Hybrid methods for radiation transport using diagonally implicit Runge-Kutta and space-time discontinuous Galerkin time integration. (English) Zbl 1416.65343
Summary: In this work, we describe extensions of a hybrid method for time-dependent linear, kinetic radiation transport problems to high-order time integration schemes of the diagonally-implicit Runge-Kutta (DIRK) and space-time discontinuous Galerkin (STDG) types. The hybrid methods are constructed by splitting the radiation flux into collided and uncollided components to which low- and high-resolution discrete ordinates approximations are applied. The efficiency of hybrid methods constructed using DIRK, STDG, integral deferred correction (IDC), and implicit Euler schemes is compared using a test problem in one-dimensional slab geometry containing material discontinuities. It is observed that (i) higher-order methods are more efficient than the implicit Euler method, often by an order of magnitude or more; (ii) third-order methods yield solutions of a given error in roughly half the time of the second-order method of the same type; and (iii) for a given order of accuracy it is found that the most efficient class of time integration scheme is STDG, followed by DIRK, with IDC the least efficient, for the test problem considered. Two test problems in two-dimensional \(xy\)-geometry are used to compare the computational efficiency of hybrid and standard discrete ordinates methods constructed with DIRK and STDG integrators. We observe that replacing a standard discrete ordinates method using an angular quadrature of order \(N\) with a hybrid discrete ordinates method using angular quadratures of order \(2N\) and \(N / 2\) for the uncollided and collided fluxes, respectively, usually reduces overall solution time by a factor of 2 or more while simultaneously reducing the resulting solution error by a factor of 2 or more for the test problems considered.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 35R09 | Integro-partial differential equations |
| 45K05 | Integro-partial differential equations |
| 65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
References:
| [1] | Adams, Marvin L.; Larsen, Edward W., Fast iterative methods for discrete-ordinates particle transport calculations, Prog. Nucl. Energy, 40, 1, 3-159 (2002) |
| [2] | Ahrens, C.; Beylkin, G., Rotationally invariant quadratures for the sphere, Proc. R. Soc. A, Math. Phys. Eng. Sci., 465, 2110, 3103-3125 (Jul 2009) · Zbl 1178.65030 |
| [3] | Alcouffe, R. E.; O’Dell, R. D.; Brinkey, F. W., A first-collision source method that satisfies discrete \(S_N\) transport balance, Nucl. Sci. Eng., 105, 198-203 (1990) |
| [4] | Alcouffe, Raymond E., A first collision source method for coupling Monte Carlo and discrete ordinates for localized source problems, (Alcouffe, Raymond; Dautray, Robert; Forster, Arthur; Ledanois, Guy; Mercier, B., Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics. Monte-Carlo Methods and Applications in Neutronics, Photonics and Statistical Physics, Lecture Notes in Physics, vol. 240 (1985), Springer: Springer Berlin, Heidelberg), 352-366 |
| [5] | Alexander, Roger, Diagonally implicit Runge-Kutta methods for stiff O.D.E.’s, SIAM J. Numer. Anal., 14, 6, 1006-1021 (1977) · Zbl 0374.65038 |
| [6] | Asher, Uri M.; Petzold, Linda R., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations (1998), SIAM · Zbl 0908.65055 |
| [7] | Atkinson, Kendall; Han, Weimin, Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Lecture Notes in Mathematics, vol. 2044 (2012), Springer: Springer Berlin, Heidelberg · Zbl 1254.41015 |
| [8] | Balay, Satish; Abhyankar, Shrirang; Adams, Mark F.; Brown, Jed; Brune, Peter; Buschelman, Kris; Dalcin, Lisandro; Eijkhout, Victor; Gropp, William D.; Kaushik, Dinesh; Knepley, Matthew G.; McInnes, Lois Curfman; Rupp, Karl; Smith, Barry F.; Zampini, Stefano; Zhang, Hong; Zhang, Hong, PETSc web page (2016) |
