Linear wave propagation

(Difference between revisions)
 Revision as of 06:25, 14 January 2006 (view source)← Older edit Revision as of 07:14, 14 January 2006 (view source)Newer edit → Line 3: Line 3: [/itex] [/itex] == Domain == == Domain == - x=[0,1] + :$x=[-10,10]$ == Initial Condition == == Initial Condition == - :$u(x,0)=e^{-360*{(x-0.25)}^2}$ + :$u(x,0)=exp[-ln(2){(\frac{x-x_c}{r})}^2]$ == Boundary condition == == Boundary condition == - u[0]=0,u[imax]=u[imax-1](x[imax]=1.0) + :$u(-10)=0$ == Exact solution == == Exact solution == - :$u(x,t)=e^{-360*{((x-c*t)-0.25)}^2}$ + :$u(x,0)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]$ == Numerical method == == Numerical method == - c=1,t=0.25 + :$c=1,dx=1/6,dt=0.5dx,t=7.5$ + :$\mbox{Long wave :}\frac{r}{dx}=20$ + :$\mbox{Medium wave: }\frac{r}{dx}=6$ + :$\mbox{Short wave : } \frac{r}{dx}=3$ + === 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 + :$\mbox{Domain} \alpha=0.25 , a=\frac{2}{3}(\alpha+2)$ + :$\mbox{Boundary} \alpha=2 ,a=-(\frac{11+2\alpha}{6}),b=\frac{6-\alpha}{2},c=\frac{2\alpha-3}{2},d=\frac{2-\alpha}{6}$ + == Results == == Results == + [[Image:Initial_condition.png|450px]] [[Image:Initial_condition.png|450px]] [[Image:Result_wave.png|450px]] [[Image:Result_wave.png|450px]] == Reference == == Reference ==

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

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