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Positive numerical integration methods for chemical kinetic systems. (English) Zbl 0984.65070
Summary: Chemical kinetics conserves mass and renders nonnegative solutions: a good numerical simulation would ideally produce mass-balanced, positive concentration vectors. Many time-stepping methods are mass conservative; however, unconditional positivity restricts the order of a traditional method to one. The projection method presented in this paper ensures mass conservation and positivity. First, a numerical approximation is computed with one step of a mass-preserving traditional scheme. If there are negative components, the nearest vector in the reaction simplex is found by solving a quadratic optimization problem; this vector is shown to better approximate the true solution. A simpler version involves just one projection step and stabilizes the reaction simplex. This technique works best when the underlying time-stepping scheme favors positivity. Projected methods are more accurate than clipping and allow larger time steps for kinetic systems which are unstable outside the positive quadrant.

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
80A30 Chemical kinetics in thermodynamics and heat transfer
Full Text: DOI
[1] Ascher, U.M.; Chin, H.; Reich, S., Stabilization of DAEs and invariant manifolds, Numer. math., 67, 131, (1994) · Zbl 0791.65051
[2] Bolley, C.; Crouzeix, M., Conservation de la positivite lors de la discretization des problemes d’evolution parabolique, R.A.I.R.O. numer. anal., 12, 237, (1978) · Zbl 0392.65042
[3] Carmichael, G.R.; Peters, L.K.; Kitada, T., A second generation model for regional-scale transport/chemistry/deposition, Atmos. environ., 20, 173, (1986)
[4] Damian-Iordache, V.; Sandu, A.; Damian-Iordache, M.; Carmichael, G.R.; Potra, F.A., Symbolic preprocessor for chemistry kinetics—user’s guide, 52246, (1995)
[5] Goldfarb, D.; Idnani, A., A numerically stable dual method for solving strictly convex quadratic programs, Math. progr., 27, 1, (1983) · Zbl 0537.90081
[6] Hairer, E.; Norsett, S.P.; Wanner, G., Solving ordinary differential equations I. nonstiff problems, (1993) · Zbl 0789.65048
[7] Hairer, E.; Wanner, G., Solving ordinary differential equations II. stiff, and differential-algebraic problems, (1991) · Zbl 0729.65051
[8] Hundsdorfer, W., Numerical solution of advection – diffusion – reaction equations, (1996)
[9] D. E. Kinnison, NASA HSRP/AESA Stratospheric Models Intercomparison, for NASA ftp site, contact kinnison1@llnl.gov.
[10] Owren, B.; Simonsen, H.H., Alternative integration methods for problems in structural dynamics, Comput. meth. appl. mech. eng., 122, 1, (1995) · Zbl 0851.73076
[11] Sandu, A., Numerical aspects of air quality modeling, (1997)
[12] Sandu, A.; Blom, J.G.; Spee, E.; Verwer, J.; Potra, F.A.; Carmichael, G.R., Benchmarking stiff ODE solvers for atmospheric chemistry equations II—rosenbrock solvers, Atmos. environ., 31, 3459, (1997)
[13] Shampine, L.F., Conservation laws and the numerical solution of odes, Comput. math. appl., 12B, 1287, (1986) · Zbl 0641.65057
[14] Verwer, J.G.; Hunsdorfer, W.; Blom, J.G., Numerical time integration of air pollution models, modeling, analysis and simulations report, (1998)
[15] Verwer, J.; Spee, E.J.; Blom, J.G.; Hunsdorfer, W., A second order rosenbrock method applied to photochemical dispersion problems, SIAM J. sci. comput., 20, 1456, (1999) · Zbl 0928.65116
[16] J. G. Blom, and, J. Verwer, A comparison of integration methods for atmospheric transport-chemistry problems, J. Comput. Appl. Math, to appear. · Zbl 1010.76066
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