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Wave propagation

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== Introduction ==
== Introduction ==
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The 1-D wave propagation problem is used as a test case for studying the dissipation and dispersion errors in a given finite difference scheme.There are a number of schemes which can be used to solve the problem. The usage of a higher order compact stencil with a low storage 4th order Runga Kutta scheme to solve the current problem is discussed.  
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The 1-D wave propagation problem is used as a test case for studying the dissipation and dispersion errors in a given finite difference scheme.There are a number of schemes which can be used to solve the problem. The usage of a higher order compact stencil with a low storage 4th order Runge-Kutta scheme to solve the current problem is discussed.  
== Compact scheme ==
== Compact scheme ==
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== Runga Kutta ==  
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== Runge-Kutta ==  
Consider  
Consider  
:<math> \frac {\partial U}{\partial t}=H </math>
:<math> \frac {\partial U}{\partial t}=H </math>
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:<math> U^{M+1}=U^M+b^Mdtf^M </math>
:<math> U^{M+1}=U^M+b^Mdtf^M </math>
:<math> f^M=a^Mf^{M-1}+H </math>
:<math> f^M=a^Mf^{M-1}+H </math>
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: where M refers to the stages ,dt is the time step and the coefficients a and b are given by  
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where M refers to the stages ,dt is the time step and the coefficients a and b are given by  
:a[5]={0,-0.41789047,-1.19215169,-1.69778469,-1.51418344}
:a[5]={0,-0.41789047,-1.19215169,-1.69778469,-1.51418344}
:b[5]={0.149665602,0.37921031,0.82295502,0.69945045,0.15305724}
:b[5]={0.149665602,0.37921031,0.82295502,0.69945045,0.15305724}
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== Sample result ==  
== Sample result ==  
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[[Image:wp_result.jpg]]
[[Image:wp_result.jpg]]
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== Reference ==
== Reference ==
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{{reference-paper|author=Williamson, Williamson|year=1980|title=Low Storage Runge-Kutta Schemes|rest=Journal of Computational Physics, Vol.35, pp.48–56}}
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<hr>
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*{{reference-paper|author=Williamson, 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=Lele, Lele, S. K.|year=1992|title=Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics|rest=Journal of Computational Physics, Vol. 103, pp 16–42}}
+
 
 +
*{{reference-paper|author=Lele, Lele, S. K.|year=1992|title=Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics|rest=Journal of Computational Physics, Vol. 103, pp 16–42}}

Revision as of 03:37, 20 September 2005

Contents

Introduction

The 1-D wave propagation problem is used as a test case for studying the dissipation and dispersion errors in a given finite difference scheme.There are a number of schemes which can be used to solve the problem. The usage of a higher order compact stencil with a low storage 4th order Runge-Kutta scheme to solve the current problem is discussed.

Compact scheme

Runge-Kutta

Consider

 \frac {\partial U}{\partial t}=H

The low storage scheme is implemented as follows

 U^{M+1}=U^M+b^Mdtf^M
 f^M=a^Mf^{M-1}+H

where M refers to the stages ,dt is the time step and the coefficients a and b are given by

a[5]={0,-0.41789047,-1.19215169,-1.69778469,-1.51418344}
b[5]={0.149665602,0.37921031,0.82295502,0.69945045,0.15305724}

Sample result

Wp result.jpg

Reference

  • Williamson, Williamson (1980), "Low Storage Runge-Kutta Schemes", Journal of Computational Physics, Vol.35, pp.48–56.
  • Lele, Lele, S. K. (1992), "Compact Finite Difference Schemes with Spectral-like Resolution,” Journal of Computational Physics", Journal of Computational Physics, Vol. 103, pp 16–42.
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