The Scheduled Relaxation Jacobi (SRJ) method is a linear solver algorithm which greatly improves the convergence of the Jacobi iteration through the use of judiciously chosen relaxation factors (an SRJ scheme) which attenuate the solution error. Until now, the method has primarily been used to accelerate the solution of elliptic PDEs (e.g. Laplace, Poisson's equation) as the currently available schemes are restricted to solving this class of problems. The goal of this paper is to present a methodology for constructing SRJ schemes which are suitable for solving non-elliptic PDEs (or equivalent, nonsymmetric linear systems arising from the discretization of these PDEs), thereby extending the applicability of this method to a broader class of problems. These schemes are obtained by numerically solving a constrained minimization problem which guarantees the solution error will not grow as long as the linear system has eigenvalues which lie in certain regions of the complex plane. We demonstrate that these schemes are able to accelerate the convergence of standard Jacobi iteration for the nonsymmetric linear systems arising from discretization of the 1D and 2D steady advection-diffusion equations.

翻译：雅各比(SRJ)法是一种线性求解算法,它通过使用明智选择的放松因数(SRJ办法),大大改进了雅各变异的趋同,从而减轻了解决办法错误;到目前为止,这种方法主要用于加速解决椭圆式PDE(例如Laplace,Poisson的等式),因为现有办法仅限于解决这类问题;本文件的目的是提出一种方法,用以构建适合解决非电子化PDE(或等效的这些PDE离散产生的非对称线性线性系统)的SRJ办法,从而将这种方法的适用性扩大到更广泛的问题类别。这些办法是通过从数字上解决一个有限的最小化问题获得的,保证解决办法错误不会随着线性系统在复杂平面的某些地区的叶素价值而扩大。我们证明,这些办法能够加速因离散式1D和2稳定D等离析产生的非对称线性线性线性线性系统的标准雅集成。