Rolling mapping/mesh: a mapping method/interconnection structure for a mesh with various densities

In order to solve the partial differential equations (PDEs), the target physical region is divided into a nearest neighbor mesh (NNM). Although a dense mesh improves the accuracy of the solution, the computation time is stretched with a large amount of calculation. Thus, a mesh with partially varied densities improves the accuracy locally without explosion of the computation cost. For calculations of PDEs, parallel machines in which processing units (PUs) are connected with an NNM connection structure have been utilized because the required communication is usually matched with the connection structure. However, if a mesh with various densities is utilized, the PU-assigning algorithm (mapping) becomes difficult. Conventional mapping methods increase communication between remote PUs and require the cyclic links for the PU array. Here, we propose a mapping method called the rolling mapping for assigning tasks of the PDE into multiprocessor with the NNM structure. Using this method, the mesh with various densities are mapped into PU array without cyclic links nor increasing the data exchange between remote PUs. With the scheduling method, the overhead caused by remaining remote communication can be hidden as possible. An extended NNM structure called the rolling mesh is also proposed. With this structure, every message can be communicated directly between PUs