What's the difference between grid-based and mesh-based methods for PDEs?

Question

I am into computational fluid dynamics and so far I've found that the most common approaches to solve for the governing equations are Eulerian and Lagrangian. The former samples the domain at fixed locations while the latter samples the domain using the particles moving freely across the domain.

Reading through some pdf's over the net someone mentioned about mesh-based methods but as far as I could see they were very close to what the Eulerian approach is (if not the same).

Could someone clarify is there is indeed a difference between what is called Grid-Based methods (like Finite differences) and Mesh-Based methods (like ???) ?

Answer 1

There is no firm distinction between grids and meshes. However, they are often used in slightly different ways. The following definitions are more guidelines of common usage than actual rules and you may hear people use them interchangeably in many cases.

Grids are typically a set of simulation elements that have a well defined structure to their alignment with square or rectangular grids being the most prototypical.

Meshes are often more general. They may be unstructured and use various shapes of elements, sometimes even mixing elements of different types in the same mesh.

Finally, both Eulerian and Lagrangian methods may use meshes and grids. There is even a third class of method called Arbitrary Lagrangian Eulerian Methods which uses a grid that may be either or neither Eulerian or Lagrangian. Sometimes people use the term mesh-based methods or grid-based methods to refer to the class of all methods that use meshes or grids. In this case I don't believe there is any intention to distinguish between meshes and grids, rather the intention is to distinguish methods that use a mesh or a grid as a key part of the method from those that require neither.

Answer 2

I agree with all the above replies and comments. Since this is about common practice adopted in natural English laguage rather than a formal defintion of a word in math, different people may adopt different conventions. Some practices that I saw:

• meshes are more genernal than grids in two senses:

1. Many people tend to call a rectangular grid in real space as grid and a curved grid in real space as mesh.
2. When one says "a grid", they may refer to the grid-points in the grid. When one says "a mesh", they may refer to both the mesh-points and the surfaces and volumes formed by the mesh-points.

As to Euler vs. Lagrangian, the original motivation is to distinguish static points and moving points. Since it is usually hard to maintain a mesh for moving points, Largrangina methods tend to not making use of mesh (i.e., neighborhood) in thier computations (such as computing derivatives) and use Mote-Carlo random samplings (usually in velocity space), in which case, one will not say the sampling points form a mesh, i.e., the methods are meshless.