GROK:  Why don't I see Einstein's C in Plasma Dynamics Research.

Einstein's constant c, which represents the speed of light in a vacuum, is not directly used in the mathematical modeling of plasma dynamics for several key reasons:

Plasma Dynamics' Complexity: Plasma dynamics involves the behavior of charged particles under the influence of electromagnetic fields, where collective effects and particle interactions are significant. The speed of light comes into play primarily in the context of electromagnetic wave propagation in plasma, but the direct application of c in the bulk dynamics of plasmas is less relevant because the velocities of particles in plasma can vary widely from near-zero to relativistic speeds. Instead, models often deal with the thermal, drift, or collective velocities of particles, which do not necessarily approach c.

Magnetohydrodynamics (MHD): In MHD, which is a common framework for studying plasma dynamics, the speed of light isn't a direct parameter because MHD equations are typically formulated in terms of fluid-like variables rather than individual particle velocities. 

Here, plasma is treated as a fluid with properties like density, velocity, pressure, and magnetic field, where the speed of light might appear in the context of electromagnetic field equations but not as a constant defining the system's behavior.

Kinetic Models: In kinetic models like the Vlasov equation or particle-in-cell (PIC) simulations, while individual particle velocities can be relativistic, the focus is on distribution functions and collective behavior rather than treating each particle's velocity as a fraction of c.

Regarding the methods of handling time elements in mathematical modeling of plasmas:

Discrete Time Models:
Time-Stepping Methods: These involve numerical integration where time is treated as a series of discrete steps.

 Common methods include the Euler method, Runge-Kutta methods, or more sophisticated ones like the Boris algorithm for particle motion in electromagnetic fields. 

Discrete time models are crucial in simulations where plasma evolution is tracked step-by-step, allowing for detailed control over how physical processes evolve over time.

Continuous Time Models (though not explicitly asked, it's important for context):

Analytical Solutions: When possible, equations are solved analytically, treating time as a continuous variable. This approach is less common due to the complexity of plasma dynamics but is used in simpler scenarios or steady-state analyses.

Quasi-static or Slow-Time Approximations: Here, certain aspects of plasma behavior are assumed to change on a slower time scale compared to others, allowing for simplification in the modeling process. 

This method can blend discrete and continuous time elements by considering long-term averages or stable states while still using discrete methods for dynamic changes.

These methods allow for the modeling of plasma dynamics at different scales and complexities, from rapid particle interactions to long-term stability and evolution of plasma structures.