High order spline interpolations in the particle simulation

In the conventional plasma simulation models using particles, in which the total momentum is conserved, the total energy is not conserved due to nonuniformity of space caused by the spatial grids. The grid spacing was severely limited on the order of the Debye length. It is found that the use of the high order spline spatial interpolation removes this limit without a significant increase of the computation and complexity of the algorithm in the system that only the long-wavelength, collective phenomena are important. The theoretical analysis of the total energy is improved compared with the previous work and its scaling law is related to the so-called aliasing error explicitly. The coefficient eta introduced previously is referred as the model index and is used again as a measure of the accuracy of the models for the energy conservation.