# Glossary

*Advection-Diffusion equation*

**Advection–Diffusion equation** (cited from Wikipedia^{[15]}) is a partial differential equation, which describes physical phenomena where particles or energy (or other physical quantities) are transported inside a physical system due to two processes: advection and diffusion.

*Algorithm*

**Algorithm** (cited from Wikipedia^{[18]}) is a sequence of instructions, often used for calculation and data processing. It is a type of effective method in which a list of well-defined instructions for completing a task will, when given an initial state, proceed through a series of successive states, eventually terminating in an end-state.

*Asymptote*

In analytic geometry, an **asymptote** (cited from Wikipedia^{[33]}) of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. In some contexts, such as algebraic geometry, an asymptote is defined as a line which is tangent to a curve at infinity.

*Backtracking*

**Backtracking** is a general algorithm that, given a descent direction, will seek the increment , minimizing the residual.

*Boundary conditions*

**Boundary conditions** are conditions, which must be satisfied by the degrees of freedom on the boundary of the simulation domain^{[8]}.

*Connectivities*

**Connectivities** of a mesh are the ordered lists of all the node numbers belonging to an element.

*Convergence criterion and stopping criterion*

A simulation has the objective of determining some quantity such as velocities, water levels or concentrations. When numerical iteration methods are employed, the convergence of the simulation is defined by acceptable errors:

- on parameters relevant to the solver, like the residual or the increment over an iteration;

- on the physical values, where one could tolerate a higher error on the velocities than on the water level. It is often the case that certain quantities may reach convergence at a different rate than other quantities.

The former is the **convergence criterion**, the latter is the **stopping criterion**.

*Degrees of freedom*

**Degrees of freedom** represent the unknowns of the problem for which one seeks a solution. Depending on the problem, they will refer to concentrations, to sediment transport rates, to water level, to specific discharges, etc…

*Delta T*

**ΔT**, **Delta T**, **delta-T**, **deltaT**, or **DT** is a time increment.

*Direct resolution method (Direct solver)*

A **direct solver** is a matrix resolution algorithm (method), which computes the solution in a single step^{[8]}.

*Drying/Wetting*

**Drying/Wetting** is an expression describing the capacity of the hydrodynamic model to predict the zones covered and uncovered by water ^{[8]}.

*Elements*

**Elements** are simple geometric forms such as cubes for 3D, triangles and quadrilaterals for 2D and lines for 1D. The collection of elements is called a mesh^{[2]}.

*Error*

- __Error introduced by the mathematical model __

So as to approach a physical system, one uses approximation. In fact, a physical system *u _{ex}*(

*x*) is mathematically approached by

*u*(

*x*) and it is rare to find exactly

*u*(

*x*) =

*u*(

_{ex}*x*).The

**error**is the distance between the exact mathematic model and the behavior of the physical system, which is represented by the following equation:

where || represents the chosen __norm__*, u*_{ex} represents the physical system and* u*, the approximation of *u*_{ex}*.*

- __Approximation ____error ____on one element__

The approximation **error** is related to the element size and to the approximation degree. It is bounded by the following inequality:

where *h* is the mesh size, *c* a constant unrelated to approximation, and *α*, a constant related to the approximation degree. In order to decrease the error for the linear element , one can lower *h* or increase the approximation order.

*Exner equation*

The **Exner equation** (cited from Wikipedia^{[32]}) is a statement of conservation of mass that applies to sediment in a fluvial system such as a river. It was developed by the Austrian meteorologist and sedimentologist Felix Maria Exner, from whom it derives its name.

*Finite element mesh*

A **finite element mesh** is a homogeneous assembly of identical or complementary geometric entities of one, two or three dimensions (elements), made of a finite number of nodes placed inside or on the contour of the domain. It is a discretization of the computational domain^{[8]}.

*Finite element method (FEM)*

**Finite Element method** (FEM) is a numerical technique used to solve partial differential equations. The FEM is based on discretizing a domain into elements (and nodes) and constructing basis (or interpolation) functions across the elements. From these functions, a system of linear or non-linear equations is generated and solved on the computer^{[9]}. This unique homogeneous structure of calculation, gathers all the data of the problem. In H2D2, the meshes are unstructured, composed of triangles of different sizes, well-suited to describe with ease complex geometries and to be refined where needed.

*Formulation*

**Formulation** is the problem definition type.

*Global properties*

**Global properties** are properties that do vary neither in time, nor in space^{[8]}.

*Homotopy*

In topology, two continuous functions from one topological space to another are called **homotopic** (cited from Wikipedia^{[35]}) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

In H2D2, it is used to describe the load path on the system, from an initial solution to the final solution.

