 Research article
 Open Access
A review of some geometric integrators
 Dina Razafindralandy^{1}Email authorView ORCID ID profile,
 Aziz Hamdouni^{1} and
 Marx Chhay^{2}
https://doi.org/10.1186/s403230180110y
© The Author(s) 2018
 Received: 9 December 2016
 Accepted: 18 May 2018
 Published: 13 June 2018
Abstract
Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler–Lagrange equations) and the conservation laws (Nœther’s theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Liesymmetrypreserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structurepreserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.
Keywords
 Geometric integration
 Symplectic integrator
 Multisymplectic
 Variational integrator
 Liesymmetry preserving scheme
 Discrete Exterior Calculus
Introduction
With the increasing performance of computers (speed, parallel processing, storage capacity, ...), one might think that it is not worth to design completely new algorithms to solve numerically basic problems in mechanics. It is tempting to think that to simulate physics with a fair precision, one just has to take small enough time and space steps. Yet, experiments show that, even with an academic problem such as an harmonic oscillator, classical numerical schemes which rely on a direct discretization of the equations are unable to predict correctly the solution over a long time period, even with a fine time and space grids. In fact, the equation of a system may hide physically very important properties, such as conservation laws, which are destroyed by these schemes, leading to a meaningless prediction or a blow up.
Many physical properties of a system are encoded within a geometric structure of the equation. A natural way to correctly predict these properties is then to preserve exactly the geometric structure during the simulation. This is the foundation principle of geometric integrators. This approach of discretization generally enables a better representation of the physics of the system and a particular robustness for a long time or/and large space simulation.
The aim of this article is to draw the attention of computational mechanics specialists to geometric integrators. However, since many geometric structures, and consequently many geometric integrators, exist, it is not conceivable to describe all of them. Instead, we will focus on the most widely used. The main properties of these integrators will be listed. Examples of construction will be presented and illustration on mechanical systems will be given. Many references will be provided throughout the article for each part of the article for more complete details.
One of the oldest geometric structures used in physics is the symplecticity of Hamiltonian systems. When it can be written in a symplectic framework, an ordinary differential equation (ODE) exhibits a preservation of a differential form, the symplectic form, over time. The symplectic formulation also permits to obtain conservation laws via Nœther’s theorem [1–3]. Symplectic integrators are built to preserve the symplecticity of the flow at the discrete level. One of the first works on symplectic integrators is that of Vogeleare in 1956 (see [4]), followed by many papers and books in the 1980’s [5–11]. Since then, an abundant literature can be found on the subject.
Many symplectic systems come from a variational problem. In this case, the equation represents a trajectory which minimizes a Lagrangian action. A way of building a symplectic integrator for such a problem is to discretize the Lagrangian function and solve the corresponding discrete variational problem. This approach yields variational integrators [4, 12–14]. Variational integrators are automatically symplectic and momentum preserving. They have been developed since the 1960’s in optimal control theory and the 1970’s in mechanics ([15–22], see [13] for a more complete historical overview). Note that variational integrators can be extended to solve efficiently nonvariational problems by embeeding the latter into a larger Lagrangian system [23, 24].
Attempts on the extension of the symplectic geometry to fields appeared from the 1950’s with the works of Gallisot [25] and Dedecker (see [26–28]), followed by many others a decade later [28–31]. That gave rise to two major theories. The first one is based on the polysymplectic approach in which the symplectic form is extended into a vectorvalued form [32–37]. The other theory led to the multisymplectic notion where the cotangent bundle is replaced by the bundle of kforms [26, 30, 38, 39]. Nœther’s theorem has been extended to the polysymplectic and multisymplectic approaches [40–43]. At the computational side, the extension of symplectic integrators to partial differential equations (PDEs) led to multisymplectic integrators [44–49]. They have been used to solve a wide range of problems in physics [16, 50–53].
