- #1

popiol

- 1

- 0

_{s}x [0,t

_{max}]), so the transport equation looks generally like this

div (

*u*[

*v*; 1 ] -

*D*[ grad

_{s}

*u*; 0 ] ) =

*Q*,

where

*u*is either density (of one of the species), velocity, temperature or pressure;

*v*is velocity;

*D*is diffusion coefficient; grad

_{s}is gradient over the space domain Ω

_{s}; and

*Q*is either reaction rate, pressure gradient plus buoyancy force (-grad

_{s}p + f

_{b}), energy release or 0.

The approximation scheme is

*u*

_{i}= Σ

_{j<i}(

*D*(

*d*

_{ij}

^{-1}-

*d*

_{i.}

^{-1}) - [

*v*

_{j}; 1 ] . (

*x*

_{j}-

*x*

_{i}) /

*d*

_{ij})

*w*

_{ij}

*u*

_{j}

where

*u*

_{i}is a value in the

*i*-th mesh node;

*d*

_{ij}= ||

*x*

_{j}-

*x*

_{i}||;

*d*

_{i.}

^{-1}= Σ

_{j<i}

*w*

_{ij}/

*d*

_{ij}; Σ

_{j<i}

*w*

_{ij}= 1; and

*x*

_{i}is a mesh node in Ω. Mesh nodes are sorted by time, so t(

*x*

_{i}) > t(

*x*

_{j}) =>

*i*>

*j*.

It is a bit difficult to define stability in this case, but the following condition seems reasonable

lim

_{i -> ∞}(

*u*

_{i}- Σ

_{j<i}

*u*

_{j}

*w*

_{ij}) = 0,

which implies

lim

_{i -> ∞}

*D*(

*d*

_{ij}

^{-1}-

*d*

_{i.}

^{-1}) - [

*v*

_{j}; 1 ] . (

*x*

_{j}-

*x*

_{i}) /

*d*

_{ij}= 1.

It should also be true that for each

*j*> 0

Σ

_{i>j}(

*D*(

*d*

_{ij}

^{-1}-

*d*

_{i.}

^{-1}) - [

*v*

_{j}; 1 ] . (

*x*

_{j}-

*x*

_{i}) /

*d*

_{ij})

*w*

_{ij}= 1

Any idea how to verify those conditions?