zbMATH — the first resource for mathematics

Multiscale high-order/low-order (HOLO) algorithms and applications. (English) Zbl 1380.65316
Summary: We review the state of the art in the formulation, implementation, and performance of so-called high-order/low-order (HOLO) algorithms for challenging multiscale problems. HOLO algorithms attempt to couple one or several high-complexity physical models (the high-order model, HO) with low-complexity ones (the low-order model, LO). The primary goal of HOLO algorithms is to achieve nonlinear convergence between HO and LO components while minimizing memory footprint and managing the computational complexity in a practical manner. Key to the HOLO approach is the use of the LO representations to address temporal stiffness, effectively accelerating the convergence of the HO/LO coupled system. The HOLO approach is broadly underpinned by the concept of nonlinear elimination, which enables segregation of the HO and LO components in ways that can effectively use heterogeneous architectures. The accuracy and efficiency benefits of HOLO algorithms are demonstrated with specific applications to radiation transport, gas dynamics, plasmas (both Eulerian and Lagrangian formulations), and ocean modeling. Across this broad application spectrum, HOLO algorithms achieve significant accuracy improvements at a fraction of the cost compared to conventional approaches. It follows that HOLO algorithms hold significant potential for high-fidelity system scale multiscale simulations leveraging exascale computing.
Reviewer: Reviewer (Berlin)

65M99 Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems
76N15 Gas dynamics, general
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
86A05 Hydrology, hydrography, oceanography
65C05 Monte Carlo methods
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
Full Text: DOI
[1] Gol’din, V. Y., A quasi-diffusion method of solving the kinetic equation, USSR Comput. Math. Math. Phys., 4, 136-149, (1967) · Zbl 0149.11804
[2] Alcouffe, R. E., Diffusion synthetic acceleration methods for the diamond differenced discrete ordinates equations, Nucl. Sci. Eng., 66, 344-355, (1977)
[3] Winkler, K.-H. A.; Norman, M. L.; Mihalas, D., Implicit adaptive-grid radiation hydrodynamics, (Multiple Time Scales (A86-47618 23-64), (1985), Academic Press, Inc. Orlando, FL), 145-184
[4] Larsen, E., A grey transport acceleration method for time-dependent radiative transfer problems, J. Comput. Phys., 78, 459-480, (1988) · Zbl 0661.65146
[5] Mason, R. J., Implicit moment particle simulation of plasmas, J. Comput. Phys., 41, 2, 233-244, (1981) · Zbl 0469.76121
[6] Denavit, J., Time-filtering particle simulations with \(\omega_{p e} \operatorname{\Delta} t \gg 1\), J. Comput. Phys., 42, 2, 337-366, (1981) · Zbl 0467.76119
[7] Brackbill, J. U.; Forslund, D. W., An implicit method for electromagnetic plasma simulation in two dimensions, J. Comput. Phys., 46, 271, (1982) · Zbl 0489.76127
[8] Brackbill, J.; Forslund, D., Simulation of low-frequency electromagnetic phenomena in plasmas, (Brackbill, J. U.; Cohen, B. I., Multiple Time Scales, (1985), Academic Press)
[9] Dukowicz, J.; Smith, R., Implicit free-surface method for the Bryan-Cox-semtner Ocean model, J. Geophys. Res., 99, C4, 7991-8014, (1994)