Other equations of mechanics cannot straightforwardly formulated with a Lagrangian functional nor a (multi)symplectic structure. It is the case of many mechanical dissipative systems. Yet, many of them have important invariance properties, called symmetries [54], under some transformations. Symmetries play a fundamental role since they may encode exact model solutions (selfsimilar, vortex, shock solutions, ...), conservation laws via Nother’s theorem or physical principles (Galilean invariance, scale invariance, ...) [55–68]. They have also been used for modelling purposes such as the establishment of wall laws in turbulent flows or the development of turbulence models [69–76]. It is then important that numerical schemes do not break symmetries if one wishes to reproduce numerically the cited model solutions and properties. Preserving the symmetries of the equations at the discrete level is the founding key of invariant integrators [77–83], as will be seen in “Invariant integrators” section. Note that the symmetry structure does not exclude simplecticity structure. So, combining symplectic/variational algorithm and invariant scheme is feasible, but not done in this article.
Recently, a new generation of geometric integrators, called discrete exterior calculus (DEC), has been developed. It appeared from the observation that equations of physics, especially electromagnetism, are better written with differential forms (instead of vector fields) and exterior calculus operators (see [84–87] and the series of papers of Bossavit [88–92]). DEC aims at developing a discrete version of the theory of exterior calculus, and more generally differential geometry, where most equations of physics are formulated. This framework offers naturally coherent discretization of derivation operators (divergence, gradient, curl) since, in exterior calculus, they are represented by a single operator \({\text {d}}\), the exterior derivative. The basic tools in DEC theory are discrete differential forms, seen as simplicial cochains. This approach of discrete form could be found in the work of Tonti [93] (see also [94]) in the 1970’s, and later in [95–98]. The current development of DEC has been inspired by the works of Bossavit in electromagnetism [99–105]. More recent works in electromagnetism include [106–109]. In the field of mechanics, DEC has been employed to solve Darcy, Euler and Navier–Stokes equations in some basic configurations [110–112], to geometrizes elasticity problems [113, 114] or to solve portHamiltonian systems [115]. DEC belongs to the family of cochainbased mimetic discretization methods described in [116]. Among the members of this family, we can cite the covolume method [117], splinebased cochain discretization [118], mimetic spectral elements [119–121] or spectral DEC [122, 123]. Other compatible discretization techniques include the Finite Element Exterior Calculus (FEEC) [124–126], mimetic finite differences ([127] and references therein) and works in [128–132].
In the next section, we recall the basic principle of symplectic geometry. We then show why and how to build symplectic and multisymplectic integrators. In “Variational integrators” section, variational systems and variational integrators are presented. A total variation approach, which extends the usual presentation of Lagrangian mechanics, is used. Indeed, with a total variation consideration, an energy equation (and momentum equation in case of PDE) naturally arises. This equation can be used to build, for instance, energypreserving schemes. Invariant integrators are dealt with in “Invariant integrators” section. The approach used is that of Kim [79, 133] and Chhay and Hamdouni [80] which consists in modifying classical schemes to make them invariant. At last, discrete exterior calculus is described in “Discrete exterior calculus” section. Simulations of fluid flows convecting a passive scalar are presented.
Some theoretical tools of differential geometry are used in this article to highlight the geometry structures. Their definition can be found, for example, in [1, 54, 134–143]. However, their use has been limited (at the risk of using formal definitions) and the examples are simple enough for readers more interested in the computational aspect.
Symplectic integrators
We begin with a brief introduction to Hamiltonian mechanics. Interested readers can find deeper presentations in [1, 2, 144–146].
Hamiltonian system and flow
Hamiltonian system
Flow of a numerical scheme
Failure of classical integrators
In this section, we show, through elementary examples, that the use of nonstructurepreserving algorithms may have dramatic consequences when simulating a longtime evolution problem.
Harmonic oscillator
The growth of the initial volume can be interpreted as a numerical production of energy, whereas the diminution of the volume is associated to a numerical dissipation. Hence, through this elementary example, it was shown that the non preservation of an intrinsic property of the system yields an accumulation of numerical errors, modfying the initial area and contradicting Liouville’s theorem.
Kepler’s problem
The equation, with initial conditions \((q_1=1/2,q_2=0,p_1=0,p_2=\sqrt{3})\) is solved with the second order Runge–Kutta scheme. The flow is described by Eq. (14). It can be shown that this scheme is not symplectic.
Other examples, including molecular dynamics and population evolution problems, which show the failure of classical methods are presented in [150]. In the next section, we present some symplectic schemes and show how they are suitable for the resolution of longtime evolution problem having a Hamiltonian structure.
