Consider a simple time derivative at a point P:

$$\frac{\mathbf{u}_P^{(n+1)} - \mathbf{u}_P^{(n)}}{\Delta t}$$

The momentum equation, which then is used to construct the coefficient matrix of u_p is discretized using neighbouring cell velocities as:

$$a_p u_p = \sum a_i u_i + S_{up} + S_u$$

The large time steps Δt in a SIMPLE algorithm effectively reduce the time derivative to very small values. The time derivative contributes very directly to the diagonal of a matrix, since the time derivative does not require any neighbouring points, atleast in the way the discretization is shown above. The time derivative is responsible for the diagonal dominance of the matrix, which contributes heavily to the stability of the solution. How does SIMPLE get around not having significant time derivative? — underrelaxation.

You might have heard of iteration schemes with “over-relaxation”. Underrelaxation in SIMPLE is an unrelated concept. It is a way to artificially amplify the diagonal coefficient a_p to achieve diagonal dominance.

The following term (with 1- α) is added to both sides of the discretization equation:

$$a_p u_p + \frac{1 - \alpha}{\alpha} a_p u_p = \sum a_i u_i + S_{up} + S_u + \frac{1 - \alpha}{\alpha} u_p^{(n-1)}$$

You may ask, is it really the same? For a converged solution, the terms are indeed the same, and they cancel out. Underrelaxation needs to take effect for a non-converged solution. Grouping the terms:

$$\frac{1}{\alpha} a_p u_p = \sum a_i u_i + S_{up} + S_u + \frac{1 - \alpha}{\alpha} u_p^{(n-1)} $$

The diagonal coefficient is being effectively ampified via a 1/α multiplier, where α < 1

Lets check the relax function invoked in UEqn.relax()

forAll(D, celli)
{
    D[celli] = max(mag(D[celli ]), sumOff[celli ]);
}

D /= alpha;

D is the diagonal matrix, or the a_p coefficient. Diagonal dominance is first checked when compared to the off-diagonal elements sumOff. The central coefficient is then multiplied by the 1/α multiplier.

This mechanism for stability allows SIMPLE to use the “t=1 second” timestep that we are familiar with. Underrelaxation is not applied to PISO algorithms (with Courant Number <1).

For reference, you set your underrelaxaton factors in fvSolutions

relaxationFactors
{
    equations
    {
        U           0.9;    // 0.9 is more stable but 0.95 more convergent
        ".*"        0.9;    // 0.9 is more stable but 0.95 more convergent
    }
}

References:

1. Jasak, Hrvoje. “Error analysis and estimation for the finite volume method with applications to fluid flows.” (1996)