GePUP swMATH ID: 15834 Software Authors: Zhang, Qinghai Description: GePUP: generic projection and unconstrained PPE for fourth-order solutions of the incompressible Navier-Stokes equations with no-slip boundary conditions. A generic projection maps one vector to another such that their difference is a gradient field and the projected vector does not have to be solenoidal. Via a commutator of Laplacian and the generic projection, the projected velocity is formulated as the sole evolutionary variable with the incompressibility constraint enforced by a pressure Poisson equation so that the dissipation of velocity divergence is governed by a heat equation. Different from previous projection methods, the GePUP formulation treats the time integrator as a black box. This prominent advantage is illustrated by straightforward formations of a semi-implicit time-stepping scheme and another explicit time-stepping scheme. Apart from its stability, the GePUP schemes have an optimal efficiency in that within each time step the solution is advanced by solving a sequence of linear systems with geometric multigrid. A key component of the GePUP schemes is a fourth-order discrete projection for no-penetration domains. Results of numerical tests in two and three dimensions demonstrate that the GePUP schemes are fourth-order accurate both in time and in space. To facilitate efficiency comparison to other methods, a simple formula is introduced. Systematic arguments and timing results show that the GePUP schemes could be vastly superior over lower-order methods in terms of efficiency and accuracy. In some cases, the GePUP schemes running on the author’s personal desktop would be faster than a second-order method running on the fastest supercomputer in the world! This paper contains enough details so that one can reproduce the numerical results by following the exposition. Homepage: http://link.springer.com/article/10.1007%2Fs10915-015-0122-4 Keywords: incompressible Navier-Stokes equations; no-slip boundary conditions; generic projection; unconstrained pressure Poisson equation; lid-driven cavity; Laplace-Leray commutator Related Software: MARS; AMRCFI; SADoRe; FEniCS; Gmsh; FIAT; BoxLib; BiCGstab; mctoolbox; PROST; Gerris; Matlab; Chombo; MARS Cited in: 8 Publications all top 5 Cited by 12 Authors 6 Zhang, Qinghai 1 Fogelson, Aaron L. 1 Hussaini, M. Yousuff 1 Li, Zhixuan 1 Lin, Zhi 1 Pei, Chaoxu 1 Rosales, Rodolfo Ruben 1 Seibold, Benjamin 1 Shirokoff, David 1 Sussman, Mark 1 Vahab, Mehdi 1 Zhou, Dong all top 5 Cited in 6 Serials 2 SIAM Journal on Numerical Analysis 2 Journal of Scientific Computing 1 Computer Methods in Applied Mechanics and Engineering 1 Journal of Computational Physics 1 Mathematics of Computation 1 SIAM Journal on Scientific Computing all top 5 Cited in 7 Fields 8 Fluid mechanics (76-XX) 7 Numerical analysis (65-XX) 2 Ordinary differential equations (34-XX) 2 Partial differential equations (35-XX) 1 Mathematical logic and foundations (03-XX) 1 Algebraic geometry (14-XX) 1 Biology and other natural sciences (92-XX) Citations by Year