form, note that the
solvepde function solves problems of the form
So the standard wave equation has coefficients , , , and .
c = 1; a = 0; f = 0; m = 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); 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.
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);
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
Set the following initial conditions:
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);
This choice avoids putting energy into the higher vibration modes and permits a reasonable time step size.
Specify the solution times as 31 equally-spaced points in time from 0 to 5.
n = 31; tlist = linspace(0,5,n);
model.SolverOptions.ReportStatistics ='on'; result = solvepde(model,tlist);
457 successful steps 38 failed attempts 992 function evaluations 1 partial derivatives 113 LU decompositions 991 solutions of linear systems
u = result.NodalSolution;
Create an animation to visualize the solution for all time steps. 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 -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"); 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