Which one of the following is suitable for judging convergence in Two-Fluid Flow Solver?

1) Absolute Residual (L^2-Norm).

2) Relative Residual.

3) Fraction Change in Velocity, Pressure and Volume Fraction.

I am asking this question because due to large value of momentum exchange coefficients one can get very high value of absolute residual even when solutions are close to actual value.

Is there any other criterion used?

  • $\begingroup$ It seems there may be two parts to your question: 1) whether the numerical convergence of an iterative method has reached an acceptable accuracy, or 2) whether the fluid flow has reached a steady state (as your part 3 suggests). The third answer by @NamRakes addresses your last part, and the other two answers seem to be options for your first two parts. $\endgroup$
    – Charles
    Commented Apr 25, 2016 at 7:02

1 Answer 1


Normally in any numerical solution of PDEs, you have the conjugate pairs, say, (displacement, force), OR (pressure, velocity) that contribute to the energy of the system. Let's notate these abstractly as (degree of freedom, its conjugate), or (DOF, CJG). In order to cover a wide range of application, one should check for convergence of

  1. Relative residual (of the conjugate) values against some reference value, say L2-norm of the residual in system,

  2. Iterate value of the DOF against some reference value, could be L2 or L1-norm of DOF in the system, and

  3. Verify that $\Delta $Energy due to "incremental DOF times residual" is small compared to, say, the energy input to the system.

1 verifies that the residual goes to zero relative to some norm, 2 helps when large changes in DOFs are seen even at low values of residuals, and 3 ensures there is energy balance in the system.

It is very rare to check for absolute value of residuals (at least in solid mechanics, my field).


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