Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. It only takes a minute to sign up.
Sign up to join this community
I have a PDE which looks like Helmholtz wave equation on one dimensional domain. $$\dfrac{d^2u(x)}{dx^2}+\pi^2u(x)=f(x)$$ where $-\infty <x<\infty $
Also, $f(x)= 1$ for $-0.25<x<0.25$, I call this region "inner region"
$f(x)=0$ elsewhere
I truncated the computational domain using absorbing boundary conditions into $-1<x<1$
I chose mesh size as $\lambda/10$ at 1 GHz.
Finally I plotted the answer as shown below. How can I test its trueness of its? Why do I observe decrease in the inner region?
Using the given ODE, the boundary conditions $u(x)=0$ at $x=\pm1$, and the symmetry of the solution it is easy to write the exact analytic solution:
$u(x)= a \sin(\pi x)$, for x $\in$ [1/4,1]
$u(x)=- a \sin(\pi x)$, for x $\in$ [-1,-1/4]
$u(x) = 1/\pi^2 + b \cos(\pi x)$, for x $\in$ [-1/4,1/4]
The matching condition for the function derivative at x=1/4 yields $b=-a$, and the matching condition for the function value at x=1/4 yields $a \sqrt{2}/2 = b \sqrt{2}/2 + 1/\pi^2$, so we find $a=1/(\sqrt{2} \pi^2)$ and $b=-1/(\sqrt{2} \pi^2)$. At the maximum point x=1/2, u(1/2)=a $\sim$0.07 which visually matches the plot; and at the minimum point x=0, u(0)=$(1/\pi^2+b) \sim$ 0.03 which visually matches the plot; a more detailed comparison could be done comparing the numerical values from the code against the analytic solution.
