Stochastic goal-oriented error estimation with memory. (English) Zbl 1422.76131
Summary: We propose a stochastic dual-weighted error estimator for the viscous shallow-water equation with boundaries. For this purpose, previous work on memory-less stochastic dual-weighted error estimation is extended by incorporating memory effects. The memory is introduced by describing the local truncation error as a sum of time-correlated random variables. The random variables itself represent the temporal fluctuations in local truncation errors and are estimated from high-resolution information at near-initial times. The resulting error estimator is evaluated experimentally in two classical ocean-type experiments, the Munk gyre and the flow around an island. In these experiments, the stochastic process is adapted locally to the respective dynamical flow regime. Our stochastic dual-weighted error estimator is shown to provide meaningful error bounds for a range of physically relevant goals. We prove as well as show numerically that our approach can be interpreted as a linearized stochastic-physics ensemble.
MSC:
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35Q86 | PDEs in connection with geophysics |
| 86A05 | Hydrology, hydrography, oceanography |
Keywords:
computational fluid dynamics; oceanography; adjoint; ensemble method; Mori-Zwanzig formalism; error estimationReferences:
| [1] | Prudhomme, S.; Oden, J. T., On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Eng., 176, 1, 313-331 (1999) · Zbl 0945.65123 |
| [2] | Becker, R.; Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 2001, 10, 1-102 (2001) · Zbl 1105.65349 |
| [3] | Giles, M., On adjoint equations for error analysis and optimal grid adaptation in CFD, (Frontiers of Computational Fluid Dynamics 1998 (1998), World Scientific Pub. Co. Inc.), 155-169 · Zbl 0989.76050 |
| [4] | Giles, M. B.; Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality, Acta Numer., 11, 145-236 (2002) · Zbl 1105.65350 |
| [5] | Fidkowski, K. J.; Darmofal, D. L., Review of output-based error estimation and mesh adaptation in computational fluid dynamics, AIAA J., 49, 4, 673-694 (2011) |
| [6] | Grätsch, T.; Bathe, K.-J., Review: a posteriori error estimation techniques in practical finite element analysis, Comput. Struct., 83, 235-265 (Jan. 2005) |
| [7] | Venditti, D. A.; Darmofal, D. L., Adjoint error estimation and grid adaptation for functional outputs: application to quasi-one-dimensional flow, J. Comput. Phys., 164, 1, 204-227 (2000) · Zbl 0995.76057 |
| [8] | Rauser, F.; Marotzke, J.; Korn, P., Ensemble-type numerical uncertainty information from single model integrations, J. Comput. Phys., 292, 30-42 (2015) · Zbl 1349.86011 |
| [9] | Roy, C. J., Review of Discretization Error Estimators in Scientific Computing (2010), AIAA Paper 2010-0126 |
| [10] | Mori, H.; Fujisaka, H.; Shigematsu, H., A new expansion of the master equation, Prog. Theor. Phys., 51, 1, 109-122 (1974) |
| [11] | Mori, H., Transport, collective motion, and Brownian motion, Prog. Theor. Phys., 33, 3, 423-455 (1965) · Zbl 0127.45002 |
| [12] | Zwanzig, R., Nonlinear generalized Langevin equations, J. Stat. Phys., 9, 3, 215-220 (1973) |
| [13] | Givon, D.; Kupferman, R.; Stuart, A., Extracting macroscopic dynamics: model problems and algorithms, Nonlinearity, 17, 6, R55 (2004) · Zbl 1073.82038 |
| [14] | Chorin, A. J.; Hald, O. H., Estimating the uncertainty in underresolved nonlinear dynamics, Math. Mech. Solids, Article 1081286513505465 pp. (2013) |
| [15] | Bonaventura, L.; Ringler, T., Analysis of discrete shallow-water models on geodesic Delaunay grids with C-type staggering, Mon. Weather Rev., 133, 8, 2351-2373 (2005) |
| [16] | Rípodas, P.; Gassmann, A.; Förstner, J.; Majewski, D.; Giorgetta, M.; Korn, P.; Kornblueh, L.; Wan, H.; Zängl, G.; Bonaventura, L., Icosahedral Shallow Water Model (ICOSWM): results of shallow water test cases and sensitivity to model parameters, Geosci. Model Dev., 2, 2, 231-251 (2009) |
| [17] | Rauser, F.; Riehme, J.; Leppkes, K.; Korn, P.; Naumann, U., On the use of discrete adjoints in goal error estimation for shallow water equations, Proc. Comput. Sci., 1, 1, 107-115 (2010) |
| [18] | Giles, M. B.; Pierce, N. A., Adjoint error correction for integral outputs, (Barth, T.; Deconinck, H., Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics. Error Estimation and Adaptive Discretization Methods in Computational Fluid Dynamics, Lecture Notes in Computational Science and Engineering, vol. 25 (2003), Springer: Springer Berlin, Heidelberg), 47-95 · Zbl 1128.76356 |
| [19] | Munk, W. H., On the wind-driven ocean circulation, J. Meteorol., 7, 2, 80-93 (1950) |
| [20] | Vallis, G. K., Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation (2006), Cambridge University Press · Zbl 1374.86002 |
| [21] | Hecht, M. W.; Petersen, M. R.; Wingate, B. A.; Hunke, E.; Maltrud, M., Lateral mixing in the eddying regime and a new broad-ranging formulation, (Ocean Modeling in an Eddying Regime (2013), American Geophysical Union), 339-352 |
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