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Numerical effects are known to plague adaptive mesh refinement (AMR) codeswhen treating massive particles, e.g. representing massive black holes (MBHs).In an evolving background, they can experience strong, spurious perturbationsand then follow unphysical orbits. We study by means of numerical simulationsthe dynamical evolution of a pair MBHs in the rapidly and violently evolvinggaseous and stellar background that follows a galaxy major merger. We confirmthat spurious numerical effects alter the MBH orbits in AMR simulations, andshow that numerical issues are ultimately due to a drop in the spatialresolution during the simulation, drastically reducing the accuracy in thegravitational force computation. We therefore propose a new refinementcriterion suited for massive particles, able to solve in a fast and precise wayfor their orbits in highly dynamical backgrounds. The new refinement criterionwe designed enforces the region around each massive particle to remain at themaximum resolution allowed, independently upon the local gas density. Suchmaximally-resolved regions then follow the MBHs along their orbits, andeffectively avoids all spurious effects caused by resolution changes. Our suiteof high resolution, adaptive mesh-refinement hydrodynamic simulations,including different prescriptions for the sub-grid gas physics, shows that thenew refinement implementation has the advantage of not altering the physicalevolution of the MBHs, accounting for all the non trivial physical processestaking place in violent dynamical scenarios, such as the final stages of agalaxy major merger.

Massive black hole and gas dynamics in galaxy nuclei mergers. I. Numerical implementation