[10] Lanzdron, P.; Rose, D.; Wilkes, J., An analysis of approximate nonlinear elimination, SIAM J. Sci. Comput., 17, 538-559, (1996) · Zbl 0855.65054
[11] Vu, H. X.; Brackbill, J. U., CELEST1D: an implicit, fully kinetic model for low-frequency, electromagnetic plasma simulation, Comput. Phys. Commun., 69, 253-276, (1992)
[12] Lapenta, G.; Brackbill, J., CELESTE 3D: implicit adaptive grid plasma simulation, (International School/Symposium for Space Simulation, Kyoto, Japan, March 13-19, (1997))
[13] Taitano, W. T.; Knoll, D. A.; Chacón, L.; Chen, G., Development of a consistent and stable fully implicit moment method for Vlasov-Ampère particle in cell (pic) system, SIAM J. Sci. Comput., 35, 5, S126-S149, (2013) · Zbl 1282.82038
[14] Knoll, D.; Park, H.; Smith, K., Application of the Jacobian-free Newton-Krylov method to nonlinear acceleration of transport source iteration in slab geometry, Nucl. Sci. Eng., 167, 122-132, (2011)
[15] Park, H.; Knoll, D. A.; Newman, C. K., Nonlinear acceleration of transport criticality problem, Nucl. Sci. Eng., 172, 52-65, (2012)
[16] Park, H.; Knoll, D. A.; Rauenzahn, R. M.; Wollaber, A. B.; Densmore, J. D., A consistent, moment-based, multiscale solution approach for thermal radiative transfer problems, Transp. Theory Stat. Phys., 41, 284-303, (2012) · Zbl 1278.82051
[17] Taitano, W.; Knoll, D.; Chacón, L.; Reisner, J.; Prinja, A., Moment-based acceleration for neutral gas kinetics with BGK collision operator, J. Comput. Theor. Transp., 43, 1-7, 83-108, (2014)
[18] Chen, G.; Chacón, L.; Leibs, C. A.; Knoll, D. A.; Taitano, W., Fluid preconditioning for Newton-Krylov-based, fully implicit, electrostatic particle-in-cell simulations, J. Comput. Phys., 258, 555, (2014) · Zbl 1349.78091
[19] Chen, G.; Chacón, L., An energy- and charge-conserving, nonlinearly implicit, electromagnetic 1D-3V Vlasov-Darwin particle-in-cell algorithm, Comput. Phys. Commun., 185, 10, 2391-2402, (2014) · Zbl 1360.78014
[20] Chen, G.; Chacón, L., A multi-dimensional, energy- and charge-conserving, nonlinearly implicit, electromagnetic Vlasov-Darwin particle-in-cell algorithm, Comput. Phys. Commun., 197, 73-87, (2015) · Zbl 1352.65405
[21] Brown, P. N.; Saad, Y., Hybrid Krylov methods for nonlinear systems of equations, SIAM J. Sci. Stat. Comput., 11, 3, 450-481, (1990) · Zbl 0708.65049
[22] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397, (2004) · Zbl 1036.65045
[23] Anderson, D. G., Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Mach., 12, 547, (1965) · Zbl 0149.11503
[24] Walker, H.; Ni, P., Anderson acceleration for fixed-point iterations, SIAM J. Numer. Anal., 49, 1715, (2013) · Zbl 1254.65067
[25] Duderstadt, J. J.; Martin, W. R., Transport theory, (1979), Wiley New York · Zbl 0407.76001
[26] Pomraning, G. C., The equations of radiation hydrodynamics, (1973), Pergamon Press Oxford, New York
[27] Park, H.; Knoll, D. A.; Rauenzahn, R. M.; Newman, C. K.; Densmore, J. D.; Wollaber, A. B., An efficient and time accurate, moment-based scale-bridging algorithm for thermal radiative transfer problems, SIAM J. Sci. Comput., 35, S18-S41, (2013) · Zbl 1285.65092
[28] Willert, J.; Park, H., Using residual Monte Carlo to solve the high-order problem within moment-based accelerated thermal radiative transfer equations, J. Comput. Phys., 276, 405-421, (2014) · Zbl 1349.82130
