# Unsteady flow calculation parameters

### Discretization settings

**Discretization method:**

- The method called "Classic" is the one described in the theoretically documentation.
- The "homogeneous permanent" method limits the "waves" that can appear when starting a unsteady flow calculation from a water line computed by the steady flow calculation (Fluvia). Indeed, in this case, the discretization of the transient equations gives exactly the equations of the steady state when one removes the derivative terms with respect to time. It is less conventional because it can lead to not very normal hydraulic phenomena in the case of change of geometry (cf Cunge book 1980). The small "waves" mentioned below can be explained by the fact that the stabilization regime of the classical scheme does not lead exactly to the initial permanent water line. The new equilibrium will not be very far, but to achieve this, a balancing of levels and therefore mass transfers will be realized.

**Initialisation method:**

- Zero Variation: for a given time step, the calculation of the new water line is initialized from the water line calculated at the previous time.
- Previous Variation: The calculation of the new water line is initialized from the water line calculated at the previous time at which we apply the flow and water elevation variations observed in the previous time step. This option can accelerate the convergence if hydraulic variables are often progressive. However, our experience indicates that convergence is very fast, and this option is not essential.

**Theta coefficient of the Preissmann scheme:**

This coefficient sets the implicit elements of the numerical scheme. It should be set between 0.5 (stability limit) and 1 (total implicitation). If this condition is met, the Preissmann implicit scheme is unconditionally stable. It is set to 0.6 by default. It is possible to modulate it automatically according to the Froude (cf below). This can be useful because a high Theta coefficient will generate more numerical dissipation and thus limit the numerical oscillations that can appear with high Froude. Putting it large (close to 1) all the time is not very good numerically, so the option to set it variable, just according to the Froude, is interesting.

Note: by default HEC-RAS advises to start modeling with Theta=1, which is quite violent anyway, because it generates a lot of numerical dissipation.

**Simple super critical**

This option allows, in the case of classical discretization as well as the homogeneous to steady flow discretization, to gradually eliminate the inertia terms of the dynamic equation of Saint-Venant when the Froude number exceeds a certain value and for example becomes close to 1. This is a classical method to better manage high Froude numbers, as well as supercritical flows in some cases. This stems from the fact that it is these terms that generate numerical problems in the Preissmann scheme at high Froude numbers. We delete these terms progressively and linearly: 0% for Fr = FrMin (default 0.6) up to 100% for Fr >= FrMax (default 0.9). The default values of FrMin and FrMax can be changed in the advanced settings of this option ("Detail" button). This option is to be used with moderation because it presents certain risks of numerical stability with Froude high and especially with low water depths.

- No: The option is not applied and the calculation is stopped as soon as the Froude is greater than or equal to 1 on one of the sections of the model.
- Yes local: The option is only applied to sections with high Froude, at the moment it happens and gradually as explained above.
- Global Yes: The option is applied to all sections of the model and removing 100% of the selected terms during all the simulation time.
- Sart-Preissmann: more advanced method with modification of the Preissmann scheme cf article Sart et al 2010 (http://hal.ird.fr/hal-00632009/document).

One can also, in the advanced options ("Detail" button), choose the term(s) to be deleted: the local acceleration term ${dQ/dt}$, and/or the term d convective acceleration ${\frac{d\frac{Q^2}{S}}{dx}}$. If we remove these two terms, we have the model of the diffusive wave. Older versions of SIC (prior to 5.37b) allowed to remove only the second term, convective acceleration which is the one that poses the problems of numerical stability at high Froude numbers. These terms are progressively removed locally or globally (depending on the option chosen above). For the local option, this or these terms are removed for Froude values between FrMin (0.6 by default) and FrMax (0.9 by default), but nothing prevents to put FrMin=0 and FrMax=0, which implies that these terms will always be removed whatever the Froude values. In this case we have an implementation of the diffusive wave model. The global option will give the same thing. Attention in case of variable Theta, this reduction will also apply on this coefficient Theta, in this case one would have then Theta=ThetaMax.

We currently do not allow to remove the term ${\frac{dy}{dx}}$ , which would then lead to the kinematic wave model. See for example http://www.agroparistech.fr/coursenligne/hydraulique/degoutte1.pdf for information on these simplified variants of the Saint-Venant equations.

Another option is to propose a variable Preissmann Theta implicitation coefficient, increasing when the Froudes are high. It is varied with the same logic as for the terms of inertia, locally, and linearly between Theta and ThetaMax depending on the Froude for the local option. For the global option one has in this case systematically Theta = ThetaMax, everywhere and all the time.

Another activatable option is to recalculate the torrential water line in steady state. Flow rates and water elevations are first calculated with the conventional method, and then the water elevations are recalculated with the steady state algorithm. This option is to be used without reserve and without guarantee, especially in the case of meshed and/or branched networks, and is no longer conservative in volume, since the dimensions of the water line are recalculated.

### Numerical resolution method

The calculation can be done in linear (without iterations) or non-linear with the Newton or quasi-Newton method (the derivative is not calculated at each iteration). In the case of non-linear calculation, the iterations convergence criteria should be defined:

**Precision of the non linear system**

The convergence of the linear system is tested on flows of structures and boundary conditions. The iterations are also made on the Saint-Venant equations.

The calculation succeeds when the deviations of flow and water elevation are below the requested precisions. The requsted precisions can be defined in absolute terms (in m^{3}/s for flows and in m for water elevations) or in relative values (in proportion of the flow rate or in proportion of the water elevation with reference to the lowest elevation in the studied model).

This test can be very demanding if the precision is set to a small value (eg.: 0.0001 m^{3}/s). On systems with significant flow rates, it is desirable, even mandatory, to increase it. For example, the Rhône, which has flows of about 2000 m^{3}/s the system will fail to converge to an accuracy of 0.0001 m^{3}/s. However with a value of 0.01 m^{3}/s the calculation is done smoothly and with a largely sufficient accuracy. If you have any messages indicating that convergence is not possible, decrease the coefficient within reasonable limits (ex .: 0.1% of nominal capacity).

**Max iterations of the non-linear system**

Maximum number of iterations to converge the system. If a significant accuracy is required on a system having difficulties to converge, it may be necessary to increase it.

Note that if the system fails to converge, SIRENE restarts the calculation again by halving the time step of calculation. This is repeated if necessary until divide by 8 the time step calculation chosen by the user.

**Derivative calculation frequency (Quasi-Newton)**

- 0: the calculation is linear (without iterations).
- 1: the calculation is performed by non-linear with Newton’s method
- >1: the calculation is performed by quasi-Newton (the calculation of the derivative is not performed at each iteration)

**Do not stop the calculation in case of non convergence**

By default, if the calculation did not reach the precision required after the maximum number of iterations and after the divisions of the time step of calculation, the simulation is stopped. Enable this option allows in this case to only display a warning message and switch to the calculation of the next time step.

### Write in all calculation sections (to allow a hot start unsteady flow computation)

If selected, we can use the results of the simulation as the initial condition for a new unsteady flow simulation. For networks with many sections of calculation, it may be appropriate to increase the write frequency results in order not to burden the XML file and associated treatments (see Time Settings).