Maximum principle preserving space and time flux limiting for diagonally implicit Runge-Kutta discretizations of scalar convection-diffusion equations. (English) Zbl 1503.65198
The authors present two techniques to obtain maximum principle preserving (MPP) numerical schemes for scalar nonlinear convection-diffusion partial differential equations. The approach followed here is similar to that one of D. Kuzmin et al. [J. Sci. Comput. 91, No. 1, Paper No. 21, 34 p. (2022; Zbl 1539.65095)] which dealt with explicit methods for hyperbolic problems. Both methodologies are based on combining a low-order MPP scheme with a high-order scheme, limiting the contribution from their difference. The study presented here is focused on using finite volumes in space and Runge-Kutta methods in time but the limiters developed here could be used with a wide range of space and time discretizations. Using these limiters with appropriate discretizations allows to obtain a scheme whose local error is of any desired order. That is, the methods are MPP for time steps of any size. Some numerical tests are presented to show the desirable properties of the resulting schemes.
Reviewer: Abdallah Bradji (Annaba)
MSC:
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
| 76R50 | Diffusion |
| 76M12 | Finite volume methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
Keywords:
scalar convection-diffusion equations; positivity-preserving implicit schemes; diagonally implicit Runge-Kutta time stepping; flux-corrected transport; monolithic convex limitingCitations:
Zbl 1539.65095References:
| [1] | Anderson, R.; Dobrev, V.; Kolev, T.; Kuzmin, D.; de Luna, MQ; Rieben, R.; Tomov, V., High-order local maximum principle preserving (MPP) discontinuous Galerkin finite element method for the transport equation, J. Comput. Phys., 334, 102-124 (2017) · Zbl 1380.65245 · doi:10.1016/j.jcp.2016.12.031 |
| [2] | Arbogast, T.; Huang, C-S; Zhao, X.; King, DN, A third order, implicit, finite volume, adaptive Runge-Kutta WENO scheme for advection-diffusion equations, Comput. Methods Appl. Mech. Eng., 368, 113155 (2020) · Zbl 1506.76108 · doi:10.1016/j.cma.2020.113155 |
| [3] | Bolley, C.; Crouzeix, M., Conservation de la positivité lors de la discrétisation des problémes d’évolution paraboliques, R.A.I.R.O. Anal. Numérique, 12, 3, 237-245 (1978) · Zbl 0392.65042 · doi:10.1051/m2an/1978120302371 |
| [4] | Boris, JP; Book, DL, Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 1, 38-69 (1973) · Zbl 0251.76004 · doi:10.1016/0021-9991(73)90147-2 |
| [5] | Chen, Z.; Huang, H.; Yan, J., Third order maximum-principle-satisfying direct discontinuous Galerkin methods for time dependent convection diffusion equations on unstructured triangular meshes, J. Comput. Phys., 308, 198-217 (2016) · Zbl 1351.76052 · doi:10.1016/j.jcp.2015.12.039 |
| [6] | Feng, D.; Neuweiler, I.; Nackenhorst, U.; Wick, T., A time-space flux-corrected transport finite element formulation for solving multi-dimensional advection-diffusion-reaction equations, J. Comput. Phys., 396, 31-53 (2019) · Zbl 1452.76087 · doi:10.1016/j.jcp.2019.06.053 |
| [7] | Gottlieb, S., Ketcheson, D.I., Shu, C.-W.: Strong Stability Preserving Runge-Kutta And Multistep Time Discretizations. WORLD SCIENTIFIC, (2011) · Zbl 1241.65064 |
| [8] | Guermond, J-L; Nazarov, M.; Popov, B.; Yang, Y., A second-order maximum principle preserving lagrange finite element technique for nonlinear scalar conservation equations, SIAM J. Numer. Anal., 52, 4, 2163-2182 (2014) · Zbl 1302.65225 · doi:10.1137/130950240 |