[29] Willert, J.; Park, H.; Knoll, D., A comparison of acceleration methods for solving the neutron transport k-eigenvalue problem, J. Comput. Phys., 274, 681-694, (2014) · Zbl 1351.82125
[30] Park, H.; Knoll, D. A.; Rauenzahn, R. M.; Wollaber, A. B.; Lowrie, R. B., Moment-based acceleration of Monte Carlo solution for multifrequency thermal radiative transfer problems, J. Comput. Theor. Transp., 43, 1-7, 314-335, (2014)
[31] Densmore, J.; Park, H.; Wollaber, A.; Rauenzahn, R.; Knoll, D., Monte Carlo simulation methods in moment-based scale-bridging algorithms for thermal radiative-transfer problems, J. Comput. Phys., 284, 40-58, (2015) · Zbl 1351.80007
[32] Willert, J.; Park, H.; Taitano, W., Using Anderson acceleration to accelerate the convergence of neutron transport calculations with anisotropic scattering, Nucl. Sci. Eng., 181, 342-350, (2015)
[33] Willert, J.; Park, H.; Taitano, W., Applying nonlinear diffusion acceleration to the neutron transport k-eigenvalue problem with anisotropic scattering, Nucl. Sci. Eng., 181, 351-360, (2015)
[34] Aristova, E. N.; Gol’din, V. Y.; Kolpakov, A., Multidimensional calculations of radiation transport by nonlinear quasi-diffusion method, (Proceedings of ANS International Conference on Mathematics and Computation, Reactor Physics and Environmental Analysis in Nuclear Applications, (1999)), 27-30
[35] Aristova, E. N., Simulation of radiation transport in a channel based on the quasi-diffusion method, Transp. Theory Stat. Phys., 37, 5-7, 483-503, (2008) · Zbl 1156.82385
[36] Anistratov, D., Consistent spatial approximation of the low-order quasi-diffusion equation on coarse grids, Nucl. Sci. Eng., 149, 138-161, (2005)
[37] Larsen, E. W.; Young, J., A functional Monte Carlo method for k-eigenvalue problems, Nucl. Sci. Eng., 159, 107-126, (2008)
[38] Anistratov, D. Y.; Gol’din, V. Y., Multilevel quasidiffusion methods for solving multigroup neutron transport k-eigenvalue problems in one-dimensional slab geometry, Nucl. Sci. Eng., 169, 2, 111-132, (2011)
[39] Wieselquist, W. A.; Anistratov, D. Y.; Morel, J. E., A cell-local finite difference discretization of the low-order quasidiffusion equations for neutral particle transport on unstructured quadrilateral meshes, J. Comput. Phys., 273, 343-357, (2014) · Zbl 1351.82068
[40] Smith, R.; Gent, P., Reference manual for the parallel Ocean program (POP), (2002), Los Alamos National Laboratory, Tech. Rep. Los Alamos Technical Report LA-UR-02-2484
[41] Lee, D.; Downar, T. J.; Kim, Y., Convergence analysis of the nonlinear coarse-mesh finite difference method for one-dimensional fixed-source neutron diffusion problem, Nucl. Sci. Eng., 147, 127-147, (2004)
[42] Kim, K.-S.; DeHart, M. D., Unstructured partial- and net-current based coarse mesh finite difference acceleration applied to the extended step characteristics method in NEWT, Ann. Nucl. Energy, 38, 2-3, 527-534, (2011)
[43] Wolters, E.; Larsen, E.; Martin, W., Hybrid Monte-Carlo-CMFD methods for accelerating fission source convergence, Nucl. Sci. Eng., 174, 286-299, (2013)
[44] Larsen, E. W.; Kelley, B. W., The relationship between the coarse-mesh finite difference and the coarse-mesh diffusion synthetic acceleration methods, Nucl. Sci. Eng., 178, 1-15, (2014)
