# Linear wave propagation

(Difference between revisions)
 Revision as of 07:14, 14 January 2006 (view source)← Older edit Revision as of 07:17, 14 January 2006 (view source)Newer edit → Line 27: Line 27: :$\mbox{Domain} \alpha=0.25 , a=\frac{2}{3}(\alpha+2)$ :$\mbox{Domain} \alpha=0.25 , a=\frac{2}{3}(\alpha+2)$ :$\mbox{Boundary} \alpha=2 ,a=-(\frac{11+2\alpha}{6}),b=\frac{6-\alpha}{2},c=\frac{2\alpha-3}{2},d=\frac{2-\alpha}{6}$ :$\mbox{Boundary} \alpha=2 ,a=-(\frac{11+2\alpha}{6}),b=\frac{6-\alpha}{2},c=\frac{2\alpha-3}{2},d=\frac{2-\alpha}{6}$ - + 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 ==

## Problem definition

$\frac{\partial u}{\partial t}+ c \frac{\partial u}{\partial x}=0$

## Domain

$x=[-10,10]$

## Initial Condition

$u(x,0)=exp[-ln(2){(\frac{x-x_c}{r})}^2]$

## Boundary condition

$u(-10)=0$

## Exact solution

$u(x,0)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]$

## Numerical method

$c=1,dx=1/6,dt=0.5dx,t=7.5$
$\mbox{Long wave :}\frac{r}{dx}=20$
$\mbox{Medium wave: }\frac{r}{dx}=6$
$\mbox{Short wave : } \frac{r}{dx}=3$

### Space

#### Explicit Scheme (DRP)

${(\frac{\partial u}{\partial x})}_i=\frac{1.0}{dx}\sum_{k=-3}^3 a_k u_{i+k}$

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: $\alpha v_{i-1} + v_i + \alpha v_{i+1}=\frac{a}{2h}(u_{i+1}-u_{i-1})$
Boundaries: $v_1+\alpha v_2=\frac{1}{h}(au_1+bu_2+cu_3+du_4)$

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

$\mbox{Domain} \alpha=0.25 , a=\frac{2}{3}(\alpha+2)$
$\mbox{Boundary} \alpha=2 ,a=-(\frac{11+2\alpha}{6}),b=\frac{6-\alpha}{2},c=\frac{2\alpha-3}{2},d=\frac{2-\alpha}{6}$

Both the schemes are 4th order accurate in the domain.The compact scheme has third order accuracy at the boundry.