Some symplectic integrators
There are some ways to build symplectic integrators [151–155]. One of them is through a generating function. Another way is to adapt existing integrators. In the present paper, we adopt the latter method. Two types of integrators are investigated, those based on Euler schemes and those belonging to the Runge–Kutta method family.
Eulertype symplectic integrators
Symplectic Runge–Kutta methods
In fact, a backward analysis shows that symplectic integrators solve exactly a Hamiltonian system, with a Hamiltonian function \(\tilde{H}\) which is a perturbation of H [160]. This explains the bounded error on H.
Numerical tests
Harmonic oscillator
With the centered symplectic scheme, the error on H is below the machine precision (Fig. 5b). In fact, it can be shown that, since this scheme is a Runge–Kutta one, it preserves H exactly.
Kepler’s problem
As a second test, we consider again Kepler’s problem. The equation is solved with the classical 4thorder Runge–Kutta scheme (RK4) and its symplectic version (RK4sym).
As for the symplectic scheme, the error of RK4sym on H stays almost the same over a long time period, as could be predicted with estimation (29).
The above results show that symplectic integrators preserve many first integrals (in the sense that the error is bounded). However, not necessarily all first integrals are preserved. For example, standard symplectic integrators do not necessarily preserve the Laplace–Runge–Lenz vector. Recall that this vector is an invariant of Kepler’s problem which makes the equation superintegrable (see [161]). It is possible to build specific schemes which preserve this invariant. It is done in [162] using the canonical LeviCivita transformation [163].
Vortex dynamic
If the initial position of two opposite vortices are shifted from the square configuration with a small angle \({\alpha }\) as in Fig. 8 then the system is quasiintegrable. Each particle has a quasiperiodic trajectory. But if only one vortex (instead of two opposite ones) is shifted from the square configuration, symmetries are lost and the trajectory becomes chaotic (see [168–170]).
These results show the importance of the preservation of the equation’s structure when simulating Hamiltonian ODEs. In order to extend symplectic integrators to PDEs, multisymplectic geometry will be introduced in the next subsection.
Multisymplectic integrators
Mulstisymplectic integrators have been intensively developped for PDE, specifically in the framework of water waves [47, 171]. Indeed, a large class of models for water waves inherits a Hamiltonian structure of infinite dimension. Thus a multisymplectic structure can be exhibited, for example, for Serretype equations in deep water configuration [172], and for the Serre–Green–Naghdi equations in shallow water configuration [173]. Therefore multisymplectic schemes appear as natural structure preserving integrators applied to these models. The efficiency of such geometric numerical methods is presented in [174] where a longtime simulation of the Korteweg–de Vries dynamics is performed with more robustness and more accuracy using a multisymplectic scheme than a Fouriertype pseudospectral method.
A precise presentation of the multisymplectic structure on a manifold necessitates the introduction of many mathematical tools. As our article aims to be a review paper, we do not wish to do it here. The reader can refer to [175, §1.1–1.3] for a good mathematical approach. Instead, we propose a simplified framework, in a vector space.
Conservation laws
Examples
Eulertype multisymplectic schemes
Multisymplectic Runge–Kutta schemes
It was shown in “Symplectic Runge–Kutta methods” section that a symplectic Runge–Kutta scheme can be obtained with a rather simple condition of the coefficients in the Butcher tableau which guarantees the symplecticity. However, no extension has been established yet in the generic multisymplectic case. Multisymplectic RK schemes were presented and studied in [178–180] for the partitioned case. Another particular RK scheme is the implicit Gauss–Legendre integrator [181]. It will be illustrated hereafter on the SineGordon equation. Other multisymplectic schemes, based on the Runge–Kutta–Nyström method, can be found in [182–184].
