Wave propagation

== Introduction ==
The 1-D wave propagation problem is used as a test case for studying the dissipation and dispersion errors in a given finite difference scheme.There are a number of schemes which can be used to solve the problem. The usage of a higher order compact stencil with a low storage 4th order Runge-Kutta scheme to solve the current problem is discussed. 

== Compact scheme ==

== Runge-Kutta == 
Consider 
:<math> \frac {\partial U}{\partial t}=H </math>
The low storage scheme is implemented as follows 
:<math> U^{M+1}=U^M+b^Mdtf^M </math>
:<math> f^M=a^Mf^{M-1}+H </math>
where M refers to the stages ,dt is the time step and the coefficients a and b are given by 
:a[5]={0,-0.41789047,-1.19215169,-1.69778469,-1.51418344}
:b[5]={0.149665602,0.37921031,0.82295502,0.69945045,0.15305724}

== Sample result == 

[[Image:wp_result.jpg]]

== Reference ==

*{{reference-paper|author=Williamson, Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}}

*{{reference-paper|author=Lele, Lele, S. K.|year=1992|title=Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics|rest=Journal of Computational Physics, Vol. 103, pp 16–42}}