Linear wave propagation

== Problem definition ==
:<math> \frac{\partial u}{\partial t}+ c \frac{\partial u}{\partial x}=0
</math>
== Domain == 
:<math> x=[-10,10] </math>
== Initial Condition == 
:<math> u(x,0)=exp[-ln(2){(\frac{x-x_c}{r})}^2]</math>

== Boundary condition == 
:<math>u(-10)=0</math>

== Exact solution ==
:<math> u(x,t)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]</math>

== Numerical method == 
:<math>c=1,dx=1/6,dt=0.5dx,t=7.5</math>
:<math> \mbox{Long wave  :}\frac{r}{dx}=20</math> 
:<math> \mbox{Medium wave: }\frac{r}{dx}=6</math> 
:<math> \mbox{Short wave : } \frac{r}{dx}=3</math> 
=== Space ===
==== Explicit Scheme (DRP)====
:<math> {(\frac{\partial u}{\partial x})}_i=\frac{1.0}{dx}\sum_{k=-3}^3 a_k u_{i+k} </math>
The coefficients can be found in Tam(1993).At the right boundaries use fourth order central difference and fourth backward difference.At left boundaries use second order central difference for i=2 and fourth order central difference for i=3.The Dispersion relation preserving (DRP) finite volume scheme can be found in Popescu (2005).
====Implicit Scheme(Compact)====
:Domain: <math>\alpha v_{i-1} + v_i + \alpha v_{i+1}=\frac{a}{2h}(u_{i+1}-u_{i-1}) </math>
:Boundaries: <math> v_1+\alpha v_2=\frac{1}{h}(au_1+bu_2+cu_3+du_4) </math>
where v refers to the first derivative.For a general treatment of compact scheme refer to Lele (1992).In this test case the following values are used 
:<math> \mbox{Domain:} \alpha=0.25 , a=\frac{2}{3}(\alpha+2) </math>
:<math> \mbox{Boundary:} \alpha=2 ,a=-(\frac{11+2\alpha}{6}),b=\frac{6-\alpha}{2},c=\frac{2\alpha-3}{2},d=\frac{2-\alpha}{6} </math>
Both the schemes are 4th order accurate in the domain.The compact scheme has third order accuracy at the boundary.
===Time (4th Order Runga-Kutta)===
:<math>\frac{\partial u}{\partial t}=f </math>

:<math>u^{M+1} =u^M + b^{M+1}dtH^M </math>

:<math> H^M=a^MH^{M-1}+f^M </math>
,M=1,2..5 .The coefficients a and b can be found in Williamson(1980)

== Results ==
[[Image:Initial_condition.png|450px]]
[[Image:Result_wave.png|450px]]
== Reference ==
{{reference-paper|author=Mihaela Popescu, Wei Shyy , Marc Garbey|year=2005|title=Finite volume treatment of dispersion-relation-preserving and optimized prefactored compact schemes for wave propagation|rest=Journal of Computational Physics, Vol. 210, pp. 705-729}}

{{reference-paper|author=Tam and Webb|year=1993|title=Dispersion-relation-preserving finite difference schemes for computational acoustics|rest=Journal of Computational Physics, Vol. 107, pp. 262-281}}

{{reference-paper|author=SK Lele|year=1992|title=Compact finite difference schemes with spectrum-like resolution|rest=Journal of Computational Physics, Vol.103, pp.16-42}}

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