Numerical test

A leapfrog (LF) scheme:This scheme is in fact a symplectic (not multisymplectic) scheme in the sense that it preserves a spatial symplectic twoform over time.$$\begin{aligned} \frac{u^{n+1}_i2u^n_i+u^{n1}_i}{{\Delta }t^2}\frac{u^n_{i+1}2u^n_i+u^n_{i1}}{{\Delta }x^2}+V'(u^n_i)=0.\end{aligned}$$

An energy conserving but not multisymplectic scheme developped in [185] that we call EC:$$\begin{aligned} \frac{u^{n+1}_i2u^n_i+u^{n1}_i}{{\Delta }t^2}\frac{u^n_{i+1}2u^n_i+u^n_{i1}}{{\Delta }x^2}+\frac{V(u^{n+1}_i)V(u^{n1}_i)}{u^{n+1}_iu^{n1}_i}=0.\end{aligned}$$

A ninepoint box multisymplectic (MS) scheme, which is a Runge–Kutta scheme, simplified by variable substitution [186]:where the time and space discretization operators are defined by$$\begin{aligned}&{\Delta }_t^2\left( u^{n+1}_i2u^n_i+u^{n1}_i\right) {\Delta }^2_x\left( u^n_{i+1}2u^n_i+u^n_{i1}\right) \\&\quad +V'\left( u^{n+\frac{1}{2}}_{i+\frac{1}{2}}\right) +V'\left( u^{n\frac{1}{2}}_{i+\frac{1}{2}}\right) +V'\left( u^{n+\frac{1}{2}}_{i\frac{1}{2}}\right) +V'\left( u^{n\frac{1}{2}}_{i\frac{1}{2}}\right) =0. \end{aligned}$$$$\begin{aligned} {\Delta }_t^2z^n_i=\frac{z^{n+1}_i2z^n_i+z^{n1}_i}{{\Delta }t^2}, \quad \quad {\Delta }_x^2z^n_i=\frac{z^n_{i+1}2z^n_i+z^n_{i1}}{{\Delta }x^2}. \end{aligned}$$
Some results on the quality of the multisymplectic scheme regarding conservation laws are given in [170]. It can be concluded that symplectic and multisymplectic schemes are particularly suitable to the numerical resolution of long time evolution problems.
In the next section, geometric integrators for variational ODEs and PDEs are presented.
Variational integrators
In this section, we deal with Lagrangian systems, that are systems coming from a calculus of variation.
Reminders on Lagrangian mechanics
In Lagrangian mechanics, a Lagrangian system is described by a Lagrangian density. Taking the variation of the corresponding Lagrangian action over the configuration variable \(\mathbf q\), the Euler–Lagrange equation of the system is deduced from Hamilton’s principle of least action. This is the traditional approach of presenting the Euler–Lagrange equation [1, 187, 188].
In a similar way, variational integrators are obtained by taking the variation of a discrete Lagrangian action over the discrete variable \(\mathbf q^n\). Abundant literature on the traditional presentation of variational integrators exists [13, 14, 150].
Generally, the numerical solution of the discrete Euler–Lagrange equation preserves the evolution law of energy of the system with a good precision, but not exactly. In [189], Kane et al. proposed to associate an ad hoc equation to the discrete equations in order to satisfy a discrete energy conservation.
In fact, the algorithm obtained in [189] can be viewed as a generalisation of variational calculus, where variation along time is permitted [170, 190]. In this new approach, the energy equation is not defined as in [189] but results naturally from the variation of the action in time direction. In the present paper, this second approach is adopted.
Total variation calculus
In the standard way of deducing the Euler–Lagrange equation, only tangent vectors \({\delta }\mathbf q\in T_{\mathbf q}Q\) are considered. However, by considering the time as a variable, and thus the tangent vectors \(({\delta }t,{\delta }\mathbf q)\), conservation laws, as defined in Nœther’s theorem [191–194], appear naturally.
Nœther’s theorem
Variational integrators
A variational integrator is a scheme which verifies a discrete Hamiltonian’s variational principle. The most popular way of obtaining a variational integrator is from a discretization of the Lagrangian density and a derivation of a discrete version of Eqs. (44), (45) by mimicking the variational procedure used in “Total variation calculus” section. This is the approach of Marsden and West [13, 14], based on a finite difference discretization of the Lagrangian density. Other approaches can be found, for instance, in [22, 195–200]. As examples, a higherorder, spectral method is presented in [22]. A finite element method is described in [195]. Another approach, where at each time iteration an exact variational problem is solved, is developped in [198, 201].
Examples
Rectangle rule
for \(n=1,\ldots ,N1\).
Midpoint rule
Some other variational integrators, such as Newmark scheme or symplectic Runge–Kutta written for separated Lagrangian, can be found in [13, 150].
