Scheduled Relaxation Jacobi Schemes

Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the below specified tolerances. The in the reduction of the residual increases with the number levels employed in the algorithm. Applying original methodology to compute the algorithm parameters with more 5 levels notably hinders obtaining SRJ schemes, as the mixed (non-linear) algebraic-differential from which they result become stiff. Here we present a new methodology for obtaining the of SRJ schemes that overcomes the limitations of the original algorithm and provide parameters for SRJ with up to 15 levels and resolutions of up to 215 points per dimension, allowing for acceleration factors than several hundreds with respect to the Jacobi method for typical resolutions and, in some high cases, close to 1000. Furthermore, we extend the original algorithm to apply it to certain systems of non-linear ePDEs.
Below we include in a single tarball the files needed to use directly the weights ω over a complete M-cycle for different resolutions (/N/) and assuming that Neumann boundary conditions are employed in a two-dimensional problem. If different boundary conditions (e.g., Dirichlet) or number of dimensions (/d/) are employed, please see our paper (Eqs. 10 and 11) to compute the effective value of /N/ you need to use