| [9] | Balay, Satish; Abhyankar, Shrirang; Adams, Mark F.; Brown, Jed; Brune, Peter; Buschelman, Kris; Dalcin, Lisandro; Eijkhout, Victor; Gropp, William D.; Kaushik, Dinesh; Knepley, Matthew G.; Curfman McInnes, Lois; Rupp, Karl; Smith, Barry F.; Zampini, Stefano; Zhang, Hong; Zhang, Hong, PETSc Users Manual (2016), Argonne National Laboratory, Technical Report ANL-95/11 - Revision 3.7 |
| [10] | Balay, Satish; Gropp, William D.; Curfman McInnes, Lois; Smith, Barry F., Efficient management of parallelism in object oriented numerical software libraries, (Arge, E.; Bruaset, A. M.; Langtangen, H. P., Modern Software Tools in Scientific Computing (1997), Birkhäuser Press), 163-202 · Zbl 0882.65154 |
| [11] | Berninger, H.; Frénod, E.; Gander, M.; Liebendörfer, M.; Michaud, J., Derivation of the isotropic diffusion source approximation (IDSA) for supernova neutrino transport by asymptotic expansions, SIAM J. Math. Anal., 45, 6, 3229-3265 (Jan 2013) · Zbl 1307.35181 |
| [12] | Berninger, Heiko; Frénod, Emmanuel; Gander, Martin J.; Liebendörfer, Matthias; Michaud, Jérôme; Vasset, Nicolas, A mathematical description of the IDSA for supernova neutrino transport, its discretization and a comparison with a finite volume scheme for Boltzmann’s equation, (ESAIM: Proceedings, vol. 38 (Dec 2012)), 163-182 · Zbl 1329.85003 |
| [13] | Brunner, Thomas A., Forms of Approximate Radiation Transport (2002), Sandia National Laboratories: Sandia National Laboratories Albuquerque, NM, Technical Report SAND2002-1778 |
| [14] | Christlieb, Andrew; Guo, Wei; Morton, Maureen; Qiu, Jing-Mei, A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations, J. Comput. Phys., 267, 7-27 (2014) · Zbl 1349.76443 |
| [15] | Christlieb, Andrew; Ong, Benjamin, Implicit parallel time integrators, J. Sci. Comput., 49, 167-179 (2011) · Zbl 1243.65076 |
| [16] | Christlieb, Andrew J.; Haynes, Ronald D.; Ong, Ben W., A parallel space-time algorithm, SIAM J. Sci. Comput., 34, 5, C233-C248 (2012) · Zbl 1259.65143 |
| [17] | Christlieb, Andrew J.; Liu, Yuan; Xu, Zhengfu, High order operator splitting methods based on an integral deferred correction framework, J. Comput. Phys., 294, 224-242 (2015) · Zbl 1349.65210 |
| [18] | Christlieb, Andrew J.; MacDonald, Colin B.; Ong, Benjamin W., Parallel high-order integrators, SIAM J. Sci. Comput., 32, 2, 818-835 (2010) · Zbl 1211.65089 |
| [19] | Crockatt, Michael M.; Christlieb, Andrew J.; Garrett, C. Kristopher; Hauck, Cory D., An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction, J. Comput. Phys., 346, 1, 212-241 (October 2017) · Zbl 1380.65300 |
| [20] | Deakin, Tom; McIntosh-Smith, Simon; Martineau, Matt; Gaudin, Wayne, An improved parallelism scheme for deterministic discrete ordinates transport, Int. J. High Perform. Comput. Appl., 32, 555-569 (2016) |
| [21] | Delfour, M.; Hager, W.; Trochu, F., Discontinuous Galerkin methods for ordinary differential equations, Math. Comput., 36, 154, 455-473 (May 1981) · Zbl 0469.65053 |
| [22] | Dutt, Alok; Greengard, Leslie; Rokhlin, Vladimir, Spectral deferred correction methods for ordinary differential equations, BIT Numer. Math., 40, 2, 241-266 (2000) · Zbl 0959.65084 |
| [23] | Evans, Lawrence C., Partial Differential Equations, Graduate Studies in Mathematics, vol. 19 (2010), The American Mathematical Society · Zbl 1194.35001 |
| [24] | Filippone, W. L.; Morel, Jim E.; Walters, Wallace F., An extended first collision source method for electron beam source problems, Nucl. Sci. Eng., 112, 1, 1-15 (1992) |
| [25] | Golse, François; Jin, Shi; Levermore, C. David, The convergence of numerical transfer schemes in diffusive regimes I: discrete-ordinate method, SIAM J. Numer. Anal., 36, 5, 1333-1369 (Jan 1999) · Zbl 1053.82030 |
| [26] | (Graziani, Frank, Computational Methods in Transport. Computational Methods in Transport, Lecture Notes in Computational Science and Engineering, vol. 48 (2006), Springer-Verlag: Springer-Verlag Berlin, Heidelberg) |