[45] Yuk, S.; Cho, N., Whole core transport solution with 2-d/1-d fusion kernel via p-CMFD acceleration and p-CMFD embedding of nonoverlapping local/global iterations, Nucl. Sci. Eng., 181, 1-16, (2015)
[46] Honrubia, J. J., A synthetically accelerated scheme for radiative-transfer calculations, J. Quant. Spectrosc. Radiat. Transf., 49, 5, 491-515, (1993)
[47] Morel, J.; Wareing, T.; Smith, K., A linear-discontinuous spatial difference scheme for sn radiative transfer calculations, J. Comput. Phys., 128, 445-462, (1996) · Zbl 0864.65095
[48] Ramone, G. L.; Adams, M. L.; Nowak, P. F., A transport synthetic acceleration method for transport iterations, Nucl. Sci. Eng., 125, 3, 257-283, (1997)
[49] Adams, M. L.; Larsen, E. W., Fast iterative methods for discrete-ordinates particle transport calculations, Prog. Nucl. Energy, 40, 1, 3-159, (2002)
[50] Warsa, J. S.; Wareing, T.; Morel, J., Krylov iterative methods and the degraded effectiveness of diffusion synthetic acceleration for multidimensional SN calculations in problem with material discontinuities, Nucl. Sci. Eng., 147, 218-248, (2004)
[51] Willert, J.; Kelley, C. T.; Knoll, D. A.; Park, H., A hybrid deterministic/Monte Carlo method for solving the k-eigenvalue problem with a comparison to analog Monte Carlo solutions, J. Comput. Theor. Transp., 43, 1-7, 50-67, (2014)
[52] Fleck, J.; Cummings, J., Implicit Monte Carlo scheme for calculating time and frequency dependent nonlinear radiation transport, J. Comput. Phys., 8, 3, 313-342, (1971) · Zbl 0229.65087
[53] Gentile, N., Implicit Monte Carlo diffusion - an acceleration method for Monte Carlo time-dependent radiative transfer simulations, J. Comput. Phys., 172, 2, 543-571, (2001) · Zbl 0986.65148
[54] Densmore, J. D.; Urbatsch, T. J.; Evans, T. M.; Buksas, M. W., A hybrid transport-diffusion method for Monte Carlo radiative-transfer simulations, J. Comput. Phys., 222, 2, 485-503, (2007) · Zbl 1111.82308
[55] Bhatnagar, P.; Gross, E.; Krook, M., A model for collision processes in gases. I. small amplitude processes in charged and neutral one-component systems, Phys. Rev. Lett., 94, 3, 511-525, (1954) · Zbl 0055.23609
[56] Kelly, C., Iterative methods for linear and nonlinear equations, (1995), Society for Industrial and Applied Mathematics
[57] Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31, (1978) · Zbl 0387.76063
[58] Brilliantov, N.; Porschel, T., Kinetic theory of granular gases, (2004), Oxford University Press
[59] Birdsall, C. K.; Langdon, A. B., Plasma physics via computer simulation, (2005), McGraw-Hill New York
[60] Hockney, R. W.; Eastwood, J. W., Computer simulation using particles, (1988), Taylor & Francis, Inc. Bristol, UK · Zbl 0662.76002
[61] Chen, G.; Chacón, L.; Barnes, D. C., An energy- and charge-conserving, implicit, electrostatic particle-in-cell algorithm, J. Comput. Phys., 230, 7018-7036, (2011) · Zbl 1237.78006
[62] Nielson, C. W.; Lewis, H. R., Particle-code models in the nonradiative limit, Methods Comput. Phys., 16, 367-388, (1976)
[63] Friedman, A.; Langdon, A. B.; Cohen, B. I., A direct method for implicit particle-in-cell simulation, Comments Plasma Phys. Control. Fusion, 6, 6, 225-236, (1981)
