American Institute of Mathematical Sciences

August  2016, 36(8): 4247-4270. doi: 10.3934/dcds.2016.36.4247

High-order finite-volume methods on locally-structured grids

 1 Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, United States

Received  June 2015 Revised  December 2015 Published  March 2016

We present an approach to designing arbitrarily high-order finite-volume spatial discretizations on locally-rectangular grids. It is based on the use of a simple class of high-order quadratures for computing the average of fluxes over faces. This approach has the advantage of being a variation on widely-used second-order methods, so that the prior experience in engineering those methods carries over in the higher-order case. Among the issues discussed are the basic design principles for uniform grids, the extension to locally-refined nest grid hierarchies, and the treatment of complex geometries using mapped grids, multiblock grids, and cut-cell representations.
Citation: Phillip Colella. High-order finite-volume methods on locally-structured grids. Discrete and Continuous Dynamical Systems, 2016, 36 (8) : 4247-4270. doi: 10.3934/dcds.2016.36.4247
References:
 [1] M. Aftosmis, M. Berger and J. Melton, Robust and efficient Cartesian mesh generation for component-based geometry, AIAA Journal, 6 (1998), 952-960. doi: 10.2514/6.1997-196. [2] M. Barad and P. Colella, A fourth-order accurate local refinement method for Poisson's equation, Journal of Computational Physics, 209 (2005), 1-18. doi: 10.1016/j.jcp.2005.02.027. [3] P. Basu, M. Hall, S. Williams, B. Van Straalen, L. Oliker and P. Colella, Compiler-directed transformation for higher-order stencils, in Proceedings of the Parallel and Distributed Processing Symposium (IPDPS), Institute for Electrical and Electronics Engineers (2015), 313-323. doi: 10.1109/IPDPS.2015.103. [4] J. B. Bell, P. Colella and M. Welcome, A conservative front-tracking for inviscid compressible flow, in Proceedings of the Tenth AIAA Computational Fluid Dynamics Conference, American Institute for Aeronautics and Astronautics, (1991), 814-822. doi: 10.2514/6.1991-1599. [5] M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82 (1989), 64-84. doi: 10.1016/0021-9991(89)90035-1. [6] M. J. Berger and A. Jameson, Automatic adaptive grid refinement for the Euler equations, AIAA Journal, 23 (1985), 561-568. doi: 10.2514/3.8951. [7] M. J. Berger and R. J. LeVeque, An adaptive Cartesian mesh algorithm for the euler equations in arbitrary geometries, in Proceedings of the AIAA 9th Computational Fluid Dynamics Conference, American Institute for Aeronautics and Astronautics, (1989), 1-7. doi: 10.2514/6.1989-1930. [8] J. P. Boris and D. L. Book, Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, Journal of Computational Physics, 11 (1973), 38-69. doi: 10.1016/0021-9991(73)90147-2. [9] A. Bourlioux, A. T. Layton and M. L. Minion, Higher-order multi-implicit spectral deferred correction methods for problems of reacting flow, Journal of Computational Physics, 189 (2003), 651-675. doi: 10.1016/S0021-9991(03)00251-1. [10] C. Chaplin and P. Colella, A single stage flux-corrected transport algorithm for high-order finite-volume methods, preprint, arXiv:1506.02999. [11] I.-L. Chern and P. Colella, A conservative front tracking method for hyperbolic conservation laws, Technical Report UCRL-97200, Lawrence Livermore National Laboratory, 1987. [12] P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, Journal of Computational Physics, 87 (1990), 171-200. doi: 10.1016/0021-9991(90)90233-Q. [13] P. Colella, Volume-of-fluid methods for partial differential equations, In Godunov Methods: Theory and Applications, pages 161-177. Kluwer, 2001. [14] P. Colella, M.R. Dorr, J. A. F. Hittinger and D. F. Martin, High-order, finite-volume methods in mapped coordinates, Journal of Computational Physics, 230 (2011), 2952-2976. doi: 10.1016/j.jcp.2010.12.044. [15] P. Colella and M. D. Sekora, A limiter for PPM that preserves accuracy at smooth extrema, Journal of Computational Physics, 227 (2008), 7069-7076. doi: 10.1016/j.jcp.2008.03.034. [16] P. Colella and P. R. