2-D linearised Euler equation

== Problem Definition ==
:<math> \frac{\partial u}{\partial t}+M \frac{\partial u}{\partial x}+\frac{\partial p}{\partial x}=0 </math>
:<math> \frac{\partial v}{\partial t}+M \frac{\partial v}{\partial x}+\frac{\partial p}{\partial y}=0 </math>
:<math> \frac{\partial p}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial p}{\partial x}=0 </math>
:<math> \frac{\partial \rho}{\partial t}+\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+M\frac{\partial \rho}{\partial x}=0 </math>
where  M is the mach number , speed of sound is assumed to be 1, all the variabled refer to acoustic perturbations over the mean flow.
== Domain ==
[-50,50]*[-50,50]
== Initial Condition ==
:<math> p(x,0)=a*exp(-ln(2)*((x-xc)^2+(y-yc)^2)/b^2) </math>
== Boundary Condition ==
Characteristic Boundary Condition 
== Numerical Method ==
4th Order Compact scheme in space 
4th order low storage RK in time 
== Results ==
Pressure 
:No mean flow 
[[Image:Nomeanflow.jpg]]
:Mean Flow to left at U=0.5 (c assumed to be 1 m/s)
[[Image:Meanflow.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}}