A Large-Scale Benchmark for the Incompressible Navier-Stokes Equations

Numerical methods for solving incompressible Navier-Stokes equations are widely applied and are among the most extensively studied. A plethora of approaches have been introduced targeting different applications and regimes; these methods vary in space discretization, choice of time integration scheme, or reduction of the non-linear equations to a sequence of simpler problems. While the theoretical properties of most of these variants are known, it is still difficult to pick the best option for a given problem, as practical performance of these methods has never been systematically compared over a varied set of geometries and boundary conditions. We introduce a collection of benchmark problems in 2D and 3D (geometry description and boundary conditions), including simple cases with known analytic solution, classical experimental setups, and complex geometries with fabricated solutions for evaluation of numerical schemes for incompressible Navier-Stokes equations in laminar flow regime. We compare the performance of a representative selection of most broadly used algorithms for Navier-Stokes equations on this set of problems. Where applicable, we compare the most common spatial discretization choices (unstructured triangle/tetrahedral meshes and structured or semi-structured quadrilateral/hexahedral meshes). The study shows that while the type of spatial discretization used has a minor impact on the accuracy of the solutions, the choice of time integration method, spatial discretization order, and the choice of solving the coupled equations or reducing them to simpler subproblems have very different properties. Methods that are directly solving the original equations tend to be more accurate than splitting approaches for the same number of degrees of freedom, but numerical or computational difficulty arise when they are scaled to larger problem sizes. Low-order splitting methods are less accurate, but scale more easily to large problems, while higher-order splitting methods are accurate but require dense time discretizations to be stable. We release the description of the experiments and an implementation of our benchmark, which we believe will enable statistically significant comparisons with the state of the art as new approaches for solving the incompressible Navier-Stokes equations are introduced.