Linear wave propagation
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(Difference between revisions)
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:<math> \mbox{Domain} \alpha=0.25 , a=\frac{2}{3}(\alpha+2) </math> | :<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> | :<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 boundry. | |
| + | ===Time (4th Order Runga-Kutta)=== | ||
== Results == | == Results == | ||
Revision as of 07:17, 14 January 2006
Contents |
Problem definition
Domain
![x=[-10,10]](/W/images/math/1/b/1/1b14c536c4db5da52bbdcfe07f70db81.png)
Initial Condition
![u(x,0)=exp[-ln(2){(\frac{x-x_c}{r})}^2]](/W/images/math/b/3/8/b38fab63c854b1f256b78d925988a116.png)
Boundary condition
Exact solution
![u(x,0)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]](/W/images/math/a/b/0/ab0b51feabf18adf6c4177f6cc507732.png)
Numerical method
Space
Explicit Scheme (DRP)
The coefficients can be found in Tam(1992).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:
- Boundaries:
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
Both the schemes are 4th order accurate in the domain.The compact scheme has third order accuracy at the boundry.
Time (4th Order Runga-Kutta)
Results