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations, Journal of Computational Physics, 54 (1989), 174-201. doi: 10.1016/0021-9991(84)90143-8. [17] D. Devendran, D. T. Graves and H. Johansen, A higher-order finite-volume discretization method for Poisson's equation in cut cell geometries, preprint, arXiv:1411.4283. [18] C. Gatti-Bono and P. Colella, An anelastic allspeed projection method for gravitationally stratified flows, Journal of Computational Physics, 216 (2006), 589-615. doi: 10.1016/j.jcp.2005.12.017. [19] S. M. Guzik, X. Gao, L. D. Owen, P. McCorquodale and P. Colella, A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive mesh refinement, Computers and Fluids, 123 (2015), 202-217. doi: 10.1016/j.compfluid.2015.10.001. [20] J. Hilditch and P. Colella, A Projection Method for Low Mach Number Fast Chemistry Reacting Flow, Technical Report AIAA-97-0263, American Institute of Aeronautics and Astronautics, 1997. doi: 10.2514/6.1997-263. [21] H. Johansen and P. Colella, A Cartesian grid embedded boundary method for Poisson's equation on irregular domains, Journal of Computational Physics, 147 (1998), 60-85. doi: 10.1006/jcph.1998.5965. [22] C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Applied Numerical Mathematics, 44 (2003), 139-181. doi: 10.1016/S0168-9274(02)00138-1. [23] H.-O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24 (1972), 199-215. doi: 10.1111/j.2153-3490.1972.tb01547.x. [24] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communications on Pure and Applied Mathematics 7 (1954), 159-193. doi: 10.1002/cpa.3160070112. [25] P. D. Lax, On Discontinuous Initial-Value Problems and Finite-Difference Schemes, Technical Report LAMS-1332, Los Alamos Scientific Laboratory, 1952. [26] P. D. Lax and B. Wendroff, Systems of conservation laws, Communications on Pure and Applied Mathematics, 13 (1960), 217-237. doi: 10.1002/cpa.3160130205. [27] R. Malladi, J. A. Sethian and B. C. Vemuri, Shape modeling with front propagation: A level set approach, IEEE Transactions on Pattern Anal. Machine Intell, 17 (1995), 158-175. doi: 10.1109/34.368173. [28] P. McCorquodale and P. Colella, A high-order finite-volume method for conservation laws on locally refined grids, Communications in Applied Mathematics and Computational Science 6 (2011), 1-25. doi: 10.2140/camcos.2011.6.1. [29] P. McCorquodale, P. Colella and H. Johansen, A Cartesian grid embedded boundary method for the heat equation on irregular domains, Journal of Computational Physics, 173 (2001), 620-635. doi: 10.1006/jcph.2001.6900. [30] P. McCorquodale, M. R. Dorr, J. A. F. Hittinger and P. Colella, High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids, Journal of Computational Physics, 288 (2015), 181-195. doi: 10.1016/j.jcp.2015.01.006. [31] L. I. Millett and S. H. Fuller, et al., The Future of Computing Performance: Game Over or Next Level?, National Academies Press, 2011. [32] J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, Journal of Applied Physics, 21 (1950), 232-237. doi: 10.1063/1.1699639. [33] W. F. Noh, CEL: A time-dependent, two-space-dimensional, coupled Eulerian - Lagrangian code, Methods in Computational Physics, 3 (1964), 117-180. [34] R. B. Pember, J. B. Bell, P. Colella, W. Y. Crutchfield and M. L. Welcome, An adaptive Cartesian} grid method for unsteady compressible flow in irregular regions, Journal of Computational Physics, 120 (1995), 278-304. doi: 10.1006/jcph.1995.1165. [35] R. B. Pember, L. H. Howell, J. B. Bell, P. Colella, W. Y. Crutchfield, W. A. Fiveland and J. P. Jessee, An adaptive projection method for unsteady, low-Mach-number combustion, Combustion Science and Technology, 140 (1998), 123-168. doi: 10.1080/00102209808915770. [36] J. S. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, Journal of Computational Physics, 115 (1994), 153-168. doi: 10.1006/jcph.1994.1184. [37] P. Schwartz, J. Percelay, T. Ligocki, H. Johansen, D. Graves, D. Devendran, P. Colella and E. Ateljevich, High-accuracy embedded boundary grid generation using the divergence theorem, Communications in Applied Mathematics and Computational Science, 10 (2015), 83-96. doi: 10.2140/camcos.2015.10.83. [38] D. Trebotich, M. F. Adams, S. Molins, C. I. Steefel and C. Shen, High-resolution simulation of pore-scale reactive transport processes associated with carbon sequestration, Computing in Science and Engineering, 16 (2014), 22-31. doi: 10.1109/MCSE.2014.77. [39] B. van Leer, Towards the ultimate conservative differences scheme IV: a new approach to numerical convection, Journal of Computational Physics, 23 (1977), 263-275. [40] S. Williams, A. Waterman and D. Patterson, Roofline: an insightful visual performance model for multicore architectures, Communications of the ACM, 52 (2009), 65-76. doi: 10.1145/1498765.1498785. [41] P. R. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics, 54 (1984), 115-173. doi: 10.1016/0021-9991(84)90142-6. [42] S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 31 (1979), 335-362. doi: 10.1016/0021-9991(79)90051-2. [43] S. T. Zalesak, A physical interpretation of the Richtmyer two-step Lax-Wendroff scheme and its generalization to higher spatial order, in Advances in Computer Methods for Partial Differential Equations, IMACS, (1984), 19-21.

