Grad-Shafranov方程快速求解优化研究

发布时间：2018-03-07 23:20

本文选题：Grad-Shafranov方程　切入点：平衡重建　出处：《中国科学技术大学》2017年硕士论文　论文类型：学位论文

【摘要】：在等离子体平衡重建迭代过程中,我们需要快速求解Grad-Shafranov(G-S)方程。在目前的EAST等离子体平衡重建PEFIT代码中,采用五点差分方法通过离散正弦变换(DST)在65×65网格中进行快速求解,当网格进一步扩展到更精细的129x129或257x257时,其需要耗费的时间将成倍增长,将严重影响等离子体平衡重建的实时性。本文开展了多种改进算法的研究,并采用具有四阶精度的紧致差分格式离散化G-S方程,并在此基础上通过DST方法快速求解,获得更好的加速比,并成功应用到PEFIT代码中。本文首先描述了低网格下基于GPU的多重网格方法加速,在串行条件下,多重网格方法要优于DST方法,但多次实践证明,在使用GPU加速后多重网格算法并没有提供好的加速比。通过经验总结,证明多重网格算法无法满足实时要求。本文着重描述了如何构造具有四阶精度的紧致差分格式的Grad-Shafranov离散方程,并使用DST技术进行快速求解,实现基于紧致差分格式的快速G-S方程求解。结果表明,在65 × 65网格下,在给定方程右端项电流分布的前提下,使用GPU求解G-S方程所需时间大约为34μs。此外,本文还提出了一种扩展网格剩余节点值的估算方法。目前Grad-Shafranov方程用于实时算法是在二维空间下进行的,而Grad-Shafranov方程属于线性椭圆偏微分方程的一种,本文尝试了将离散正弦变换技术应用到规则区域内三维空间下椭圆偏微分方程的快速求解方法中,并使用了 CUDA进行了加速。
[Abstract]:In the iterative process of plasma equilibrium reconstruction, we need to solve the Grad-Shafranov-G-S) equation quickly. In the current EAST plasma equilibrium reconstruction PEFIT code, the five-point difference method is used to solve the problem in 65 脳 65 grids by discrete sinusoidal transform. When the grid is further extended to the finer 129x129 or 257x257, the amount of time it takes will increase exponentially, which will seriously affect the real-time performance of plasma equilibrium reconstruction. The G-S equation is discretized by a compact difference scheme with fourth-order accuracy. On this basis, a better speedup can be obtained by solving the G-S equation quickly by the DST method. And successfully applied to PEFIT code. Firstly, this paper describes the acceleration of multigrid method based on GPU in low grid. Under serial condition, multigrid method is superior to DST method, but it has been proved by practice many times. The multigrid algorithm with GPU acceleration does not provide a good speedup. It is proved that the multigrid algorithm can not meet the real-time requirements. This paper mainly describes how to construct Grad-Shafranov discrete equations with fourth-order precision compact difference scheme, and use DST technique to solve them quickly. The fast solution of G-S equation based on compact difference scheme is realized. The results show that under the condition of 65 脳 65 grid and given current distribution at the right end of the equation, the time required to solve G-S equation using GPU is about 34 渭 s. In this paper, a new method for estimating the residual node value of extended grid is proposed. At present, the Grad-Shafranov equation is used in two dimensional space, and the Grad-Shafranov equation is a kind of linear elliptic partial differential equation. In this paper, the discrete sinusoidal transform technique is applied to the fast solution of elliptic partial differential equations in three dimensional space, and CUDA is used to accelerate it.
【学位授予单位】：中国科学技术大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.8;O53

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