4 Computational Modelling in Hydraulic and Coastal Engineering
2. It has to be convergent, that is, the numerical solution must tend
asymptotically towards the analytical solution, as the discretization
steps (Δx, Δy, Δz, Δt) tend to zero. A non-convergent method is of no
practical use.
3. It has to be numerically stable. For stable methods, the inevitably
introduced errors during the solution procedure do not increase indefinitely but decay and become negligible after some solution steps.
1.1.1.3 Reliability of mathematical models
Using a numerical method and producing just a solution for a mathematical model is not the ultimate goal. A full application of a mathematical
model requires involvement of three subsequent phases of (1) calibration,
(2) verification and (3) validation. Model calibration is the quantitative determination of the model parameters. Model parameters are certain unknown
variables and physical sub-processes that need to be identified a priori for
the model to be operable. The variables are mostly expressed as lumped constants or known mathematical expressions. Regarding the sub-processes
that are not described in detail by the model in order to avoid unnecessary
computational complexity, they are approximated by parameters or relationships, taking specific values under specific conditions. Determination
of the model parameters is based on available data sets of input–output
values obtained from relevant physical models, field measurements or any
available analytical solutions of the model. Those data are used only for
the calibration phase of the model. A typical example of parameterization in fluid flows is the determination of the wall friction coefficient. This
coefficient quantifies the effects of the boundary layer of the velocity profile. Thus, by excluding this layer from the model, the model solves for the
bulk flow velocity outside of the boundary layer, while the boundary-layer
effects are expressed by a wall-friction coefficient (e.g. Darcy-Weisbach
friction coefficient).
Model verification is the proof of model truthfulness. Verification is conducted by using sets of known input–output data, different from those used
for calibration, which should also be reproduced by the calibrated model.
Finally, model validation is the explicit recognition and delineation of
the limits of model applicability, so that the users do not use the model outside those limits, because they may obtain non-realistic results. As already
mentioned, the model formulation is based on simplifying approximations
and parameterizations. Those assumptions impose (implicitly) the limits of
model applicability, beyond which the assumptions for the model formulation are no longer valid. For example, if the model is based on the neglect
of nonlinear terms (linearized model) the model is not valid for applications
where the nonlinear terms become significant.
2. It has to be convergent, that is, the numerical solution must tend
asymptotically towards the analytical solution, as the discretization
steps (Δx, Δy, Δz, Δt) tend to zero. A non-convergent method is of no
practical use.
3. It has to be numerically stable. For stable methods, the inevitably
introduced errors during the solution procedure do not increase indefinitely but decay and become negligible after some solution steps.
1.1.1.3 Reliability of mathematical models
Using a numerical method and producing just a solution for a mathematical model is not the ultimate goal. A full application of a mathematical
model requires involvement of three subsequent phases of (1) calibration,
(2) verification and (3) validation. Model calibration is the quantitative determination of the model parameters. Model parameters are certain unknown
variables and physical sub-processes that need to be identified a priori for
the model to be operable. The variables are mostly expressed as lumped constants or known mathematical expressions. Regarding the sub-processes
that are not described in detail by the model in order to avoid unnecessary
computational complexity, they are approximated by parameters or relationships, taking specific values under specific conditions. Determination
of the model parameters is based on available data sets of input–output
values obtained from relevant physical models, field measurements or any
available analytical solutions of the model. Those data are used only for
the calibration phase of the model. A typical example of parameterization in fluid flows is the determination of the wall friction coefficient. This
coefficient quantifies the effects of the boundary layer of the velocity profile. Thus, by excluding this layer from the model, the model solves for the
bulk flow velocity outside of the boundary layer, while the boundary-layer
effects are expressed by a wall-friction coefficient (e.g. Darcy-Weisbach
friction coefficient).
Model verification is the proof of model truthfulness. Verification is conducted by using sets of known input–output data, different from those used
for calibration, which should also be reproduced by the calibrated model.
Finally, model validation is the explicit recognition and delineation of
the limits of model applicability, so that the users do not use the model outside those limits, because they may obtain non-realistic results. As already
mentioned, the model formulation is based on simplifying approximations
and parameterizations. Those assumptions impose (implicitly) the limits of
model applicability, beyond which the assumptions for the model formulation are no longer valid. For example, if the model is based on the neglect
of nonlinear terms (linearized model) the model is not valid for applications
where the nonlinear terms become significant.