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References:
 [1] M. Aftosmis, M. Berger and J. Melton, Robust and efficient Cartesian mesh generation for component-based geometry, AIAA Journal, 6 (1998), 952-960. doi: 10.2514/6.1997-196. [2] M. Barad and P. Colella, A fourth-order accurate local refinement method for Poisson's equation, Journal of Computational Physics, 209 (2005), 1-18. doi: 10.1016/j.jcp.2005.02.027. [3] P. Basu, M. Hall, S. Williams, B. Van Straalen, L. Oliker and P. Colella, Compiler-directed transformation for higher-order stencils, in Proceedings of the Parallel and Distributed Processing Symposium (IPDPS), Institute for Electrical and Electronics Engineers (2015), 313-323. doi: 10.1109/IPDPS.2015.103. [4] J. B. Bell, P. Colella and M. Welcome, A conservative front-tracking for inviscid compressible flow, in Proceedings of the Tenth AIAA Computational Fluid Dynamics Conference, American Institute for Aeronautics and Astronautics, (1991), 814-822. doi: 10.2514/6.1991-1599. [5] M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82 (1989), 64-84. doi: 10.1016/0021-9991(89)90035-1. [6] M. J. Berger and A. Jameson, Automatic adaptive grid refinement for the Euler equations, AIAA Journal, 23 (1985), 561-568. doi: 10.2514/3.8951. [7] M. J. Berger and R. J. LeVeque, An adaptive Cartesian mesh algorithm for the euler equations in arbitrary geometries, in Proceedings of the AIAA 9th Computational Fluid Dynamics Conference, American Institute for Aeronautics and Astronautics, (1989), 1-7. doi: 10.2514/6.1989-1930. [8] J. P. Boris and D. L. Book, Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, Journal of Computational Physics, 11 (1973), 38-69. doi: 10.1016/0021-9991(73)90147-2. [9] A. Bourlioux, A. T. Layton and M. L. Minion, Higher-order multi-implicit spectral deferred correction methods for problems of reacting flow, Journal of Computational Physics, 189 (2003), 651-675. doi: 10.1016/S0021-9991(03)00251-1. [10] C. Chaplin and P. Colella, A single stage flux-corrected transport algorithm for high-order finite-volume methods, preprint, arXiv:1506.02999. [11] I.-L. Chern and P. Colella, A conservative front tracking method for hyperbolic conservation laws, Technical Report UCRL-97200, Lawrence Livermore National Laboratory, 1987. [12] P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, Journal of Computational Physics, 87 (1990), 171-200. doi: 10.1016/0021-9991(90)90233-Q. [13] P. Colella, Volume-of-fluid methods for partial differential equations, In Godunov Methods: Theory and Applications, pages 161-177. Kluwer, 2001. [14] P. Colella, M.R. Dorr, J. A. F. Hittinger and D. F. Martin, High-order, finite-volume methods in mapped coordinates, Journal of Computational Physics, 230 (2011), 2952-2976. doi: 10.1016/j.jcp.2010.12.044. [15] P. Colella and M. D. Sekora, A limiter for PPM that preserves accuracy at smooth extrema, Journal of Computational Physics, 227 (2008), 7069-7076. doi: 10.1016/j.jcp.2008.03.034. [16] P. Colella and P. R. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations, Journal of Computational Physics, 54 (1989), 174-201. doi: 10.1016/0021-9991(84)90143-8. [17] D. Devendran, D. T. Graves and H. Johansen, A higher-order finite-volume discretization method for Poisson's equation in cut cell geometries, preprint, arXiv:1411.4283. [18] C. Gatti-Bono and P. Colella, An anelastic allspeed projection method for gravitationally stratified flows, Journal of Computational Physics, 216 (2006), 589-615. doi: 10.1016/j.jcp.2005.12.017. [19] S. M. Guzik, X. Gao, L. D. Owen, P. McCorquodale and P. Colella, A freestream-preserving fourth-order finite-volume method in mapped coordinates with adaptive mesh refinement, Computers and Fluids, 123 (2015), 202-217. doi: 10.1016/j.compfluid.2015.10.001. [20] J. Hilditch and P. Colella, A Projection Method for Low Mach Number Fast Chemistry Reacting Flow, Technical Report AIAA-97-0263, American Institute of Aeronautics and Astronautics, 1997. doi: 10.2514/6.1997-263. [21] H. Johansen and P. Colella, A Cartesian grid embedded boundary method for Poisson's equation on irregular domains, Journal of Computational Physics, 147 (1998), 60-85. doi: 10.1006/jcph.1998.5965. [22] C. A. Kennedy and M. H. Carpenter, Additive Runge-Kutta schemes for convection-diffusion-reaction equations, Applied Numerical Mathematics, 44 (2003), 139-181. doi: 10.1016/S0168-9274(02)00138-1. [23] H.