Symplecticity of a variational integrator
Numerical test
Variational integrator on PDE
So far, we only considered variational integrator for ODEs. As we did for Hamiltonian mechanics, the Lagrangian approach can be extended to PDEs.
Lagrangian approach of PDE
Example of variational integrator
In this section, we presented geometric integrators for ODEs and PDEs having a symplectic structure, coming from a variational problem. In the next section, we consider more general equations which may not have any symplectic structure but a Lie symmetry group. We then show how to construct geometric integrator for such equations.
Invariant integrators
We first set some theoretical background, by defining symmetry of an equation and precising the notion of invariance.
Symmetry group
Computing the Lie symmetry groups of an equation is often a tremendous task. Fortunately, it can be made algorithmic with use of infinitesimal generators [54], such that many computer algebra packages can be used [204–207]. Some of them even compute conservation laws.
The knowledge of symmetries of an equation gives precious information on the equation and on the physical phenomenon it modelises. For example, from one known solution, symmetries enable to find other solutions. Symmetries may also be used to lower the order of the equation and to compute selfsimilar solutions [54]. More fundamentaly, as stated by Nœther’s theorem [191–193], symmetries are linked to conservation laws.
As introduced, preserving the symmetry group through the discretization process is necessary if one wishes not to loose selfsimilar solutions and conservation laws during simulations. A way of building integrators that are compatible with, or invariant under, the symmetry group is to make use of independent differential invariants of the equation [77]. However, combining these invariants into a numerically stable scheme is rather complicated. Instead, we follow the idea in [79–81], which consists in modifying classical schemes so as to make them invariant under the symmetry group. To show how to do this, we need to formalize the notion of a discretization.
Numerical scheme
Starting from any existing scheme \((N,\Phi )\), our aim is to derive a new scheme \((\tilde{N},\tilde{\Phi })\) which is invariant under the symmetry group of the equation. This will be done using the concept of moving frame.
Invariantization by moving frame
Let us illustrate the invariantization process (91) on Burgers’ equation.
Application

time translations:$$\begin{aligned} g_1\ :\ (t,x,u) \longmapsto (t+a_1,x, u), \end{aligned}$$(92)

space translations:$$\begin{aligned} g_2\ :\ (t,x,u) \longmapsto (t,x+a_2, u), \end{aligned}$$(93)

scaling transformations:$$\begin{aligned} g_3\ :\ (t,x,u) \longmapsto (te^{2a_3},xe^{a_3}, ue^{a_3}), \end{aligned}$$(94)

projections:$$\begin{aligned} g_4\ :\ (t,x,u) \longmapsto \left( \frac{t}{1a_4t},\frac{x}{1a_4t}, (1a_4t)u+ a_4x\right) , \end{aligned}$$(95)

and Galilean boosts:$$\begin{aligned} g_5\ :\ (t,x,u) \longmapsto (t,x+a_5t, u+a_5). \end{aligned}$$(96)
A moving frame \(\rho \) associated to \(g_0\) is an element of \(G_0\). This means that \(\rho [\underline{\mathbf m}]{\cdot }\underline{\mathbf m}\) is of the form (98), with particular values of the parameters \(a_i\), depending on \(\underline{\mathbf m}\). Determining \(\rho [\underline{\mathbf m}]\) is then equivalent to deciding the values of the \(a_i\)’s.
Transformation of the grid
Invariantization of the scheme
Determination of \(a_4\) and \(a_5\)
Since g, \({\rho }[\mathbf m^n_j]\) and \(\rho [g{\cdot }\mathbf m^n_j]\) all have the same form (112) but with different values of the parameters \(a_4\) and \(a_5\), we have to call the parameters of each of them differently in order to avoid confusion. To this aim, we keep \(a_4\) and \(a_5\) for \({\rho }[\mathbf m^n_j]\); we denote \({\mu }\) and \(\eta \) the parameters of g, and \(\overline{a}_4\) and \(\overline{a}_5\) those of \(\rho [g{\cdot }\mathbf m^n_j]\).
Order of accuracy
A fundamental result guarantees that the invariantization of a numerical scheme using the moving frame technique preserves the consistency. However, the order of consistency may change.