[64] Cohen, B. I.; Langdon, A. B.; Friedman, A., Implicit time integration for plasma simulation, J. Comput. Phys., 46, 1, 15-38, (1982) · Zbl 0495.76105
[65] Langdon, A. B.; Cohen, B. I.; Friedman, A., Direct implicit large time-step particle simulation of plasmas, J. Comput. Phys., 51, 1, 107-138, (1983) · Zbl 0572.76123
[66] Barnes, D. C.; Kamimura, T.; Leboeuf, J. N.; Tajima, T., Implicit particle simulation of magnetized plasmas, J. Comput. Phys., 52, 3, 480-502, (1983) · Zbl 0529.76117
[67] Langdon, A. B.; Barnes, D. C., Direct implicit plasma simulation, (Brackbill, J. U.; Cohen, B. I., Multiple Time Scales, (1985), Academic Press New York), 335-375
[68] Cohen, B. I., Multiple time-scale methods in particle simulations of plasma, Part. Accel., 19, 227-236, (1986)
[69] Mason, R. J., An electromagnetic field algorithm for 2d implicit plasma simulation, J. Comput. Phys., 71, 2, 429-473, (1987) · Zbl 0679.76117
[70] Hewett, D. W.; Langdon, A. B., Electromagnetic direct implicit plasma simulation, J. Comput. Phys., 72, 1, 121-155, (1987) · Zbl 0636.76126
[71] Friedman, A., A second-order implicit particle mover with adjustable damping, J. Comput. Phys., 90, 2, 292-312, (1990) · Zbl 0701.76121
[72] Kamimura, T.; Montalvo, E.; Barnes, D. C.; Leboeuf, J. N.; Tajima, T., Implicit particle simulation of electromagnetic plasma phenomena, J. Comput. Phys., 100, 1, 77-90, (1992) · Zbl 0758.76042
[73] Gibbons, M.; Hewett, D., The Darwin direct implicit particle-in-cell (DADIPIC) method for simulation of low frequency plasma phenomena, J. Comput. Phys., 120, 231-247, (1995) · Zbl 0841.76066
[74] Cohen, B. I.; Langdon, A. B.; Hewett, D. W.; Procassini, R. J., Performance and optimization of direct implicit particle simulation, J. Comput. Phys., 81, 1, 151-168, (1989) · Zbl 0664.65111
[75] Markidis, S.; Lapenta, G., The energy conserving particle-in-cell method, J. Comput. Phys., 230, 18, 7037-7052, (2011) · Zbl 1231.82067
[76] Lapenta, G.; Markidis, S., Particle acceleration and energy conservation in particle in cell simulations, Phys. Plasmas, 18, (2011)
[77] Hewett, D. W., Elimination of electromagnetic radiation in plasma simulation: the Darwin or magnetoinductive approximation, Space Sci. Rev., 42, 29-40, (1985)
[78] Degond, P.; Raviart, P.-A., An analysis of the Darwin model of approximation to Maxwell’s equations, Forum Math., 4, 4, 13-44, (1992) · Zbl 0755.35137
[79] Raviart, P.-A.; Sonnendrücker, E., A hierarchy of approximate models for the Maxwell equations, Numer. Math., 73, 3, 329-372, (1996) · Zbl 0862.35122
[80] Krause, T. B.; Apte, A.; Morrison, P., A unified approach to the Darwin approximation, Phys. Plasmas, 14, (2007)
[81] Chen, G.; Chacón, L., An analytical particle mover for the charge- and energy-conserving, nonlinearly implicit, electrostatic particle-in-cell algorithm, J. Comput. Phys., 247, 79-87, (2013) · Zbl 1349.76593
[82] Chacón, L.; Chen, G.; Barnes, D. C., A charge- and energy-conserving implicit, electrostatic particle-in-cell algorithm on mapped computational meshes, J. Comput. Phys., 233, 1-9, (2013) · Zbl 1286.78005
[83] Chacón, L.; Chen, G., A curvilinear, fully implicit, conservative electromagnetic PIC algorithm in multiple dimensions, J. Comput. Phys., 316, 578-597, (2016) · Zbl 1349.82139
