JFNK方法在SN输运计算中的应用

JFNK(Jacobian-free Newton-Krylov)方法是一种求解非线性方程的高效迭代算法。传统输运计算中的负通量修正与k-特征值迭代使得原本线性的输运计算转变为非线性问题数值求解。为提高非线性输运问题的计算效率,将这两类非线性问题离散成残差形式的非线性方程组,并采用JFNK方法对其进行迭代求解。分析不同约束条件对JFNK方法性能的影响,并将其与NK(Newton-Krylov)方法进行对比。针对JFNK方法的内迭代过程,分析两类子空间方法(GMRES(m)与LGMRES)对整体计算效率的影响。数值结果表明：①相比于传统的幂迭代方法,JFNK方法具有更高的计算效率;②Jacobian矩阵向量积的差分近似对结果没有影响,且基于物理的约束条件比标准的数学约束更加高效;③LGMRES可以充分利用子空间的信息,从而使得JFNK方法整体表现更加高效。     

JFNK方法迭代过程与物理约束

对Jacobian-free Newton-Krylov(JFNK)方法迭代过程进行分析,通过在迭代过程中吸收物理约束信息,对JFNK方法进行改进.改进后的JFNK方法迭代过程中的迭代序列总是满足物理约束,克服了迭代过程中可能出现的非物理现象.采用改进之后的算法求解二维三温能量方程,可以保证在迭代过程中不会出现负温度问题,使JFNK方法的健壮性得到提高.     

A spectral deferred correction strategy for low Mach number reacting flows subject to electric fields

We propose an algorithm for low Mach number reacting flows subjected to electric field that includes the chemical production and transport of charged species. This work is an extension of a multi-implicit spectral deferred correction (MISDC) algorithm designed to advance the conservation equations in time at scales associated with advective transport. The fast and nontrivial interactions of electrons with the electric field are treated implicitly using a Jacobian-Free Newton Krylov approach for which a preconditioning strategy is developed. Within the MISDC framework, this enables a close and stable coupling of diffusion, reactions and dielectric relaxation terms with advective transport and is shown to exhibit second-order convergence in space and time. The algorithm is then applied to a series of steady and unsteady problems to demonstrate its capability and stability. Although developed in a one-dimensional case, the algorithmic ingredients are carefully designed to be amenable to multi-dimensional applications.