Efficiency of adaptive mesh algorithms

Transcription

1 Efficiency of adaptive mesh algorithms Jörn Behrens KlimaCampus, Universität Hamburg Model for adaptive efficiency 10 8 = 0.75, = 0.50,!= 3.00, = 10.00! Assumptions: Tree refinement (binary tree here) Overheads for o refinement criterion o Less efficient numerics o Mesh management Refinement area as fraction Computational work = no. of unknowns! : Area of refinement large (α=3/4) Overheads large: number of cells/work uniform adaptive level of refinement o Refinement criterion (χ=1/2 of a computational step) o Numerics slow (ω= 10 times slower as uniform comp.) o Mesh management demanding (µ= 3 times a comp. step) 1

2 Model for adaptive efficiency small adaptive area! Now: Area of refinement small (α=1/10) Rest unchanged 10 8 = 0.10, = 0.50,!= 3.00, = number of cells/work uniform adaptive level of refinement Model for adaptive efficiency small adaptive area! Now: Area of refinement large (α=3/4) But efficient grid management and numerics! 10 8 = 0.75, = 0.10,!= 0.50, = number of cells/work uniform adaptive level of refinement 2

3 Simulation Codes! Finite element code based on P1NC-P1 FEM pair! Discontinuous Galerkin code based on linear or quadratic triangular discr. FEM Equations (non-linear SWE): Momentum in advection form Continuity in flux form (mass conservation!) Right hand side:! Coriolis! Bottom friction! Eddy diffusion 3

4 FEM Boundary conditions (solid) Boundary conditions (open) Boundary conditions (inundation) Extrapolation of wave height Extrapolation based on neighbors Lynnet, Wu, Liu, 2002 FEM Time discretization (leap-frog): Robert-Aselin type filter Simple Damping Low-order 4

5 FEM Space discretization (P1NC-P1): Conforming linear for ζ and H: Non-Conforming linear for v Hanert et al. (2005) Advection Projection (P1 advection) Androsov, J.B., Danilov (2011) DG Equations in flux form (non-linear SWE): with 5

6 DG Flux-limited explicit RK time stepping Girlado, Hesthaven, Warburton (2002) Zhang, Xia, Zhu (2011) Beisiegel (2012) Model Problem Gravity wave propagation in simple domain Model Domain Initial values 6

7 Efficiency metrics Metrics for adaptive efficiency can be based on! # DOFs! Convergence! Computing time! Scalability (single node)! Overhead DOF based metrics # DOFs for given resolution! Depends on size of high resolution domain (fraction of total area)! Depends on efficiency of refinement criterion! Coarse level 8, fine level ! Ratio:

8 DOF based metrics # DOFs for given error! Depends on size of high resolution domain (fraction of total area)! Depends on efficiency of refinement criterion Numerical Methods: DG A grid independent numerical method A. Müller (2009/2012) 8

9 Convergence based metrics Convergence over # DOFs Convergence based metrics Error with fixed #DOFs! Depends on size of high resolution domain (fraction of total area)! Depends on efficiency of refinement criterion cells uniform cells uniform 9

10 Time based metrics Computing time for given resolution! Depends on size of high resolution domain (fraction of total area) FEM: Adaptive: 15.8 s Uniform: 173 s DG: Adaptive: 302 s Uniform: 6484 s Time based metrics Computing time for given error! Depends on size of high resolution domain (fraction of total area)! Related to DOFs for given error Error: 10-7 Adaptive: 6.1 s (5k nodes) Uniform: s (66k nodes) 10

11 Time based metrics Error for given computing time! Depends on size of high resolution domain (fraction of total area)! Related to DOFs for given error Computing time: 230 s Adaptive: 4.7e-4 Uniform: 1.8e-3 Time based metrics Computing time per node! Measures efficiency of data structures... 11

12 Scaling metrics Scaling of DOFs over resolution! Related to highest possible resolution Scaling metrics Time scaling over resolution! Related to time to solution 12

13 Scaling metrics Time scaling over error! Related to time to solution of certain accuracy! depends on covergence and refinement criterion Measuring the overhead Time for adaptation vs. total time step time Shallow Water Dx: Total computing time: 1050 s Time per step: 4.5 s Time for adaptation: 0.12 s Fraction: 3% Semi-Lagrangia Advection Dx: Total computing time: 226s Time per step: 15.2 s Time for adaptation: 10.7 s Fraction: 70% 13

14 Ongoing!! Diverse Metrics! Efficiency with respect to parallelization...! Efficiency with respect to cache efficiency... 14