[84] Langdon, A. B., Analysis of the time integration in plasma simulation, J. Comput. Phys., 30, 2, 202-221, (1979) · Zbl 0395.76081
[85] Kingham, R.; Bell, A., Nonlocal magnetic-field generation in plasmas without density gradients, Phys. Rev. Lett., 88, (2002)
[86] Kingham, R.; Bell, A., An implicit Vlasov-Fokker-Planck code to model non-local electron transport in 2-d with magnetic fields, J. Comput. Phys., 194, 1-34, (2004) · Zbl 1136.76400
[87] Ridgers, C.; Kingham, R.; Thomas, A., Magnetic cavitation and the reemergence of nonlocal transport in laser plasmas, Phys. Rev. Lett., 100, (2008)
[88] Kho, T.; Haines, M., Nonlinear kinetic transport of electrons and magnetic field in laser-produced plasmas, Phys. Rev. Lett., 55, 825-828, (1985)
[89] Luciani, J.; Mora, P.; Bendib, A., Magnetic field and nonlocal transport in laser-created plasmas, Phys. Rev. Lett., 55, 2421-2424, (1985)
[90] Epperlein, E.; Rickard, G.; Bell, A., A code for the solution of the Vlasov-Fokker-Planck equation in 1-D and 2-D, Comput. Phys. Commun., 52, 7-13, (1988)
[91] Town, R.; Bell, A.; Bell, A., Fokker-Planck simulations of short-pulse-laser-solid experiments, Rhys. Rev. E, 50, 2, part B, 1413-1421, (1994)
[92] Epperlein, E., Implicit and conservative difference scheme for the Fokker-Planck equation, J. Comput. Phys., 112, 291-297, (1994) · Zbl 0806.76050
[93] Chacón, L.; Barnes, D.; Knoll, D.; Miley, G., An implicit energy conservative 2D Fokker-Planck algorithm I. difference scheme, J. Comput. Phys., 157, 618-653, (2000) · Zbl 0961.76057
[94] Chacón, L.; Barnes, D. C.; Knoll, D. A.; Miley, G. H., An implicit energy conservative 2D Fokker-Planck algorithm II. Jacobian-free Newton-Krylov solver, J. Comput. Phys., 157, 654-682, (2000) · Zbl 0961.76058
[95] Mousseau, V.; Knoll, D., Fully implicit kinetic solution of collisional plasmas, J. Comput. Phys., 136, 308-323, (1997) · Zbl 0896.76057
[96] Gardner, L.; Gardner, G.; Zaki, S., Collisional effects in plasmas modeled by a simplified Fokker-Planck equation, J. Comput. Phys., 107, 40-50, (1993) · Zbl 0776.76069
[97] Lenard, A.; Bernstein, I., Plasma oscillations with diffusion in velocity space, Phys. Rev. Lett., 112, 1456-1459, (1958) · Zbl 0082.45301
[98] Thomas, A.; Kingham, R.; Ridgers, C., Rapid self-magnetization of laser speckles in plasmas by nonlinear anisotropic instability, New J. Phys., 11, (2009)
[99] Johnston, T., Cartesian tensor scalar product and spherical harmonic expansions in Boltzmann’s equation, Phys. Rev. Lett., 120, 1103-1111, (1960) · Zbl 0129.22301
[100] Thomas, A.; Tzoufras, M.; Robinson, A.; Kingham, R.; Ridgers, C., A review of Vlasov-Fokker-Planck numerical modeling of inertial confinement fusion plasma, J. Comput. Phys., 231, 1051-1079, (2012) · Zbl 1385.76015
[101] Taitano, W.; Chacón, L., Charge-and-energy conserving moment-based accelerator for multi-species Vlasov-Fokker-Planck-Ampère system, part I: collisionless aspects, J. Comput. Phys., 284, 718-736, (2015) · Zbl 1351.76124
[102] Taitano, W.; Knoll, D.; Chacón, L., Charge-and-energy conserving moment-based accelerator for multi-species Vlasov-Fokker-Planck-Ampère system, part II: collisional aspects, J. Comput. Phys., 284, 737-757, (2015) · Zbl 1351.76125
