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Linear wave propagation

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== Problem definition ==
== Problem definition ==
:<math> \frac{\partial u}{\partial t}+ c \frac{\partial u}{\partial x}=0
:<math> \frac{\partial u}{\partial t}+ c \frac{\partial u}{\partial x}=0
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{{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=SK Lele|year=1992|title=Compact finite difference schemes with spectrum-like resolution|rest=Journal of Computational Physics, Vol.103, pp.16-42}}
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{{reference-paper|author=Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}}
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{{reference-paper|author=Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}}

Latest revision as of 09:55, 17 December 2008

Contents

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

Initial condition.png Result wave.png

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.

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