LUSGS understanding

Gustavo

I've been reading a bit of LUSGS schemes on Blazek's book and I'm facing some difficulties to understand it ...perhaps someone can enlight me...

On LUSGS we basically want to factorise (on three matrixes) the implicit operator (A) on the deltaW, like (D+L).D^(-1).(D+U) being L a lower triangular, U an upper triangular e D a diagonal. These (L, D, U) have different definitions for structured and unstructured meshes. In a general way, calling R the residual, we could split the equation A.deltaW=-R, where A stands for the implicit and R(residual) the explicit operators, in two parts. These would be

D.deltaW(1-ijk) =-R(n-ijk) - LdeltaW(1) D.deltaW(n-ijk)=D.deltaW(1-ijk)-UdeltaW(n)

If I have correctly understood, as we are marching the solution on space by layers (i+j+k=constant, or layer composition on unstructured grids) we first solve the first equation for the whole grid as a forward step, as we would already have LdeltaW(1) from the previous steps, discovering deltaW(1-ijk) (just having to invert matrix D for this), and then we solve the second on a similar way but sweeping the whole grid on the opposite direction as now we would already have UdeltaW(n), from previous steps, and D.deltaW(1-ijk) from the previous equation. First of all, is this correct, is this the way it works ?

If so, then on the beggining of the first equation calculation, let's say i+j=0 , how would I have LdeltaW(1) , the same for the second with UdeltaW(n) ?? From boundary conditions, perhaps ??