| [27] | Hairer, Ernst; Nørsett, Syvert P.; Wanner, Gerhad, Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, vol. 8 (1993), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0789.65048 |
| [28] | Hairer, Ernst; Wanner, Gerhard, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer Series in Computational Mathematics, vol. 14 (1996), Springer-Verlag: Springer-Verlag Berlin, Heidelberg · Zbl 0859.65067 |
| [29] | Hairer, Ernst; Wanner, Gerhard, Stiff differential equations solved by Radau methods, J. Comput. Appl. Math., 111, 93-111 (1999) · Zbl 0945.65080 |
| [30] | Hangstrom, Thomas; Zhou, Ruhai, On the spectral deferred correction of splitting methods for initial value problems, Commun. Appl. Math. Comput. Sci., 1, 1, 169-205 (2006) · Zbl 1105.65076 |
| [31] | Hauck, Cory D.; McClarren, Ryan G., A collision-based hybrid method for time dependent, linear, kinetic transport equations, Multiscale Model. Simul., 11, 4, 1197-1227 (2013) · Zbl 1292.82051 |
| [32] | Henderson, H. V.; Searle, S. R., On deriving the inverse of a sum of matrices, SIAM Rev., 23, 1, 53-60 (Jan 1981) · Zbl 0451.15005 |
| [33] | Hoisie, Adolfy; Lubeck, Olaf; Wasserman, Harvey, Performance and scalability analysis of teraflop-scale parallel architectures using multidimensional wavefront applications, Int. J. High Perform. Comput. Appl., 14, 4, 330-346 (2000) |
| [34] | Huang, Jinfang; Jia, Jun; Minion, Michael, Arbitrary order Krylov deferred correction methods for differential algebraic equations, J. Comput. Phys., 221, 2, 739-760 (February 2007) · Zbl 1110.65076 |
| [35] | Huang, Jingfang; Jia, Jun; Minion, Michael, Accelerating the convergence of spectral deferred correction methods, J. Comput. Phys., 214, 633-656 (2006) · Zbl 1094.65066 |
| [36] | Shi, Jin; Levermore, David, The discrete-ordinate method in diffusive regimes, Transp. Theory Stat. Phys., 20, 5-6, 413-439 (Oct 1991) · Zbl 0760.65125 |
| [37] | Johnson, C.; Pitkäranta, J., An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation, Math. Comput., 46, 173, 1-26 (January 1986) · Zbl 0618.65105 |
| [38] | Kennedy, Christopher A.; Carpenter, Mark H., Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations. A Review (March 2016), NASA, Langley Research Center: NASA, Langley Research Center Hampton, Virginia 23681-2199, Technical Report NASA/TM-2016-219173 |
| [39] | Larsen, Edward W.; Kumar, Akansha; Morel, Jim E., Properties of the implicitly time-differenced equations of thermal radiation transport, J. Comput. Phys., 238, 82-96 (2013) · Zbl 1286.65115 |
| [40] | Larsen, Edward W.; Miller, Warren F., Convergence rates of spatial difference equations for the discrete-ordinates neutron transport equations in slab geometry, Nucl. Sci. Eng., 73, 1, 76-83 (1980) |
| [41] | Larsen, Edward W.; Morel, Jim E., Advances in discrete-ordinates methodology, (Nuclear Computational Science: A Century in Review (2010), Springer: Springer Netherlands), 1-84, Chapter 1 |
| [42] | Larsen, Edward W.; Nelson, Paul, Finite-difference approximations and superconvergence for the discrete-ordinate equations in slab geometry, SIAM J. Numer. Anal., 19, 2, 334-348 (Apr 1982) · Zbl 0482.65072 |
| [43] | Lasaint, P.; Raviart, P. A., On a finite element method for solving the neutron transport equation, (de Boor, Carl, Mathematical Aspects of Finite Elements in Partial Differential Equations (1974), Academic Press, Inc.), 89-123 · Zbl 0341.65076 |
| [44] | Lathrop, K. D., Ray effects in discrete ordinates equations, Nucl. Sci. Eng., 32, 357-369 (1968) |
| [45] | Lathrop, K. D., Remedies for ray effects, Nucl. Sci. Eng., 45, 255-268 (1971) |
| [46] | Layton, Anita T.; Minion, Michael L., Implications of the choice of quadrature nodes for Picard integral deferred corrections methods for ordinary differential equations, BIT Numer. Math., 45, 2, 341-373 (2005) · Zbl 1078.65552 |
