Title: A New Computational Method for Energetic Particle Acceleration and Transport with its Feedback

Abstract: We have developed a new computational method to explore astrophysical and heliophysical phenomena, especially those considerably influenced by non-thermal energetic particles. This novel approach considers the backreaction from these energetic particles by incorporating the non-thermal fluid pressure into Magnetohydrodynamics (MHD) equations. The pressure of the non-thermal fluid is evaluated from the energetic particle distribution evolved through Parker's transport equation, which is solved using stochastic differential equations. We implement this method in the HOW-MHD code (Seo \& Ryu 2023), which achieves 5th-order accuracy. We find that without spatial diffusion, the method accurately reproduces the Riemann solution in the hydrodynamic shock tube test when including the non-thermal pressure. Solving Parker's transport equation allows the determination of pressure terms for both relativistic and non-relativistic non-thermal fluids with adiabatic indices $\gamma_{\rm{NT}}=4/3$ and $\gamma_{\rm{NT}}=5/3$, respectively. The method also successfully replicates the Magnetohydrodynamic shock tube test with non-thermal pressure, successfully resolving the discontinuities within a few cells. Introducing spatial diffusion of non-thermal particles leads to marginal changes in the shock but smooths the contact discontinuity. Importantly, this method successfully simulates the energy spectrum of the non-thermal particles accelerated through shock, which includes feedback from the non-thermal population. These results demonstrate that this method is very powerful for studying particle acceleration when a significant portion of the plasma energy is taken by energetic particles.