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In order to address the noise problems occuring in Particle-In-Cell (PIC) simulations of intense particle beams, we have been investigating numerical methods based on the solution of the Vlasov equation on a grid of phase-space. However, especially for high intensity beam simulations in periodic or alternating gradient focusing fields, where particles are localized in phase space, adaptive strategies are required to get computationally efficient codes based on this method. To this aim, we have been developing fully adaptive techniques based on interpolating wavelets where the computational grid is changed at each time step according to the variations of the distribution function of the particles. Up to now we only had an adaptive axisymmetric code. In this talk, we are going to present a new adaptive code solving the paraxial Vlasov equation on the full 4D transverse phase space, which can handle real two-dimensional problems like alternating gradient focusing. In order to develop this code efficiently, we introduce a hierarchical sparse data structure, which enabled us not only to reduce considerably the computation time but also the required memory. All computations and diagnostics are performed on the sparse data structure so that the complexity becomes proportional to the number of points needed to describe the particle distribution function.
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