An Adaptive Multigrid Technique for Evaluating Long-Range Forces in Biomolecular Simulations

We propose an adaptive multigrid method for evaluating the nonlocal, long-range interactions in biomolecular models. The method provides O(h3) accuracy for the forces, where h is the mesh size of the finest cubic grid, even though the stencil size is minimal. Two new algorithmic components are developed: an adaptive technique for spreading a three-dimensional charge distribution onto a space mesh, and a procedure for performing coarse-to-fine grid interpolation of the potential, force components, and their derivatives (grid-to-particle interpolation at the finest level) in a consistent fashion. The adaptive mesh used in the spreading procedure contains two types of nodes: regular, positioned at nodes cubic grid, and flexible, one per cubic cell. Thus, the charges located in each grid cell are spread to these nine nodes. The method conserves charge, dipole and quadrupole moments of cells. The spreading and interpolation techniques are economical in computation time and storage due to overlap design in the stencils. Implementation is also straightforward. Our initial tests demonstrate the favorable performance for low accuracy with respect to the fast multipole technique.

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