I am trying to simulate the following system. $$\frac{\partial C}{\partial t} = D\frac{\partial ^2 C}{\partial x^2}-v\frac{\partial C}{\partial x}$$ with initial condition $$c(x,0) = C_o$$

and Neumann boundary condition. When the system under consideration is discretized, initial concentration has to be defined at all nodes.

Instead of defining equal concentration at all nodes, is there any alternative method to define the initial concentrations?

  • 1
    $\begingroup$ More information is required. If you don't want the initial condition to be a constant value for all nodes, you could maybe set it to some spatially varying function, i.e. $c(x,0) = C_0(x)$. But the exact form of that function depends on the problem you want to solve. $\endgroup$ – Savithru Dec 27 '18 at 15:25
  • $\begingroup$ @Savithru The system is the transport of species like glucose in the blood vessel network. Would it be correct to consider a Gaussian distribution? $\endgroup$ – Natasha Dec 28 '18 at 3:32
  • $\begingroup$ "is there any alternative method to define the initial concentrations?", yes, but that's a modeling choice for the modeler to make and justify. Any choice works if you justify that it's realistic. $\endgroup$ – Chris Rackauckas Dec 28 '18 at 16:52
  • $\begingroup$ @ChrisRackauckas In the literature, often all the nodes are set to equal concentration $C(x,0) = C_0$. If $C(x,0) = C_0(x)$ would it be reasonable to consider Gaussian distribution, since, in biology, many properties follow Gaussian distribution $\endgroup$ – Natasha Dec 29 '18 at 13:31
  • $\begingroup$ Gaussian distributions naturally arise from diffusive processes that start with all of the concentration at a single point source. If that makes sense for your model, then sure. $\endgroup$ – Chris Rackauckas Dec 29 '18 at 14:10

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