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Non linear wave propagation

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:<math> \frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}=0
:<math> \frac{\partial u}{\partial t}+ u \frac{\partial u}{\partial x}=0
</math>
</math>
 +
== Domain ==  
== Domain ==  
-
x=[-5,10]
+
:<math>x \in \left[-5,10\right]</math>
 +
 
== Initial Condition ==  
== Initial Condition ==  
-
:<math> u(x,0)=0 ,x <=0 </math>
+
:<math>u(x,0) =  
-
:<math> u(x,0)=1 ,x >0 </math>
+
\begin{cases}
 +
0 & x \le 0 \\
 +
1 & x > 0
 +
\end{cases}
 +
</math>
 +
 
== Boundary condition ==  
== Boundary condition ==  
-
u[0]=0
+
:<math>u(0,t)=0</math>
== Exact solution ==
== Exact solution ==
-
:<math> u(x,t)=0 ,x<=0 </math>
+
:<math>u(x,t) =  
-
:<math> u(x,t)=x/t, 0<x<t </math>
+
\begin{cases}
-
:<math> u(x,t)=1.0 ,\mbox{otherwise} </math>
+
0 & x \le 0 \\
 +
x/t & 0 < x < t \\
 +
1 & \mbox{otherwise}
 +
\end{cases}
 +
</math>
 +
 
== Numerical method ==  
== Numerical method ==  
 +
=== Space ===
 +
==== 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(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{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>
 +
 +
:<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 ==
== Results ==
-
[[Image:Nonlinear_1d.png]]
+
[[Image:Non_linear_coarse.png]]
 +
[[Image:Non_linear_medium.png]]
 +
[[Image:Non_linear_fine.png]]
== Reference ==
== 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}}

Latest revision as of 14:03, 19 July 2007

Contents

Problem definition

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

Domain

x \in \left[-5,10\right]

Initial Condition

u(x,0) = 
\begin{cases}
0 & x \le 0 \\
1 & x > 0
\end{cases}

Boundary condition

u(0,t)=0

Exact solution

u(x,t) = 
\begin{cases}
0 & x \le 0 \\
x/t & 0 < x < t \\
1 & \mbox{otherwise}
\end{cases}

Numerical method

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

Non linear coarse.png Non linear medium.png Non linear fine.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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