Linear wave propagation

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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).