Linear wave propagation

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Latest revision as of 09:55, 17 December 2008

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

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

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(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).

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 boundary.

Time (4th Order Runga-Kutta)

$\frac{\partial u}{\partial t}=f$

$u^{M+1} =u^M + b^{M+1}dtH^M$

$H^M=a^MH^{M-1}+f^M$

,M=1,2..5 .The coefficients a and b can be found in Williamson(1980)

Results

Reference

Mihaela Popescu, Wei Shyy , Marc Garbey (2005), "Finite volume treatment of dispersion-relation-preserving and optimized prefactored compact schemes for wave propagation", Journal of Computational Physics, Vol. 210, pp. 705-729.

Tam and Webb (1993), "Dispersion-relation-preserving finite difference schemes for computational acoustics", Journal of Computational Physics, Vol. 107, pp. 262-281.

SK Lele (1992), "Compact finite difference schemes with spectrum-like resolution", Journal of Computational Physics, Vol.103, pp.16-42.

Williamson (1980), "Low Storage Runge-Kutta Schemes", Journal of Computational Physics, Vol.35, pp.48–56.

@@ Line 11: / Line 11: @@
 == Exact solution ==
-:<math> u(x,0)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]</math>
+:<math> u(x,t)=exp[-ln(2){(\frac{x-x_c-ct}{r})}^2]</math>
 == Numerical method ==
 :<math>c=1,dx=1/6,dt=0.5dx,t=7.5</math>
@@ Line 20: / Line 21: @@
 ==== Explicit Scheme (DRP)====
 :<math> {(\frac{\partial u}{\partial x})}_i=\frac{1.0}{dx}\sum_{k=-3}^3 a_k u_{i+k} </math>
-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).
+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).
 ====Implicit Scheme(Compact)====
 :Domain: <math>\alpha v_{i-1} + v_i + \alpha v_{i+1}=\frac{a}{2h}(u_{i+1}-u_{i-1}) </math>
 :Boundaries: <math> v_1+\alpha v_2=\frac{1}{h}(au_1+bu_2+cu_3+du_4) </math>
 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
-:<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 boundary.
+===Time (4th Order Runga-Kutta)===
+:<math>\frac{\partial u}{\partial t}=f </math>
-== Results ==
+:<math>u^{M+1} =u^M + b^{M+1}dtH^M </math>
+:<math> H^M=a^MH^{M-1}+f^M </math>
+,M=1,2..5 .The coefficients a and b can be found in Williamson(1980)
+== Results ==
 [[Image:Initial_condition.png|450px]]
 [[Image:Result_wave.png|450px]]
 == Reference ==
+{{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}}
+{{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}}
+{{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}}
+{{reference-paper|author=Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}}