Trying to solve 1D wave equation but not getting the plot correct

General Discussion
1

I am trying to solve the 1D wave equation. When I am trying to plot the approximated solution I am getting a straight line. Can anyone please check what is wrong with the plot command.
As I am unable to attach the edp file, I am pasting the code here.

// Problem definition
//1d pde on the boundary term u_t=u_{xx} 0<x<1, 0<t<1
// u(x,0)=sin(pi*x)
// u(0,t)=0,u(1,t)=0

verbosity=0;
macro uex(t) (sin(pi*x)exp(t))// Exact Solution
macro f(t) (sin(pix)exp(t)(pi^2+1))//Source Function

int nref=2;
real[int] L2error(nref); // initialize the L2 error array
real[int] Dx(nref); // initialize the space discretization array
real[int] DT(nref); // initialize the time discretization array

for(int n=0; n<nref;n++) {
int R=2^(n+3);// grid points on x-interval [0,1]
real h=1./R;
Dx[n]=h;
real dt=h^2;
DT[n]=dt;
real t, T=1;

// Technical data to recreate OX and OY axis
border OX(t=0,1){x=t;y=0;}
border OY(t=0,1){x=0;y=t;}

meshL Th=segment(R);

plot(Th, cmm="NO ADAPTED MESH ",wait=0);//,ps=“reactiondiffusionnoadapted2dmesh.eps”);
fespace Uh(Th,P1);

Uh u, v, uold;

u=sin(pi*x);// initial condition

problem Wave(u,v)=int1d(Th)(u*v/dt+dx(u)dx(v))
-int1d(Th)(uoldv/dt)
-int1d(Th)(f(t+dt/2.)*v)
+on(1,2,u=0);

// time loop
int m, M = T/dt;
for (m = 0; m < M; m++) {
uold=u;
Wave;
t+=dt;
plot(u,cmm=“Wave Equation Appox soln. u at time t=”+t,wait=0,value=1);
}// end of time loop

L2error[n]=sqrt(int1d(Th)(abs(u-uex(t))^2));
cout << "L2 error " << L2error[n] << endl;

}// end of loop on n<nref

2

Hi,
The problem statement in the code appears to be incorrect for a 1D wave equation. The formulation shown corresponds to the weak form of a parabolic equation, such as the heat equation. For the linear wave equation with a wave speed of 1, the correct weak form should involve dx(u)*v. I have made the necessary correction and attached the updated version. Please let me know if my understanding is incorrect.
trial.edp (1.5 KB)

3

Hi,
This code is not working as the error started increasing. For my code I was getting the correct error. I was having the problem in the plot of the approximate solution. While plotting the solution I am getting a straight line which should not happen because exact solution is a sine function. Please let me know if anything wrong with the plot command for 1D problem.

4

Hi,
please check the code i am getting a sine function as output and the error is also decreasing. But i am not sure if this is the weak form for a 1D wave equation
trial.edp (1.4 KB)

5

The plot command as you write is not supposed to work for 1d.
Instead you can do as

real[int] xx(R+1),yy(R+1),yyexact(R+1);
Uh uexe=uex(t);
for (int i=0;i<R+1;i++)
{
xx[i]=i*1./R;
yy[i]=u[](i);
yyexact[i]=uexe[](i);
}
plot([xx,yy],[xx,yyexact],wait =0,bb=[[0,0],[1,3]]);

But I don’t know how to plot the axis.

About your equation, you are solving the heat equation, not the wave equation.