Understanding multidimensional decaying turbulent flows, advection-dominated flows, and convection-diffusion models requires the integration of accurate numerical structures, techniques, and simulated solutions. The choice of mathematical techniques for solving complex flow situations depends on the desired efficiency and accuracy of the numerical algorithms [1]. Solutions exist in the literature for preconditioned block-circulant systems that originate from the discretization of 3D convection [2]. But how might we design accurate multigrid algorithms for solving various diffusion equations? In particular, how do we consistently solve higher-order discontinuous Galerkin discretizations of advection-dominated flows?

The authors of this paper present a space-time efficient solution to do just that. They review the types of numerical methods for understanding the algorithms for smoothing higher-order discontinuous Galerkin discretizations of advection-dominated flows. A novice to the area of numerical mathematics will want to browse the solutions to ordinary differential equations, eigenvalues [3], and boundary value problems [4] prior to exploring the insightful ideas advocated in this remarkable paper.

The authors propose an algorithm that uses mesh sizes and a series of smaller-order discretizations to obtain the polynomial order of multigrid functions for computing these discretizations at grainier levels. In the recurrent Galerkin discretization algorithm, Legendre polynomials are concurrently used to discretize the time and space variables, to make the discrete Fourier analysis applicable to the performance analysis of the discontinuous Galerkin discretizations for advection-dominated flows. This new multigrid algorithm consists of procedures for extracting high-performance boundary layers and smoothing the solutions to higher-order discontinuous Galerkin discretizations of advection-dominated flows.

The novel multigrid algorithm was applied to a steady state machine code matrix manipulation problem, with 64-by-64 Fourier modes of operations to verify its accuracy and efficiency. The new algorithm was as equally accurate as, but more efficient in matrix manipulation than, traditional Fourier matrix manipulation code. The experimental cubic polynomials used to validate both space and time discretization of the bid multigrid algorithm are accurate and fast enough to recommend the method for solving discontinuous Galerkin discretizations of advection-dominated flows.

Will Part 2 of this paper, in which the authors advocate the additional optimization of smoothing strategies for solving higher-order discontinuous Galerkin discretizations of advection-dominated flows, be insightful? Will future multigrid algorithms for solving these discretizations be faster and applicable to more varieties of boundary layers and highly stretched problems? All applied mathematicians ought to partake in this revolutionary technological debate.