Notice that \(a_5\) does not appear when the invariantized numerical scheme is expressed in the regular and orthogonal original mesh. It is no longer the case if the mesh grid is not orthogonal, nor if the numerical solution is expressed in the transformed frame of reference.
We end up with some numerical results.
Numerical tests
The first test aims to check if the solutions given by various standard and invariant schemes are Galilean invariant.
Consistency analysis shows that the original schemes are no longer consistent with the equation when \(\lambda \ne 0\). This inconsistency introduces a numerical error which grows with \(\lambda \), independently of the step sizes. As for them, the invariantized schemes respect the physical property of the equation and provide quasiidentical solutions when \(\lambda \) changes.
Since the invariant scheme has the same invariance property as the analytical solution under projections, it does not produce an artificial error like the noninvariant FTCS. This shows the ability of invariantized scheme to respect the physics of the equation and the importance of preserving symmetries at discrete scales.
Other numerical tests are presented in [74, 81, 170].
The last geometric integrator that we shall present is the discrete exterior calculus.
Discrete exterior calculus
Discrete exterior calculus (DEC) can be seen as a differential geometry and exterior calculus theory upon a discrete manifold. The primary calculus tools of DEC are (discrete) differential forms. As we shall see, they are built by duality with the grid elements. Discrete operators on differential forms (exterior derivative, Hodge star, ...) are then defined in a way which mimics their continuous counterpart.
The first step to discretizing an equation with DEC is to define a grid, which plays the role of a discrete manifold. Then, each piece of the equation is replaced by its discrete versions. But since equations of mechanics are usually formulated with vector and tensor fields, they must beforehand rewritten in exterior calculus language, that is with differential forms.
Note that property (129) is much more general than (128). Indeed, equations (128) are meaningful only in a threedimensional space. Moreover, equations (128) are metric dependent whereas (129) is not.
In fluid mechanics, Elcott et al. [110] used DEC to solve Euler equation on a flat and a curved surface, with exact verification of Kelvin’s theorem on the preservation of circulation along a closed curve. Works on Darcy and Navier–Stokes equations were carried out in [111] and [112, 213].
In the next subsections, we recall the basis of DEC and present some test cases.
Background
We present here briefly a DEC theory which follows the approach of Hirani and his coauthors [111, 112, 214]. More details can be found in the literature on algebraic topology [215–219] and on DEC [89–92, 99–103, 214, 220–222]. Works on other related structurepreserving discretization can be found, for example, in [129, 223–225] and references therein.
Domain discretization and elements of algebraic topology
Consider a differential equation defined on a \({n_{\mathbf x}}\)dimensional spatial domain M in \({\mathbb {R}}^n\), \(n\ge n_{\mathbf x}\), written within the exterior calculus framework. In order to solve this equation numerically, the domain is meshed into an oriented simplicial complex K that we recall hereafter the definition and the associated algebraic topology elements (chain, cochain and boundary operator). Next, we shall see how to discretize differential forms and the exterior calculus operators.
Oriented simplicial simplex

for any simplex \(\sigma \in K\), any face of \(\sigma \) belongs to K,

and the intersection of two simplices \(\sigma _1,\sigma _2\in K\) is either empty or a common face of \(\sigma _1\) and \(\sigma _2\).
If \({n_{\mathbf x}}=3\), K is a set composed of tetrahedrons (3simplices), triangles (2simplices), edges (1simplices) and vertices (0simplices). In 2D, the topdimensional simplices are triangles.
Chain and cochain
Boundary operator
Discretization of differential forms and operators
Discrete differential form
Discrete exterior derivative
Wedge product
From cochain to differential form
Higher order Whitney forms are proposed, for example, in [230, 231].
As said, Whitney maps provide a single interpolation within each tetrahedron (if the mesh is threedimensional). One can also interpolate a cochain differently, for example by dividing each tetrahedron in many subregions and piecewisely construct an interpolated form in each subregion. This approach is used in covolume method and mimetic reconstruction [116].
Discrete Hodge star and codifferential operators
The orientation on the primal mesh induces an orientation on its dual [220, 221]. The boundary operator and discrete exterior derivative can also be transposed to the dual mesh. As pointed out in [112], some care has to be taken when computing the exterior derivative of a form on the dual mesh. Indeed, the dual cells situated at the boundary of the domain are not closed. So, a boundary complement to the discrete exterior derivative has to be added. This complement can be obtained from boundary conditions for many problems.