*Interpolation*

**Interpolation** estimates a value at a point *p*, as the product of given interpolation functions and of known values at specific points^{[9]}. Finite element interpolation is defined on elements as the product of basis functions and the value of the functions at nodes.

*Iterative resolution method*

I**terative resolution method** is the resolution procedure of solving the equation system in several successive steps called **iterations**^{[8]}.

*Jacobian matrix*

**Jacobian matrix** (cited from Wikipedia^{[30]}) is, in vector calculus, the matrix of all first-order partial derivatives of a vector-valued function.

*Matrix system*

M**atrix system** (cited from Wikipedia^{[24]}) is an equation system expressed in a compact way to form a single matrix equation on which mathematical operations can be performed. A matrix is, in mathematics, a rectangular array of numbers.

*Mesh*

See Finite element mesh.

*Mixing length turbulence model*

**Mixing length** is a zero equation turbulence model, where the turbulent viscosity is related to shear, i.e velocity gradients. It is controlled by the user through a mixing length coefficient, and has the same effect as a viscosity.

*Modules*

**Modules** are components of the software. In H2D2, a module is built of at least two files: a **.dll*, located in the *bin* folder, which is the computational part; and a **.cfg*, located in the *etc* folder, declaring the components to be added. It can also consist of other files (*help* files in the doc folder (text or html format) or *local* (translation)).

*Molecular viscosity*

**Molecular viscosity **(cited from Wikipedia^{[28]}) is the measure of the resistance of a fluid deformed by shear stress or extensional stress. Water viscosity is 10^{-6} m^{2}/s.

##### Navier-Stokes equations

**Navier–Stokes equations** (cited from Wikipedia^{[31]}), named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluids. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term.

*Newton’s method*

**Newton's method** (cited from Wikipedia^{[26]}), in numerical analysis, (also known as the Newton–Raphson method**,** named after Isaac Newton and Joseph Raphson) is perhaps the best-known method for finding successively better approximations to the zeroes (or roots) of a real-valued function. One may use Newton's method also to solve systems of *k* (non-linear) equations, which amounts to finding the zeroes of continuously differentiable functions *F*: **R**_{k} **→** **R**_{k}.

*Norm*

A **norm** measures a distance between entities (vectors, functions, etc…). Different norms are defined for different vector spaces. Norms can be used to estimate error, defined as the difference between an entity and a reference entity. The most usual norms are:

- the **L2 norm**

and

where *V _{e}* is the volume of the element.

The L2 norm is defined by (translated from Méthodes de résolution):

The l2 norm is the discreet version, that is to say :

This norm, also called RMSE, has the advantage of being smoother that the ∞ norm (see Root-mean-square deviation).

- the **infinity norm (∞) or max norm**

The ∞ norm is defined by:

In addition to the value, it is possible to know the *i* index of the max value. In the case of finite element, this index is linked to a node and therefore, to a location on the mesh.

- **absolute vs. relative norm**

In the case of the resolution increment *du*, the norm can be absolute, in which case one directly considers *du*:

or

or relative, in which case one instead considers *du/u*:

or

taking care for division by a small number. For the residuum, as the reference value is 0, one always uses an absolute norm.

It is possible to combine them and the expression takes the following form:

or

In H2D2, comparison with 1 is most often used.

*Numerical viscosity*

Globally, two paths can be taken to control an unstable solution: to refine the mesh, or to add viscosity. In H2D2, numerical viscosity can be added when the solution becomes unstable. Numerical viscosity adds a term in the form:

It is controlled by *ν _{n}*:

where *s* is a sensor and *γ*, a user-controlled parameter.

__Peclet__

This kind of **numerical viscosity** is controlled by the Peclet number, which is inversely proportional to the numerical viscosity.

*ν _{n}=│q/H│∆^{e}/Pe*

where *Pe*: Peclet number

*∆ ^{e}*: mesh size

*q*: specific discharge

*H*: depth

It is mostly effective in zones of high velocity, with respect to grid size.

This kind of **numerical viscosity** is controlled by the Lapidus coefficient and has a damping effect in regions of high gradient. The numerical viscosity has the form:

where *γ* is a constant value in the range of 10^{-5} and *∆ ^{e}* is the local mesh size.

*ν*

_{n}is constant per element and acts only in regions of high gradient.

*Partitioning*

**Partitioning** is the division of the elements and nodes to set up the calculation on a given number of processors.

*Peclet number*

**Peclet number** is a dimensional number also called *local Reynolds number*. It is used in an expression of numerical viscosity and its expression is:

where is the average velocity, *∆* is a characteristic dimension of the finite element and *K** ^{e}* is the viscosity. By fixing the value of

*Pe*, one can compute the viscosity.

