https://www.cfd-online.com/W/index.php?title=Special:Contributions/Rangan86&feed=atom&limit=50&target=Rangan86&year=&month=CFD-Wiki - User contributions [en]2024-03-29T14:00:19ZFrom CFD-WikiMediaWiki 1.16.5https://www.cfd-online.com/Wiki/Linear_wave_propagationLinear wave propagation2006-12-10T12:22:57Z<p>Rangan86: /* Exact solution */</p>
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<div>== Problem definition ==<br />
:<math> \frac{\partial u}{\partial t}+ c \frac{\partial u}{\partial x}=0<br />
</math><br />
== Domain == <br />
:<math> x=[-10,10] </math><br />
== Initial Condition == <br />
:<math> u(x,0)=exp[-ln(2){(\frac{x-x_c}{r})}^2]</math><br />
<br />
== Boundary condition == <br />
:<math>u(-10)=0</math><br />
<br />
== Exact solution ==<br />
:<math> u(x,t)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]</math><br />
<br />
== Numerical method == <br />
:<math>c=1,dx=1/6,dt=0.5dx,t=7.5</math><br />
:<math> \mbox{Long wave :}\frac{r}{dx}=20</math> <br />
:<math> \mbox{Medium wave: }\frac{r}{dx}=6</math> <br />
:<math> \mbox{Short wave : } \frac{r}{dx}=3</math> <br />
=== Space ===<br />
==== Explicit Scheme (DRP)====<br />
:<math> {(\frac{\partial u}{\partial x})}_i=\frac{1.0}{dx}\sum_{k=-3}^3 a_k u_{i+k} </math><br />
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).<br />
====Implicit Scheme(Compact)====<br />
:Domain: <math>\alpha v_{i-1} + v_i + \alpha v_{i+1}=\frac{a}{2h}(u_{i+1}-u_{i-1}) </math><br />
:Boundaries: <math> v_1+\alpha v_2=\frac{1}{h}(au_1+bu_2+cu_3+du_4) </math><br />
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 <br />
:<math> \mbox{Domain:} \alpha=0.25 , a=\frac{2}{3}(\alpha+2) </math><br />
:<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><br />
Both the schemes are 4th order accurate in the domain.The compact scheme has third order accuracy at the boundary.<br />
===Time (4th Order Runga-Kutta)===<br />
:<math>\frac{\partial u}{\partial t}=f </math><br />
<br />
:<math>u^{M+1} =u^M + b^{M+1}dtH^M </math><br />
<br />
:<math> H^M=a^MH^{M-1}+f^M </math><br />
,M=1,2..5 .The coefficients a and b can be found in Williamson(1980)<br />
<br />
== Results ==<br />
[[Image:Initial_condition.png|450px]]<br />
[[Image:Result_wave.png|450px]]<br />
== Reference ==<br />
{{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}}<br />
<br />
{{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}}<br />
<br />
{{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}}<br />
<br />
{{reference-paper|author=Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}}</div>Rangan86