As said, relation (146) is not the unique way to define the Hodge operator. In particular, it has the drawback that, like in covolume method [117], it does not handle nonacute triangulation. Amelioration and alternatives can be found in literature [116, 221, 227, 233–236]. For example, as presented in [221], on can choose a dual mesh based on the barycenter or on the incenter, instead of the circumcenter, to remove the angle condition. In [116], a discrete Hodge star operator is built from relation (148). A discrete Hodge, seen as a mass matrix of a Galerkin method, was also introduced in [103, 237]. The construction of efficient discrete Hodge star operators is still an open question.
In the following, as we work exclusively on acute triangulated domains, we use definition (146) of the discrete Hodge operator for our numerical experiments.
With the previously built discrete exterior derivative and discrete Hodge star, Eq. (147) enables to define the discrete codifferential operator. In fact, applying \({\delta }\) on the primal mesh boils down to applying \({\text {d}}\) on the dual mesh, and viceversa, up to a sign and multiplications by measures of simplices.
Interior product and Lie derivative
A remark has to be done on formula (150). It necessitates a metric to define the discrete Hodge star and then the interior product. In many applications, a metric is available (for instance, a metric is required to build constitutive laws). So, formula (150) can be used in these applications. However, it has the serious drawback that it defines the discrete interior product and Lie derivative, which are metric independent in the continuous case, from the metric dependant Hodge operator. Alternatives can be found in literature. For instance, a discrete Lie derivative can be built from extrusion, as developed in [238, 239].
As already noted, a finite element approach of exterior calculus exists [124–126]. Inspired by works in [233, 240–242], it makes use of the inner product (148) to build finite element approximation spaces.
Applications to fluid mechanics
Exterior calculus formulation
Numerical scheme

Equation (152c) is solved with an implicit scheme$$\begin{aligned} \frac{{\theta }^{n+1}{\theta }^n}{{\Delta }t}{\delta } (\omega ^{n+\frac{1}{2}}\wedge {\theta }^{n+1})+{\kappa }{\delta }{\text {d}}{\theta }^{n+1} = 0. \end{aligned}$$(155)
In space, a discretization with DEC is used. \({\psi }\) and \({\theta }\) are placed on primal vertices (the velocity form \(\omega \) is then on dual edges). The pressure is placed on the dual vertices.
Numerical tests
The first numerical example is a 2D channel flow. The length of the channel is \(L=2{\pi }\) and its height is \(H=1\). The viscosity is set to \({\nu }=1\) and the diffusivity to \({\kappa }=1\).
A zero flow is given as initial condition. The value of \({\theta }\) at the top wall is set to 20 and at the bottom wall and the inlet to 0. The grid is composed of 2744 triangles. The mean length of the edges is \(\Delta x_{mean}=2.989{\cdot }10^{2}\). The time step is \({\Delta }t=10^{2}\).
To show the potential of DEC in predicting more realistic problems, we deal, in what follows, with the convection of a polluant in a ventilated room. The room geometry is presented in Fig. 35. The aspect ratio is set to \(L/H=1\). The height of the inlet and outlet is \(h=H/5\).
DEC, as presented here, is used only as a spatial discretization method. The canonical metric of the space is used to define the discrete Hodge operator. This can easily be extended to a discretization on a Riemanian surface [213]. Extension to spacetime is possible. For example, for the exterior calculus discretization of problems in gauge theory, the Hodge operator has been defined in [244] relatively to the Minkowski metric. An application to general relativity is presented in [245].
Conclusion
With this article, we aimed at raising the awarness among the readers, and particularly among highperformance computing specialists, about the importance of the geometric structure of their equations. This structure traduces fundamental physical properties and its preservation at the discrete level leads to robust numerical schemes.
Without being exhaustive, we tried to give an overview of the most popular geometric structures met in mechanics. For each of them, we presented a way to build a corresponding structurepreserving integrator.
Firstly, for Hamiltonian problems, we showed that symplectic integrators are particularly robust for longtime evolution problems. Their error on the energy of the system is bounded over an exponentially large time interval. More generally, symplectic integrators present generally good properties towards the preservation of conservation laws.
