求解Hasegawa-Mima等离子体模型的Newton型方法
Newton Type Methods for solving a Hasegawa-Mima Plasma Model
摘要：
在文献[1]中，非线性空间Hasegawa-Mima等离子体方程被表示为两个线性偏微分方程的耦合系统，其解是一对（u，w）。第一个方程是双曲型的，第二个方程是椭圆型的。得到了具有周期边界条件的Hasegawa-Mima初值问题弱解的变分框架。在最近的工作[2]中，一种由有限元空间域和欧拉隐式时间格式相结合的数值方法被用于离散耦合变分Hasegawa-Mima模型。这种隐式非线性格式的半线性版本进行了测试的几种类型的初始条件。这种半线性方案被证明是缺乏效率的长时间，这需要强加一个上限的解决方案的大小。为了克服这一困难，在本文中，我们使用牛顿型方法（牛顿，弦和介绍了变形牛顿法）来解决数值的全隐式非线性格式。在FreeFEM++中测试这些方法表明有显着的改进，因为不需要长时间施加上限。在续集中，我们证明了这些方法的有效性，通过证明几个结果，特别是实施的方法的收敛性。
英文摘要：
In [1], the non-linear space-time Hasegawa-Mima plasma equation is formulated as a coupled system of two linear PDE's, a solution of which is a pair (u, w). The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa-Mima problem with periodic boundary conditions were also derived. In a more recent work [2], a numerical approach consisting of a finite element space-domain combined with an Euler-implicit time scheme was used to discretize the coupled variational Hasegawa-Mima model. A semi-linear version of this implicit nonlinear scheme was tested for several types of initial conditions. This semi-linear scheme proved to lack efficiency for long time, which necessitates imposing a cap on the magnitude of the solution. To circumvent this difficulty, in this paper, we use Newton-type methods (Newton, Chord and an introduced Modified Newton method) to solve numerically the fully-implicit non-linear scheme. Testing these methods in FreeFEM++ indicates significant improvements as no cap needs to be imposed for long time. In the sequel, we demonstrate the validity of these methods by proving several results, in particular the convergence of the implemented methods.
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