zbMATH — the first resource for mathematics

A high-order conservative Patankar-type discretisation for stiff systems of production–destruction equations. (English) Zbl 1028.80008
Summary: In the present paper, numerically robust, unconditionally positive and conservative schemes for the discretisation of stiff systems of production–destruction equations are designed. Such model systems do typically arise in geobiochemical modelling where the reproduction of these properties is vital. We suggest modified Patankar-type methods of first- and second-order in time and compare their performance by means of approximating simple linear and non-linear model problems. For the non-linear model problem, a hybrid method combining the classical Runge–Kutta scheme with a modified Patankar-type scheme gives the best numerical approximation. The classical Robertson test problem for chemical reactions which is known for its stiffness is excellently approximated with the modified Patankar-type scheme. The procedure with respect to the derivation and analysis of the modified Patankar-type schemes can be used as a guideline to develop even unconditionally positive, conservative and third-order as well as higher order methods.

80A32 Chemically reacting flows
92D25 Population dynamics (general)
Full Text: DOI
[1] Axelsson, O., Iterative solution methods, (1996), Cambridge University Press Cambridge
[2] Baretta, J.W.; Ebenhöh, W.; Ruardij, P., The European regional seas ecosystem model, a complex marine ecosystem model, Neth. J. sea res., 33, 233-246, (1995)
[3] Berzins, M., Modified mass matrices and positivity preservation for hyperbolic and parabolic pdes, Commun. numer. meth. engng., 9, 659-666, (2001) · Zbl 0986.65091
[4] Burchard, H., Applied turbulence modelling in marine waters, Lecture notes in Earth sci., 100, (2002), Springer Berlin · Zbl 1068.86001
[5] Berzins, M.; Ware, J.M., Positive cell-centered finite volume discretization methods for hyperbolic equations on irregular meshes, Appl. numer. math., 16, 417-438, (1995) · Zbl 0824.65091
[6] Deleersnijder, E.; Beckers, J.-M.; Campin, J.-M.; El Mohajir, M.; Fichefet, T.; Luyten, P., Some mathematical problems associated with the development and use of marine models, (), 41-86 · Zbl 0897.76079
[7] E. Deleersnijder, Modélisation hydrodynamique tridimensionelle de la circulation générale estivale de la région du détroit de bering, Ph.D. Thesis, Faculté des Sciences Appliquée, Université Catholique de Louvain, Louvain,1992
[8] Dippner, J.W., A Lagrangian model of phytoplankton growth dynamics for the northern adriatic sea, Cont. shelf res., 13, 331-355, (1993)
[9] Dekker, K.; Verwer, J.G., Stability of runge – kutta methods for stiff nonlinear differential equations, CWI monogr., 2, (1984), North-Holland Amsterdam · Zbl 0571.65057
[10] Fennel, W., A model of the yearly cycle of nutrients and plankton in the baltic sea, J. mar. sys., 6, 313-329, (1995)
[11] Gerisch, A.; Griffiths, D.F.; Weiner, R.; Chaplain, M.A.J., A positive splitting method for mixed hyperbolic – parabolic systems, Numer. meth. part. differential equations, 17, 2, 152-168, (2001) · Zbl 0981.65112
[12] Gerisch, A.; Verwer, J.G., Operator splitting and approximate factorization for taxis – diffusion – reaction models, Appl. numer. math., 42, 1-3, 159-176, (2002) · Zbl 0998.65102
[13] Golub, G.H.; van Loan, C., Matrix computations, (1989), The John Hopkins University Press Baltimore, MD · Zbl 0733.65016
[14] Hundsdorfer, W.; Koren, B.; van Loon, M.; Verwer, J.G., A positive finite-difference advection scheme, J. comput. phys., 117, 35-46, (1995) · Zbl 0860.65073
[15] Hairer, E.; Nørsett, S.P.; Wanner, G., Solving ordinary differential equations I, Springer ser. comput. math., (1987), Springer Berlin · Zbl 0638.65058
[16] Horváth, Z., On a second order non-negativity conserving method, Ann. univ. sci. Budapest sect. comput., 13, 237-242, (1992) · Zbl 0881.65086
[17] Horváth, Z., Consistency and stability for some nonnegativity conserving methods, Appl. numer. math., 13, 5, 371-381, (1993) · Zbl 0794.65073
[18] Horváth, Z., On higher order unconditionally nonnegative conserving methods, Ann. univ. sci. Budapest sect. comput., 15, 167-178, (1995) · Zbl 0891.65102
[19] Horváth, Z., On the unconditionally positivity of diagonally split runge – kutta methods, (), 311-320 · Zbl 0892.65047
[20] Horváth, Z., Positivity of runge – kutta and diagonally split runge – kutta methods, Appl. numer. math., 28, 309-326, (1998) · Zbl 0926.65073
[21] Hundsdorfer, W.; Trompert, R.A., Method of lines and direct discretization: A comparison for linear advection, Appl. numer. math., 13, 6, 469-490, (1994) · Zbl 0797.65067
[22] Hundsdorfer, W., Partially implicit BDF2 blends for convection dominated flows, SIAM J. numer. anal., 38, 6, 1763-1783, (2001) · Zbl 1007.76052
[23] Hairer, E.; Wanner, G., Solving ordinary differential equations II, (1991), Springer Berlin · Zbl 0729.65051
[24] Meister, A., Numerik linearer gleichungssysteme, eine einführung in moderne verfahren, (1999), Vieweg Wiesbaden · Zbl 0943.65045
[25] Neumann, T., Towards a 3D-ecosystem model of the baltic sea, J. mar. sys., 25, 405-419, (2000)
[26] Patankar, S.V., Numerical heat transfer and fluid flow, (1980), McGraw-Hill New York · Zbl 0595.76001
[27] Pietrzak, J., The use of TVD limiters for forward-in-time upstream-biased advection schemes in Ocean modeling, Monthly weather rev., 126, 812-830, (1998)
[28] Rietkerk, M.; Boerlijst, M.C.; van Langevelde, F.; HilleRisLambers, R.; van de Koppel, J.; Kumar, L.; Klausmeier, C.A.; Prins, H.H.T.; de Roos, A.M., Self-organisation of vegetation in arid ecosystems, Amer. naturalist, 160, 524-530, (2002)
[29] Robertson, H.H., The solution of a set of reaction rate equations, (), 178-182
[30] Saad, Y., Iterative methods for sparse linear systems, (1996), PWS Boston · Zbl 1002.65042
[31] Sandu, A., Positive numerical integration methods for chemical kinetic systems, J. comput. phys., 170, 589-602, (2001) · Zbl 0984.65070
[32] Varga, R.S., Matrix iterative analysis, Comput. math., 27, (2000), Springer Berlin · Zbl 0133.08602
[33] van der Houwen, P.J.; Sommeijer, B.P., The iterative solution of fully implicit discretizations of three-dimensional transport models, Appl. numer. math., 25, 243-256, (1997) · Zbl 0890.65099
[34] van der Houwen, P.J.; Sommeijer, B.P., Approximate factorization for time-dependent partial differential equations, J. comput. appl. math., 128, 447-466, (2001) · Zbl 0974.65089
[35] Verwer, J.G., Gauss – seidel iteration for stiff ODEs from chemical kinetics, SIAM J. sci. comput., 15, 5, 1243-1250, (1994) · Zbl 0804.65068
[36] Woods, J.D.; Barkmann, W., Simulating plankton ecosystems by the Lagrangian ensemble method, Philos. trans. roy. soc. London B, 343, 27-31, (1994)
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.