I am testing the convergence property of an approach. The initial conditions are likely to a real case(e.g. 10GPa in left and 100MPa in right for the one-dimensional Riemann problem). Due to inherent numerical diffusion across shock front, the shock front is shifted a little bit, 3~4 grids.

Thus, the relative errors may be large.

My question is how to apply the L1norm and L2norm in the CFD when the numerical diffusion arises.

L1norm was implemented by

DO i=1,nx

rL1norm=rL1norm+ABS( u0(0,i,1)-ua0(i,1) ) END DO

L2norm was

DO i=1,nx

rL2norm=rL2norm+((ue(0,i,1)-u0(i,1))*(ue(0,i,1)-u0(i,1))) END DO rL2norm=SQRT(rL2norm)/nx

In case of L1norm, the error values are pretty high compared with other papers mentioning the L1norm is about 10^-3 order.

In my case, L1norm was about 10^3(e.g. 127000...).

In the CFD, is there any other definition of L1norm compared with the vector norm?

Dear Jinwon,

The norm of error can be defined in numerous ways. Specifically, a suitable error norm that works for both regular and irregular grids is to use the following area(or volume) weighted norm, implemented as :

norm = 0.0 totvol = 0.0

DO i=1,nx

norm = norm +(abs(ue(0,i,1)-u0(i,1)))*volume(i)

totvol = totvol + volume(i)

END DO

norm = norm/totvol

It is easy to see that for regular grids where the cell volumes are identical this reduces to \sigma(abs(error(i))/nx, which is the discrete L1 norm of error.

Hope this helps,

Regards,

Ganesh

[Nishu]

Hi Jinwon,
Were you able to solve Riemann problem for the real case?
reply me at babuu.nishu@gmail.com