For the representation of the local mesh size *∆ ^{e}*, we choose the square root of the area of the elementary domain, where the superscript

*e*denotes the element. The lower the Peclet number, the higher the viscosity. We normally use values between 0,5 and 1.

*Picard's method*

**Picard method** (cited from Wikipedia^{[23]}) is a fixed point method in numerical analysis, also known as “simple iterative method”. More specifically, given a function *f* defined on the real numbers with real values and given a point *x _{0}* in the domain of

*f*, the fixed point iteration is:

*x _{n+1} = f (x_{n}), n = 0, 1, 2...*

which gives rise to the sequence *x _{0}, x_{1}, x_{2}, ... * which is hoped to converge to a point

*x*.

*Probes*

**Probes** are used to follow or extract a value at a particular point (node), or on a series of points.

*Reduction operation*

**Reduction operation**, in H2D2, refers to an operation done between two data sources: current and stored. There are four different types of reduction operations:

- *∑* (summation): sum all values obtained for each point of a given domain

- maxabs: keep the maximal absolute value (numerical value without regard to the sign of a real number (cited from Wikipedia^{[17]})) from all values for each point of a given domain

- max: output/keep the maximal value obtained for each point of a given domain

- min: output/keep the minimal value obtained for each point of a given domain

*Relaxation (iterative method)*

In an iterative method, **relaxation **consists in multiplying the correction *du* by a factor *α*:

*u ^{i+i} = u^{i} + α.du*

when *α** < 1*, one speaks of underrelaxation and when *α** > 1*, one speaks of overrelaxation.

*Renumbering*

**Renumbering** (cited from Wikipedia^{[13]}) is the process of assigning new node numbers to the nodes and the elements. It is automatic after partitioning. The aim of a new numbering is improving data access or reducing matrix band width.

*Residuum*

**Residuum** is the remaining values obtained when solving the matrix system, which one tries to minimize, using a given resolution algorithm, until convergence criteria are met.

*{R} = {f} – [k] {u}*

*Root-mean-square deviation*

The **root-mean-square deviation (RMSD)** or **root-mean-square error (RMSE)** (cited from Wikipedia^{[34]}) is a frequently used measure of the differences between values predicted by a model or an estimator and the values actually observed. These individual differences are called residuals when the calculations are performed over the data sample that was used for estimation, and are called prediction errors when computed out-of-sample. The RMSD serves to aggregate the magnitudes of the errors in predictions for various times into a single measure of predictive power. RMSD is a good measure of accuracy, but only to compare forecasting errors of different models for a particular variable and not between variables, as it is scale-dependent.

*Simulation*

**Simulation** is the process by which one seeks a numerical solution to a discreet problem. Simulations can be:

- stationary (time-invariant), not varying in time but in space, or

- non-stationary (time-variant), varying in time and space.

*Solicitations*

**Solicitations** refer to loads that are added to the system. Those loads can be concentrated on nodes: in this case, one refers to these as **concentrated solicitations**. The concentrated solicitations can be applied on any node of the mesh. In the concentrated solicitations file, they are associated with the third nodal degree of freedom. On the contrary, if the solicitations (loads) are distributed on the system, one refers to these as **distributed solicitations**. The sign convention states that positive (+) specifies an entering discharge and negative (-) an outgoing discharge.

*St.-Venant equation*

S**hallow water equations**, (cited from Wikipedia^{[14]}) also called **Saint Venant equations** after Adhémar Jean Claude Barré de Saint-Venant, are a set of hyperbolic partial differential equations that describe the flow below a pressure surface in a fluid (sometimes, but not necessarily, a free surface). The equations are derived from depth-integrating the Navier-Stokes equations, in the case where the horizontal length scale is much greater than the vertical length scale. Under this condition, conservation of mass implies that the vertical velocity of the fluid is small. It can be shown from the momentum equation that vertical pressure gradients are nearly hydrostatic, and that horizontal pressure gradients are due to the displacement of the pressure surface, implying that the velocity field is nearly constant throughout the depth of the fluid. Taking the vertical velocity and variations throughout the depth of the fluid to be exactly zero in the Navier-Stokes equations, the shallow water equations are derived.

*Trigger*

T**rigger** (cited from Webopedia Computer Dictionary^{[10]}) is a procedure that initiates an action (i.e., fires an action) when an event occurs.

*Turbulent viscosity*

**Turbulent viscosity** is an added viscosity, which appears due to the way turbulence is represented. See Mixing length turbulence model.

*Viscosity*

In H2D2, there are three types of **viscosity**:

Chapitre: glossary 1 ( 1 / 1 )