-O. Kreiss and J. Oliger, Comparison of accurate methods for the integration of hyperbolic equations, Tellus, 24 (1972), 199-215. doi: 10.1111/j.2153-3490.1972.tb01547.x. [24] P. D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Communications on Pure and Applied Mathematics 7 (1954), 159-193. doi: 10.1002/cpa.3160070112. [25] P. D. Lax, On Discontinuous Initial-Value Problems and Finite-Difference Schemes, Technical Report LAMS-1332, Los Alamos Scientific Laboratory, 1952. [26] P. D. Lax and B. Wendroff, Systems of conservation laws, Communications on Pure and Applied Mathematics, 13 (1960), 217-237. doi: 10.1002/cpa.3160130205. [27] R. Malladi, J. A. Sethian and B. C. Vemuri, Shape modeling with front propagation: A level set approach, IEEE Transactions on Pattern Anal. Machine Intell, 17 (1995), 158-175. doi: 10.1109/34.368173. [28] P. McCorquodale and P. Colella, A high-order finite-volume method for conservation laws on locally refined grids, Communications in Applied Mathematics and Computational Science 6 (2011), 1-25. doi: 10.2140/camcos.2011.6.1. [29] P. McCorquodale, P. Colella and H. Johansen, A Cartesian grid embedded boundary method for the heat equation on irregular domains, Journal of Computational Physics, 173 (2001), 620-635. doi: 10.1006/jcph.2001.6900. [30] P. McCorquodale, M. R. Dorr, J. A. F. Hittinger and P. Colella, High-order finite-volume methods for hyperbolic conservation laws on mapped multiblock grids, Journal of Computational Physics, 288 (2015), 181-195. doi: 10.1016/j.jcp.2015.01.006. [31] L. I. Millett and S. H. Fuller, et al., The Future of Computing Performance: Game Over or Next Level?, National Academies Press, 2011. [32] J. von Neumann and R. D. Richtmyer, A method for the numerical calculation of hydrodynamic shocks, Journal of Applied Physics, 21 (1950), 232-237. doi: 10.1063/1.1699639. [33] W. F. Noh, CEL: A time-dependent, two-space-dimensional, coupled Eulerian - Lagrangian code, Methods in Computational Physics, 3 (1964), 117-180. [34] R. B. Pember, J. B. Bell, P. Colella, W. Y. Crutchfield and M. L. Welcome, An adaptive Cartesian} grid method for unsteady compressible flow in irregular regions, Journal of Computational Physics, 120 (1995), 278-304. doi: 10.1006/jcph.1995.1165. [35] R. B. Pember, L. H. Howell, J. B. Bell, P. Colella, W. Y. Crutchfield, W. A. Fiveland and J. P. Jessee, An adaptive projection method for unsteady, low-Mach-number combustion, Combustion Science and Technology, 140 (1998), 123-168. doi: 10.1080/00102209808915770. [36] J. S. Saltzman, An unsplit 3D upwind method for hyperbolic conservation laws, Journal of Computational Physics, 115 (1994), 153-168. doi: 10.1006/jcph.1994.1184. [37] P. Schwartz, J. Percelay, T. Ligocki, H. Johansen, D. Graves, D. Devendran, P. Colella and E. Ateljevich, High-accuracy embedded boundary grid generation using the divergence theorem, Communications in Applied Mathematics and Computational Science, 10 (2015), 83-96. doi: 10.2140/camcos.2015.10.83. [38] D. Trebotich, M. F. Adams, S. Molins, C. I. Steefel and C. Shen, High-resolution simulation of pore-scale reactive transport processes associated with carbon sequestration, Computing in Science and Engineering, 16 (2014), 22-31. doi: 10.1109/MCSE.2014.77. [39] B. van Leer, Towards the ultimate conservative differences scheme IV: a new approach to numerical convection, Journal of Computational Physics, 23 (1977), 263-275. [40] S. Williams, A. Waterman and D. Patterson, Roofline: an insightful visual performance model for multicore architectures, Communications of the ACM, 52 (2009), 65-76. doi: 10.1145/1498765.1498785. [41] P. R. Woodward and P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of Computational Physics, 54 (1984), 115-173. doi: 10.1016/0021-9991(84)90142-6. [42] S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, Journal of Computational Physics, 31 (1979), 335-362. doi: 10.1016/0021-9991(79)90051-2. [43] S. T. Zalesak, A physical interpretation of the Richtmyer two-step Lax-Wendroff scheme and its generalization to higher spatial order, in Advances in Computer Methods for Partial Differential Equations, IMACS, (1984), 19-21.
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