| [9] | Guermond, J-L; Popov, B., Invariant domains and first-order continuous finite element approximation for hyperbolic systems, SIAM J. Numer. Anal., 54, 4, 2466-2489 (2016) · Zbl 1346.65050 · doi:10.1137/16M1074291 |
| [10] | Hairer, E.; Wanner, G., Solving ordinary differential equations II: Stiff and differential-algebraic problems (1996), Berlin: Springer, Berlin · Zbl 1192.65097 · doi:10.1007/978-3-642-05221-7 |
| [11] | Hajduk, H., Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws, Comput. Math. Appl., 87, 120-138 (2021) · Zbl 1524.65539 · doi:10.1016/j.camwa.2021.02.012 |
| [12] | Harten, A.; Zwas, G., Self-adjusting hybrid schemes for shock computations, J. Comput. Phys., 9, 3, 568-583 (1972) · Zbl 0244.76033 · doi:10.1016/0021-9991(72)90012-5 |
| [13] | Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 135, 2, 260-278 (1997) · Zbl 0890.65096 · doi:10.1006/jcph.1997.5713 |
| [14] | Harten, A.: Method of artificial compression. I. shocks and contact discontinuities. Technical report, New York Univ., NY (USA). AEC Computing and Applied Mathematics Center, (1974) · Zbl 0343.76023 |
| [15] | Horváth, Z., Positivity of Runge-Kutta and diagonally split Runge-Kutta methods, Appl. Numer. Math., 28, 2-4, 309-326 (1998) · Zbl 0926.65073 · doi:10.1016/S0168-9274(98)00050-6 |
| [16] | Jameson, A., Computational algorithms for aerodynamic analysis and design, Appl. Numer. Math., 13, 5, 383-422 (1993) · Zbl 0792.76049 · doi:10.1016/0168-9274(93)90096-A |
| [17] | Jiang, G-S; Shu, C-W, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 1, 202-228 (1996) · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130 |
| [18] | Kennedy, C.A., Carpenter, M.H.: Diagonally Implicit Runge-Kutta Methods for Ordinary Differential Equations, a Review. National Aeronautics and Space Administration, Langley Research Center (2016) |
| [19] | Ketcheson, DI; bin Waheed, U., A comparison of high order explicit Runge-Kutta, extrapolation, and deferred correction methods in serial and parallel, CAMCoS, 9, 2, 175-200 (2014) · Zbl 1314.65102 · doi:10.2140/camcos.2014.9.175 |
| [20] | Kurganov, A.; Petrova, G.; Popov, B., Adaptive semidiscrete central-upwind schemes for nonconvex hyperbolic conservation laws, SIAM J. Scientific Comput., 29, 6, 2381-2401 (2007) · Zbl 1154.65356 · doi:10.1137/040614189 |
| [21] | Kuzmin, D., Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws, Comput. Methods Appl. Mech. Eng., 361 (2020) · Zbl 1442.65263 · doi:10.1016/j.cma.2019.112804 |
| [22] | Kuzmin, Dmitri, de Luna, Manuel Quezada: Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws. Comput. Methods Appl. Mech. Eng. 372, 113370 (2020) · Zbl 1506.74413 |
| [23] | Kuzmin, D.; de Luna, MQ, Entropy conservation property and entropy stabilization of high-order continuous Galerkin approximations to scalar conservation laws, Comput. Fluids, 213 (2020) · Zbl 1521.76344 · doi:10.1016/j.compfluid.2020.104742 |
| [24] | Kuzmin, D.; de Luna, MQ, Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws, J. Comput. Phys., 411 (2020) · Zbl 1436.65141 · doi:10.1016/j.jcp.2020.109411 |
| [25] | Kuzmin, D., de Luna, M.Q., Ketcheson, D.I., Grüll, J.: Bound-preserving convex limiting for high-order Runge-Kutta time discretizations of hyperbolic conservation laws. arXiv:2009.01133, (2020) · Zbl 1539.65095 |
| [26] | Kuzmin, D.; Löhner, R.; Turek, S., Flux-corrected transport: principles, algorithms, and applications (2012), Berlin: Springer, Berlin · Zbl 1287.76031 · doi:10.1007/978-94-007-4038-9 |
