The wave equation in one space dimension can be written as follows:
∂2u/∂t2 =a ∂2u/∂x2+ b ∂2u/∂y2
This equation is typically described as having only one or two space dimension "x", and "y" because the only other independent variable is the time "t". The question is what is the stability criteria when using the explicit finite difference method?
This should be sufficient:
https://www.crewes.org/ForOurSponsors/ResearchReports/1998/1998-17.pdf
and
http://hplgit.github.io/num-methods-for-PDEs/doc/pub/wave/html/slides_wave-solarized.html#___sec116
There is no unique criterion but it depends on the type of discretization (symmetric or non-symmetric stencil), accuracy (first, second, etc) and single or multi-step (Euler, Adams-Bashforth, etc).
Once defined the discrete equation, you can apply the von Neumann stability analysis or you could exploit the Gershgorin eigenvalue localization theorem.
Finally, you second order PDE can be also reduced to a system of two first order equations.
The Von Neumann Stability Analysis allow to find a relation between Δt, Δx and Δy of an adequate choice. The following assumptions can be used in the finite difference scheme of the equation: un+1i,j=g.uni,j , uni+1,j = uni,j.ei.eta1 and uni,j+1 = uni,j.ei.eta2 where g: is the amplification factor , eta1=k1.Δx and eta2=k2.Δy
Thank you Vikash again for your effort. It is very much appreciated. I am trying to get caught up with my thanks!
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