Optimally efficient multigrid algorithms for incompressible Euler equations

Purpose - The aim of the paper is to achieve textbook multigrid efficiency for some flow problems. Design/methodology/approach - The steady incompressible Euler equations are decoupled into elliptic and hyperbolic subsystems. Numerous classical FAS-MG algorithms are implemented and tested for convergence. A full multigrid algorithm that costs less than 10 work units (WUs) is sufficient to reduce the algebraic error below the discretization error. A new algorithm `NUVMGP{''} is introduced. A two-step iterative procedure is adopted. First, given the pressure gradient, the convection equations are solved on the computational grid for the velocity components by performing one Gauss-Seidel iteration ordered in the flow direction. second, a linear multigrid (MG) cycle for Poisson's equation is performed to update pressure values. Findings - It is found that algorithm ``NUVMGP-FMG{''} requires less than 6 WU to attain the target solution. The convergence rates are independent on both the mesh size and the approximation order. Research limitations/implications - Lexicographic Gauss-Seidel using downstream ordering is a good solver for the advection terms and provides excellent smoothing rates for relaxation. But it is complicated to maintain downstream ordering in case the flow directions change with location. Originality/value - Although the scope of this work is limited to rectangular domains, finite difference schemes, and incompressible Euler equation, the same approaches can be extended for other flow problems. However, such relatively simple problems may provide deep understanding of the ideal convergence behavior of MG and accumulate experience to detect unacceptable performance and regain the optimal one.