  Modern astronomy has finally been able to observe protoplanetary disks in
reasonable resolution and detail, unveiling the processes happening during
planet formation. These observed processes are understood under the framework
of disk-planet interaction, a process studied analytically and modeled
numerically for over 40 years. Long a theoreticians' game, the wealth of
observational data has been allowing for increasingly stringent tests of the
theoretical models. Modeling efforts are crucial to support the interpretation
of direct imaging analyses, not just for potential detections but also to put
meaningful upper limits on mass accretion rates and other physical quantities
in current and future large-scale surveys. This white paper addresses the
questions of what efforts on the computational side are required in the next
decade to advance our theoretical understanding, explain the observational
data, and guide new observations. We identified the nature of accretion, ab
initio planet formation, early evolution, and circumplanetary disks as major
fields of interest in computational planet formation. We recommend that
modelers relax the approximations of alpha-viscosity and isothermal equations
of state, on the grounds that these models use flawed assumptions, even if they
give good visual qualitative agreement with observations. We similarly
recommend that population synthesis move away from 1D hydrodynamics. The
computational resources to reach these goals should be developed during the
next decade, through improvements in algorithms and the hardware for hybrid
CPU/GPU clusters. Coupled with high angular resolution and great line
sensitivity in ground based interferometers, ELTs and JWST, these advances in
computational efforts should allow for large strides in the field in the next
decade.
