Study Of Parallel Algorithms For The Solution Of Separable Elliptic Equation Based On GPU

In the plasma numerical simulation, it is an indispensable step to get electromagnetic field by solving elliptic partial differential equation. With the expansion of the simulation system, the time of simulation is longer than before. So, it is very significant to research the algorithms for the solution of elliptic partial differential equation and also important to research the implementation method on optimization to further improve the efficiency of plasma simulation.Therefore, parallel algorithm for the solution of separable elliptic equation based on GPU would be used to carry out a detailed study in this paper.The preface gives a detailed introduction on GPU development and the formation of CUDA (Computer Unified Device Architecture).In Chapter2, we choose the Jacobi iteration algorithm based on GPU to solve the Laplace equation over a rectangular region. On the foundation of deep analysis of Jacobi iteration algorithm, we choose the proper threads allocation and data storage mode to improve the simulation efficiency on GPU. This chapter proposes a GPU acceleration model. We adopt the Boolean value judgment in the circulation body to eliminate the data replication statement ST2of iteration algorithm to reduce the number of switches between device and host, thus reducing the computing time.Besides, we use the texture memory storage devices to reduce the amount of data reuse, thereby reducing the GPU data read in time.By solving the example of Laplace equation we find that the maximum speed-up of GTX570is24for FLOAT, and the maximum speed-up efficiency of GTX570is50%for DOUBLE. While the maximum speed-up of GT430is3for FLOAT, and the maximum speed-up efficiency of GTX570is60%for DOUBLE.The result also shows that DOUBLE could give more accurate results.In order to improve the efficiency of algorithms, we use DRM(Dimension Reduction Method) and FACR(L)(Fourier analysis-cyclic reduction) which are non-iteration generalization algorithms to solve the elliptic partial differential equations, the simulation results show that the DRM and the FACR(L) algorithm comparing with the Jacobi iteration algorithm could solve the separable elliptic equation with much less time, and could also give precise results. Then we research the DRM and the FACR(L) to solve the separable elliptic equations based on GPU.First we modified the FACR (L) calculation steps to make the operation time concentrated in FFT. Then we used CUFFT libraries to accelerate FFT part. Finally we used the algorithm based on GPU to solve the examples of Poisson equation and the Helmhotz equation in rectangular coordinate system and cylindrical coordinate system. The simulation results show that the maximum speed-up of GTX570is3.8, and the maximum speed-up efficiency of GTX570is15%. While the maximum speed-up of GT430is1.8, and the maximum speed-up efficiency of GT430is60%.With the increasing of grids, the speed-up of the total program increases. But because of the limitation of GPU memory, the grids is small, and the speed-up is much less than the theoretical speed-up.