# How to determine a PDE which is structure-preserving (energy, mass conserved)?

How to determine if a PDE is structure-preserving (energy，mass conserved)? Are there some standards in judging the preserving-structure? Or rather, how to derive the formulation of energy-preserved form of a PDE? I am new to this field, I am long for receiving some references (some books, representative papers or some experts' homepage) about these topics.

• There's the easy way which is solving it at really high accuracy on a cluster and coming back a few weeks later and just numerically checking if it conserved energy. Otherwise I think it's problem-specific how to approach this. – Chris Rackauckas Aug 23 '17 at 15:01
• Not PDEs are structuring-preserving, numerical methods are. – shuhalo Aug 23 '17 at 15:04
• @shuhalo what is your standard to obtain the conserved formulations of your bunerical method ? just the existence of the continue case may deduce what you said. – J.Xie Aug 24 '17 at 3:11
• In the analysis literature you talk about "conserved quantities", which is pure theory. The term "structure-preserving" appears only in the numerical literature. It will be helpful for you to keep in mind which term to use when. In particular, "structure-preserving" is used for some numerical methods that preserve a structure completely unrelated to conservation laws. – shuhalo Aug 24 '17 at 5:12

If a PDE is derived from a principle of least action and if it has invariance, then some quantities are conserved. For instance, in mechanics, invariance with respect to time implies conservation of a certain scalar quantity (called "energy"), invariance with respect to the origin of the frame implies conservation of a certain vector quantity (called "momentum") and invariance with respect to the orientation of the frame implies conservation of another vector quantity ("angular momentum"). There is a general theorem due to Emmy Noether that studies the structure of these symmetries and conserved quantities.

About the example of point mechanics, Landau's course [1] is crystal clear and very complete. About Emmy Noether's general theory, [2] is very enlighting and reasonably easy to read without requiring too much background.

[1] Landau and Lifshitz, Course on Theoretical Physics, Volume 1 - Mechanics

[2] Dwight Neuenschwander, Emmy Noether's Wonderful Theorem

I think that mass convervation does not belong to the same category of preserved quantities. For instance in fluid dynamics it is explicitly enforced by the continuity equation.

As Bruno Levy pointed out, certain classes of physical systems have well-known structural properties. For example, consider a Hamiltonian system

$\dot z = J\nabla H(z)$

where $J$ is a symplectic matrix and $H$ is the Hamiltonian. The Hamiltonian $H$ itself is preserved, as is any other function $F$ such that the Poisson bracket $\{H, F\} = \nabla H^\top J\nabla F = 0$ for all $z$. Symmetries of these kinds of systems give rise to conservation laws, and there are software packages using symbolic algebra that can automatically discover some symmetries of arbitrary ODE systems (see also this list). Hamiltonian systems have other useful properties not directly related to symmetries or conservation laws. For example, every equilibrium point is either a saddle point or a center, i.e. there are no asymptotically stable equilibria; the volume of a region of phase space is always preserved under Hamiltonian flow; and many others. However, it is arguably not feasible to algorithmically determine whether some totally arbitrary system is Hamiltonian if you weren't already sure to begin with, especially when you take into account non-canonical Poisson structures.

Other kinds of systems have different structural properties. For example, consider a gradient flow

$\dot z = -\nabla F(z)$;

then $dF/dt \le 0$ along the trajectories of this system. The heat equation is the prototypical example of a gradient flow. Gradient flows have equilibria at extrema of $F$ and the stability of an equilibrium point depends on the eigenvalues of the second derivative of $F$. Unlike Hamiltonian systems, a gradient flow can have stable equilibria.

Both Hamiltonian systems and gradient flows have interesting and useful structure, but in general there is no algorithm to take an arbitrary differential equation and find what class of system it belongs to. You can of course rule out some possiblities. For example, if the system clearly has an asymptotically stable equilibrium, it can't be Hamiltonian. Likewise, if the linearization of the system is not a symmetric matrix, it can't be a gradient flow by Clairaut's theorem.

Moreover, there are always weird examples from real applications that defy any classification scheme. For example, linear PDE are often classified as being either elliptic, hyperbolic, or parabolic, but there are numerous examples that don't fit neatly into these categories: the Euler-Tricomi equation changes type within the spatial domain; the PDEs that describe thin-film flows are a mixed elliptic-hyperbolic system; etc.

• @ Daniel Shapero would you please refer me some scholar who r doing well in this field? Thanks. Such as using finite difference method or finite element method, or spectral methods – J.Xie Apr 19 '18 at 14:58

A standard reference for basic topics in partial differential equations is the textbook by Lawrence C. Evans "Partial Differential Equations". His chapter 8.6 discusses Noether's theorem in a very condensed manner.

The question is very broad, so it is difficult to pinpoint a specific reference. The wikipedia article on Noether's theorem is probably a good start, including the references given there.