The meaning of the "mass matrix" on a PDE

Question

Caveat

Note, I have reviewed the question below with a similar name, and this is not a duplicate. I am asking about the Mass matrix on a PDE while the reference below is asking about the Mass matrix on a PDE discretization.

What the mass matrix represents?

Question I was trying to augment my understand of what a mass matrix does in a PDE. Now I might be using the wrong term here, but allow me to explain with a simple example using the 2D general heat equation

$$ \rho c_p \frac{\partial T}{\partial t} + \rho c_p \textbf{u}\cdot \nabla T - \nabla \cdot(K\nabla T) = f $$

I am talking specifically about the $\rho c_p$ on the first term of the equation. It always kinda bothered me when I saw this formulation, because I was not sure how this $\rho c_p$ coefficient or matrix in 2D came to be on the time derivative term. Like is it a physics thing? Or is it about expressing the equation in a unitful formulation instead of unitless formulation?

I get that having the $\rho c_p$ term on the time derivative term perhaps makes the math look nice, and we can divide both sides of the equation by $\rho c_p$ to remove the $\rho c_p$ in front of the time derivative. However, I was just wondering whether there might be some other meaning that I am missing when the $\rho c_p$ is applied to the time derivative.

Answer 1

There are two inter-related questions here:

1. What does $\rho c_p$ represent in this equation specifically? This is coming from the physical fact that the entire $\rho c_p \frac{\partial T}{\partial t}$ term is supposed to represent the volumetric heating rate of a material point, and the $\rho c_p$ term represents the appropriate material properties (density and heat capacity, respectively) to convert changes in temperature to changes in volumetric heat energy. As you mention, one could just divide by this term and get an equation for temperatures directly.
2. What is a mass matrix in general? It turns out that there are a lot of different things that are called "mass matrices" in differential equations which have different meanings. The link you posted describes a usage associated with quantifying the inner product of basis elements of certain spaces. A usage that seems more in line with what I think you're proposing, is that it is the matrix/linear operator that represents a "mass-like" quantity in a physical formulation of a PDE/ODE.

For example, it is common in introductory dynamics to refer to $M$ in the following ODE system:

$$M\vec{x}'' + C\vec{x}' + K\vec{x} = \vec{0}$$

as the mass matrix, and $K$ as the stiffness matrix, because they play those roles when interpreting this equation as modeling a system of dampened interconnected springs. This is likely to be the context in which your $c_p\rho$ term is being interpreted, since it shares a "mass-like" physical interpretation due to representing thermal inertia in that PDE.