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Burgers' equation is a fundamental PDE used in various fields such as number theory, gas dynamics, heat conduction, elasticity, etc. It is crucial especially for developing numerical models for turbulence since nonlinear viscous Burgers' equation can be considered as an approach to the Navier-Stokes equations. It can be stated as $$\frac{\partial u}{\partial t} + \frac{\partial}{\partial x} \left(\frac{u^2}{2}\right) = \nu \frac{\partial^2 u}{\partial x^2}$$ where $a<x<b$ , $t>0$ with appropriate initial and boundary conditions given. (it can be determined later)

Let assume that the finite difference numerical method are considered to obtain approximate solution of this viscous equation. In my perspective, there are two decisions need to be made to have stable solutions

  1. What kind of finite difference method can be considered for inviscid and viscous part (or both of them) i.e., Lax-Wendroff, Lax Ricthmyer, Crank-Nicholson, Lax-Friedrich's, Roe's scheme ?
  2. How one can decide the value of the artificial diffusion number ($\nu$) ? Should we consider its relation with Reynolds number since it is related with the Navier-Stokes equations ?

Note : This viscous Burgers' equation is not going to be considered as linearized. It will be considered in a nonlinear form.

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  • $\begingroup$ This is such a common problem that there is a vast number of papers and books that cover all of the questions you have. Where have you already looked? $\endgroup$ – Wolfgang Bangerth Jan 12 '17 at 4:13
  • $\begingroup$ @WolfgangBangerth the books that I have already look are "Numerical Methods for Conservation Laws", "Finite Volume Methods for Hyperbolic Problems" by R.J. LeVeque. I also would like to know some review paper which can be benificial for my questions. In your knowledge, if you know some good review papers could you please refer them ? $\endgroup$ – Loading... Jan 12 '17 at 8:13
  • $\begingroup$ It's not quite my area, but it really shouldn't be very hard to find something on google scholar, for example. $\endgroup$ – Wolfgang Bangerth Jan 12 '17 at 18:27
  • $\begingroup$ @WolfgangBangerth it is not the answer of my question. I am aware of web search tools but thanks for trying anyway. $\endgroup$ – Loading... Jan 13 '17 at 10:50
  • $\begingroup$ The books you mentioned both address your questions to varying levels of detail so I am a bit confused as to what you are looking for. Do you want a clarification on something presented there? $\endgroup$ – Kyle Mandli Jan 16 '17 at 3:21

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