| [47] | Lebedev, V. I., Quadratures on a sphere, USSR Comput. Math. Math. Phys., 16, 2, 10-24 (Jan 1976) · Zbl 0348.65023 |
| [48] | Lewis, Elmer E.; Miller, Warren F., Computational Methods of Neutron Transport (1993), Wiley |
| [49] | Liebendörfer, M.; Whitehouse, S. C.; Fischer, T., The isotropic diffusion source approximation for supernova neutrino transport, Astrophys. J., 698, 2, 1174-1190 (May 2009) |
| [50] | Maginot, Peter G.; Ragusa, Jean C.; Morel, Jim E., High-order solution methods for grey discrete ordinates thermal radiative transfer, J. Comput. Phys., 327, 719-746 (Dec 2016) · Zbl 1422.65262 |
| [51] | McClarren, Ryan G.; Hauck, Cory D., Robust and accurate filtered spherical harmonics expansions for radiative transfer, J. Comput. Phys., 229, 5597-5614 (2010) · Zbl 1193.82043 |
| [52] | Michaud, Jérôme, The IDSA and the homogeneous sphere: issues and possible improvements, Discrete Contin. Dyn. Syst., Ser. S, 9, 5, 1351-1375 (Oct 2016) · Zbl 1372.35304 |
| [53] | Mihalas, Dimitri; Weibel-Mihalas, Barbara, Foundations of Radiation Hydrodynamics (1999), Dover Publications, Inc. · Zbl 0651.76005 |
| [54] | Moustafa, Salli; Dutka-Malen, Ivan; Plagne, Laurent; Ponçot, Angélique; Ramet, Pierre, Shared memory parallelism for 3D Cartesian discrete ordinates solver, Ann. Nucl. Energy, 82, 179-187 (August 2015) |
| [55] | Petrini, F.; Fossum, G.; Fernandez, J.; Varbanescu, A. L.; Kistler, M.; Perrone, M., Multicore surprises: lessons learned from optimizing sweep3d on the cell broadband engine, (2007 IEEE International Parallel and Distributed Processing Symposium (March 2007)), 1-10 |
| [56] | Pomraning, G. C., The Equations of Radiation Hydrodynamics (1973), Pergamon Press |
| [57] | Qu, Wenzhen; Brandon, Namdi; Chen, Dangxing; Huang, Jingfang; Kress, Tyler, A numerical framework for integrating deferred correction methods to solve high order collocation formulations of ODEs, J. Sci. Comput., 68, 2, 484-520 (August 2016) · Zbl 1371.65072 |
| [58] | Radice, David; Abdikamalov, Ernazar; Rezzolla, Luciano; Ott, Christian D., A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations, J. Comput. Phys., 242, 648-669 (Jun 2013) · Zbl 1310.85002 |
| [59] | Reed, W. H.; Hill, T. R., Triangular Mesh Methods for the Neutron Transport Equation (1973), Los Alamos Scientific Laboratory: Los Alamos Scientific Laboratory Los Alamos, NM, Technical Report LA-UR-73-479 |
| [60] | Reed, William H., New difference schemes for the neutron transport equation, Nucl. Sci. Eng., 46, 2, 309-314 (1971) |
| [61] | Sloan, Ian H.; Womersley, Robert S., Constructive polynomial approximation on the sphere, J. Approx. Theory, 103, 91-118 (2000) · Zbl 0946.41007 |
| [62] | Speck, Robert; Ruprecht, Daniel; Emmett, Matthew; Minion, Michael; Bolten, Matthias; Krause, Rolf, A multi-level spectral deferred correction method, BIT Numer. Math., 55, 3, 843-867 (September 2015) · Zbl 1326.65138 |
| [63] | van der Vegt, J. J.W.; van der Ven, H., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows I. General formulation, J. Comput. Phys., 182, 2, 546-585 (Nov 2002) · Zbl 1057.76553 |
| [64] | van der Ven, H.; van der Vegt, J. J.W., Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows II. Efficient flux quadrature, Comput. Methods Appl. Mech. Eng., 191, 41-42, 4747-4780 (Sep 2002) · Zbl 1099.76521 |
| [65] | Warsa, James S.; Prinja, Anil K., Bilinear-discontinuous numerical solution of the time dependent transport equation in slab geometry, Ann. Nucl. Energy, 26, 3, 195-215 (Feb 1999) |
| [66] | Weiser, Martin, Faster SDC convergence on non-equidistant grids by DIRK sweeps, BIT Numer. Math., 55, 4, 1219-1241 (December 2015) · Zbl 1332.65103 |
| [67] | Zhao, Shan; Wei, G. W., A unified discontinuous Galerkin framework for time integration, Math. Methods Appl. Sci., 37, 7, 1042-1071 (Jun 2013) · Zbl 1292.65108 |
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