In the case of Hamiltonian PDEs, we presented a way of constructing multisymplectic integrators. It was shown that these multisymplectic schemes are more robust regarding the grid, in the sense that they allow more freedom in the choice of time and space steps, compared to classical schemes.
When the equations derive from a variational principle, we showed in section that with total variation approach, discrete energy and momentum evolution equations are obtained naturally, along with the Euler–Lagrange equation. We then presented a way to build variational integrators for ODEs and PDEs. We observed that, when the Lagrangian is time independent, these schemes preserve exactly the energy.
For more general equations, with or without symplectic or variational structure, we presented invariant schemes which preserve the Lie symmetry group of the equation. We saw that, contrarily to classical schemes, they do not introduce unphysical oscillations in presence of a pseudoshock solution. They also do not give rise to spurious solution when the grid undergoes a Galilean transformation.
At last, we presented DEC which is a space integrator based on exterior calculus and which reproduces exactly Stokes theorem and the relation \({\text {d}}^2=0\) at the discrete scale. The second property enables, for example, to verify exactly the incompressibility condition. This also ensures the exact preservation of circulation in an ideal incompressible fluid flow. The use of DEC in mechanics being very recent, we presented some applications in fluid mechanics with passive scalar convection.
Other geometrybased integrators exist. We can cite, for instance, algorithms built to preserve a Lie–Poisson structure on a discrete manifold [246, 247]. Other examples are schemes based on a portHamiltonian, or more generally, on a Dirac structure, which aim at correctly handling the interconnection between subsystems [248]. Note that a combination of port Hamiltonian structure and DEC has been considered in [249]. Liegroup integrators can also be cited [250].
The interior product or contraction of a vector field \({\mathbf {X}}\) and a differential form \({\mu }\) is often noted \(i_{{\mathbf {X}}}\omega \) instead of . It is defined as if \(\mu \) is a 2form and \({\mathbf {Y}}\) any vector field.
The pullback of \(\omega \) is defined as \(({\Phi }_t^*\omega )_{\mathbf {s}}({\mathbf {X}}^1,{\mathbf {X}}^2)=\omega _{{\Phi }_t(\mathbf {s})}\left( {{\text {d}}{\Phi }_{t}}_{\mathbf {s}}({\mathbf {X}}^1),{{\text {d}}{\Phi }_t}_{\mathbf {s}}({\mathbf {X}}^2)\right) \) for any pair of vector fields \({\mathbf {X}}^1\) and \({\mathbf {X}}^2\), \({\text {d}}{\Phi }_t\) being the differential of \({\Phi }_t\).
The exterior product of two differential 1forms \(\omega ^1\) and \(\omega ^2\) is defined as the two form \((\omega ^1\wedge \omega ^2)({\mathbf {X}}^1,{\mathbf {X}}^2)=\omega ^1({\mathbf {X}}^1)\omega ^2({\mathbf {X}}^2)\omega ^1({\mathbf {X}}^2)\omega ^2({\mathbf {X}}^1)\) for any pair of vector fields \({\mathbf {X}}^1\) and \({\mathbf {X}}^2\). More generally, the exterior product of a kform \({\alpha }\) and a lform \({\beta }\) is defined as the \((k+l)\)form [147]:
\({\alpha }\wedge {\beta }({\mathbf {X}}^1,\ldots ,{\mathbf {X}}^{k+l})=\frac{1}{k!l!}\sum _{{\tau }\in S_{k+l}}{\text {sign}}({\tau })\ {\alpha }({\mathbf {X}}^{{\tau }(1)},\ldots ,{\mathbf {X}}^{{\tau }(k)})\ {\beta }({\mathbf {X}}^{{\tau }(k+1)},\ldots ,{\mathbf {X}}^{{\tau }(k+l)})\) for any vector fields \({\mathbf {X}}^1,\ldots ,{\mathbf {X}}^{k+l}\), \(S_{k+l}\) being the permutation group of \(\{1,\ldots ,k+l\}\).
More rigorously, \(\mathcal M\) should be a fibre bundle with base \(\mathcal Z\) and a typical fiber \(\mathcal U\). The Lagrangian function is then defined in the jet bundle.
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All authors have jointly prepared the manuscript. All authors read and approved the final manuscript.
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