Suppose I am trying to formulate a multipoint model of fluid dynamics. I have a procedure for doing so the details of which is not important to this question, but only that it is based on a series expansion, the higher order terms in which series being dependent on larger number of spacial points. The equations for the first two terms in a Burger formulation is something like shown below:

\begin{align} &\frac{\partial u^0}{\partial t}(x,t)+\frac{\partial^2 u^0}{\partial x^2}(x,t) + u^0(x,t)\frac{\partial u^0}{\partial x}(x,t)\\ &\qquad\qquad\qquad\quad\;\;+\int_0^1 u^1(x,t;x_1)\frac{\partial u^1}{\partial x}(x,t;x_1) dx=0\\ &\frac{\partial u^1}{\partial t}(x,t;x_1)+\frac{\partial^2 u^1}{\partial x^2}(x,t;x_1)+u^0(x,t)\frac{\partial u^1}{\partial x}(x,t;x_1)+u^1(x,t;x_1)\frac{\partial u^0}{\partial x}(x,t)=0 \end{align}

it is however an odd formulation, the point $x_1$ appears as a mere parameter in the second formulation while it appears only in the integrand of the first equation, so that apparently there is no enough restriction on the behavior of the function $u^1$ with respect to $x_1$, so it seems there s no unique solution for $u_1$. If the flow is a homogeneous flow but $u^1$ can be written as $u^1=u^1(x-x_1,t)$, so that derivation with respect to $x$ will also take into play the point $x_1$, so that $x_1$ will now behave like it is a more important variable. This will get more importance if we further consider the higher order terms like $u^2(x,t;x_1,x_2)$ which in a homogeneous flow will be writable as $u^2(x-x_1,x-x_2,t)$ and whose $x$-derivative becomes: $$\frac{\partial u^2}{\partial x}=\frac{\partial u^2}{\partial(x-x_1)}+\frac{\partial u^2}{\partial(x-x_2)}$$ Such a treatment imports the variables $x_1$, $x_2$ and etc. in the second and higher order equations in a more proper manner, in such a way that hope to find unique solutions increase.

  1. The problem is that how the first formulation which was more general is lame in giving such unique solutions as the second formulation is apparently capable of?

  2. Or maybe there is a must in every multi-point modeling of fluid dynamics (actually turbulence) to consider one of the following two procedures?

    a- To expand the series in terms of functions of the form: $u^0(x,t)$, $u^1(x,x-x_1,t)$, $u^2(x,x-x_1,x-x_2,t)$ and etc. ?

    b- To write the equations once at the point $x$, then once at the point $x_1$ available also in the arguments of $u^1$ and higher order functions, then once at the point $x^2$ available also in the arguments of $u^2$ and higher order functions, and etc., then consider all those equations in one place, for example by adding them together. As the series is truncated somewhere at a function $u^n$ this will not contain infinitely of equations but I am not very hopeful that this is the right path to take.

    • any idea about these or other methods for multipoint formulations?

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