[103] Taitano, W.; Chacón, L.; Simakov, A.; Molvig, K., A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species rosenbluth-Fokker-Planck equation, J. Comput. Phys., 297, 357-380, (2015) · Zbl 1349.65384
[104] Taitano, W.; Chacón, L.; Simakov, A., An adaptive, conservative 0D-2V multispecies rosenbluth-Fokker-Planck solver for arbitrarily disparate mass and temperature regimes, J. Comput. Phys., 318, 391-420, (2016) · Zbl 1349.76393
[105] Rosenbluth, M.; MacDonald, W.; Judd, D., Fokker-Planck equation for an inverse-square force, Phys. Rev. Lett., 107, 1, 1-6, (1956) · Zbl 0077.44802
[106] Cheng, C.; Knorr, G., The integration of the Vlasov equation in configuration space, J. Comput. Phys., 22, 330-351, (1976)
[107] Filbet, F.; Sonnendrócker, E.; Bertrand, P., Conservative numerical scheme for the Vlasov equation, J. Comput. Phys., 172, 176-187, (2001)
[108] Willert, J.; Taitano, W.; Knoll, D., Leveraging Anderson acceleration for improved convergence of iterative solutions to transport systems, J. Comput. Phys., 273, 278-286, (2014) · Zbl 1351.82093
[109] Braginskii, S., Transport processes in plasmas, (Leontovich, M., Review of Plasma Physics, (1982), Consultants Bureau NY)
[110] Maltrud, M.; McClean, J., An eddy resolving global 1/10 Ocean simulation, Ocean Model., 8, 1, 31-54, (2005)
[111] Hallberg, R., Stable split time stepping schemes for large-scale Ocean modeling, J. Comput. Phys., 135, 1, 54-65, (1997) · Zbl 0889.76044
[112] Higdon, R.; de Szoeke, R., Barotropic-baroclinic time splitting for Ocean circulation modeling, J. Comput. Phys., 135, 1, 30-53, (1997) · Zbl 0888.76055
[113] Ringler, T.; Petersen, M.; Higdon, R.; Jacobsen, D.; Jones, P.; Maltrud, M., A multi-resolution approach to global Ocean modeling, Ocean Model., 69, 211-232, (2013)
[114] Pacanowski, R.; Dixon, K.; Rosati, A., The GFDL modular Ocean model users guide, (1993), Geophysical Fluid Dynamics Laboratory Princeton, USA, Tech. Rep. 2
[115] Newman, C.; Womeldorff, G.; Chacón, L.; Knoll, D. A., High-order/low-order methods for Ocean modeling, Proc. Comput. Sci., 51, 2086-2096, (2015)
[116] Newman, C.; Womeldorff, G.; Knoll, D. A.; Chacón, L., A communication-avoiding implicit-explicit method for a free-surface Ocean model, J. Comput. Phys., 305, 877-894, (2016) · Zbl 1349.86010
[117] Ascher, U. M.; Ruuth, S. J.; Wetton, B. T., Implicit-explicit methods for time-dependent partial differential equations, SIAM J. Numer. Anal., 32, 3, 797-823, (1995) · Zbl 0841.65081
[118] Durran, D. R.; Blossey, P. N., Implicit-explicit multistep methods for fast-wave-slow-wave problems, Mon. Weather Rev., 140, 4, 1307-1325, (2012)
[119] Kim, J.; Moin, P., Application of a fractional-step method to incompressible Navier-Stokes equations, J. Comput. Phys., 59, 2, 308-323, (1985) · Zbl 0582.76038
[120] Lambert, J., Numerical methods for ordinary differential systems: the initial value problem, (1991), John Wiley & Sons, Inc. · Zbl 0745.65049
[121] Newman, C.; Knoll, D. A., Physics-based preconditioners for Ocean simulation, SIAM J. Sci. Comput., 35, 5, S445-S464, (2013) · Zbl 1406.86005
[122] Arakawa, A.; Lamb, V., Computational design of the basic dynamical processes of the UCLA general circulation model, (Chang, J., Methods in Computational Physics, vol. 17, (1977), Academic Press New York), 173-265
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.