| [27] | Lee, J-L; Bleck, R.; MacDonald, AE, A multistep flux-corrected transport scheme, J. Comput. Phys., 229, 24, 9284-9298 (2010) · Zbl 1201.65162 · doi:10.1016/j.jcp.2010.08.039 |
| [28] | LeVeque, RJ, Numerical methods for conservation laws (1992), Berlin: Springer, Berlin · Zbl 0847.65053 · doi:10.1007/978-3-0348-8629-1 |
| [29] | Leveque, RJ, High-resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33, 2, 627-665 (1996) · Zbl 0852.76057 · doi:10.1137/0733033 |
| [30] | LeVeque, RJ, Finite volume methods for hyperbolic problems (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 1010.65040 · doi:10.1017/CBO9780511791253 |
| [31] | Liu, X-D; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 1, 200-212 (1994) · Zbl 0811.65076 · doi:10.1006/jcph.1994.1187 |
| [32] | Lohmann, C.; Kuzmin, D.; Shadid, JN; Mabuza, S., Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements, J. Comput. Phys., 344, 151-186 (2017) · Zbl 1380.65279 · doi:10.1016/j.jcp.2017.04.059 |
| [33] | Magiera, J.; Ray, D.; Hesthaven, JS; Rohde, C., Constraint-aware neural networks for riemann problems, J. Comput. Phys., 409, 109345 (2020) · Zbl 1435.76046 · doi:10.1016/j.jcp.2020.109345 |
| [34] | Nikitin, K.; Terekhov, K.; Vassilevski, Y., A monotone nonlinear finite volume method for diffusion equations and multiphase flows, Comput. Geosci., 18, 3-4, 311-324 (2014) · Zbl 1378.76076 · doi:10.1007/s10596-013-9387-6 |
| [35] | Osher, S.; Chakravarthy, S., High resolution schemes and the entropy condition, SIAM J. Numer. Anal., 21, 5, 955-984 (1984) · Zbl 0556.65074 · doi:10.1137/0721060 |
| [36] | Qiu, J.; Shu, C-W, On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes, J. Comput. Phys., 183, 1, 187-209 (2002) · Zbl 1018.65106 · doi:10.1006/jcph.2002.7191 |
| [37] | Rusanov, VV, The calculation of the interaction of non-stationary shock waves with barriers, Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 1, 2, 267-279 (1961) |
| [38] | Spijker, MN, Contractivity in the numerical solution of initial value problems, Numer. Math., 42, 271-290 (1983) · Zbl 0504.65030 · doi:10.1007/BF01389573 |
| [39] | Xiong, T.; Qiu, J-M; Zhengfu, X., High order maximum-principle-preserving discontinuous Galerkin method for convection-diffusion equations, SIAM J. Sci. Comput., 37, 2, A583-A608 (2015) · Zbl 1320.65145 · doi:10.1137/140965326 |
| [40] | Yang, P.; Xiong, T.; Qiu, J-M; Zhengfu, X., High order maximum principle preserving finite volume method for convection dominated problems, J. Sci. Comput., 67, 2, 795-820 (2016) · Zbl 1350.65095 · doi:10.1007/s10915-015-0104-6 |
| [41] | Zalesak, ST, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 3, 335-362 (1979) · Zbl 0416.76002 · doi:10.1016/0021-9991(79)90051-2 |
| [42] | Zhang, X.; Liu, Y.; Shu, C-W, Maximum-principle-satisfying high order finite volume weighted essentially nonoscillatory schemes for convection-diffusion equations, SIAM J. Sci. Comput., 34, 2, A627-A658 (2012) · Zbl 1252.65146 · doi:10.1137/110839230 |
| [43] | Zhang, X.; Shu, C-W, On maximum-principle-satisfying high order schemes for scalar conservation laws, J. Comput. Phys., 229, 9, 3091-3120 (2010) · Zbl 1187.65096 · doi:10.1016/j.jcp.2009.12.030 |
| [44] | Zhang, X.; Shu, C-W, Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments, Proc. Royal Soc. A: Math. Phys. Eng. Sci., 467, 2134, 2752-2776 (2011) · Zbl 1222.65107 · doi:10.1098/rspa.2011.0153 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
