Partitioning Ode Systems with an Application to Air Pollution Models

Abstract The numerical treatment of stiff ODE systems is carried out by using implicit methods. A long sequence of nonlinear systems has to be treated when implicit methods are used. The Newton iterative method is often used in the solution of these systems. This leads to the calculation of Jacobian matrices and to the inversion of these matrices. The computational work can in some cases be reduced considerably when some kind of partitioning is used. The conditions under which the partitioning procedures can successfully be used will be studied in this paper. An example, taken from a large air pollution model, will be given to illustrate the usefulness of the theoretical results.

[1]  I︠u︡. I. Li︠u︡bich,et al.  Matrix norms and their applications , 1988 .

[2]  Zahari Zlatev,et al.  Studying high ozone concentrations by using the Danish Eulerian model , 1993 .

[3]  Zahari Zlatev,et al.  Three-Dimensional Version of the Danish Eulerian Model , 1995, PARA.

[4]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[5]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[6]  Joke Blom,et al.  A comparison of stiff ode solvers for atmospheric chemistry problems , 1995 .

[7]  P. Deuflhard,et al.  Affine Invariant Convergence Theorems for Newton’s Method and Extensions to Related Methods , 1979 .

[8]  Peter Deuflhard,et al.  Recent progress in extrapolation methods for ordinary differential equations , 1985 .

[9]  G. Carmichael,et al.  IMPROVED QSSA METHODS FOR ATMOSPHERIC CHEMISTRY INTEGRATION , 1995 .

[10]  Peter Deuflhard,et al.  Recent Developments in Chemical Computing , 1990 .

[11]  G. R. Carmichael,et al.  The evaluation of numerical techniques for solution of stiff ordinary differential equations arising from chemical kinetic problems , 1988 .

[12]  J. G. Verwer,et al.  Explicit method for stiff ODEs from atmospheric chemistry , 1995 .

[13]  Zahari Zlatev,et al.  Exploiting the Natural Partitioning in the Numerical Solution of ODE Systems Arising in Atmospheric Chemistry , 1996, WNAA.

[14]  Naresh Kumar,et al.  A comparison of fast chemical kinetic solvers for air quality modeling , 1992 .

[15]  Zahari Zlatev,et al.  Vectorizing codes for studying long-range transport of air pollutants , 1991 .

[16]  Z. Zlatev,et al.  Relationships between emission sources and excess ozone concentrations , 1996 .

[17]  P. Deuflhard,et al.  One-step and extrapolation methods for differential-algebraic systems , 1987 .

[18]  Z. Zlatev,et al.  Studying cumulative ozone exposures in Europe during a 7‐year period , 1997 .

[19]  Adrian Sandu,et al.  Improved Quasi-Steady-State-Approximation Methods for Atmospheric Chemistry Integration , 1997, SIAM J. Sci. Comput..

[20]  Zahari Zlatev,et al.  Numerical integration of chemical ODE problems arising in air pollution models , 1997 .

[21]  Jan Verwer,et al.  An evaluation of explicit pseudo-steady-state approximation schemes for stiff ODE systems from chemical kinetics , 1993 .

[22]  Z. Zlatev.  Computational Methods for General Sparse Matrices , 1991 .

[23]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[24]  D P Chock,et al.  Comparison of stiff chemistry solvers for air quality modeling. , 1994, Environmental science & technology.

[25]  Zahari Zlatev,et al.  A Eulerian air pollution model for Europe with nonlinear chemistry , 1992 .

[26]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[27]  J. Christensen,et al.  Test of two numerical schemes for use in atmospheric transport-chemistry models , 1993 .

[28]  Adrian Sandu,et al.  Benchmarking stiff ode solvers for atmospheric chemistry problems II: Rosenbrock solvers , 1997 .

[29]  Adrian Sandu,et al.  Benchmarking Stiff ODE Solvers for Atmospheric Chemistry Problems I: Implicit versus Explicit , 1996 .

[30]  Z. Zlatev,et al.  A comparison of the predictions of an eulerian atmospheric transport — chemistry model with experimental measurements over the North sea , 1994 .

[31]  Z. Zlatev,et al.  Modeling the long-range transport of air pollutants , 1994, IEEE Computational Science and Engineering.

[32]  M. C. Dodge,et al.  A photochemical kinetics mechanism for urban and regional scale computer modeling , 1989 .

[33]  I. Duff,et al.  Direct Methods for Sparse Matrices , 1987 .

[34]  P. Deuflhard.  Order and stepsize control in extrapolation methods , 1983 .

[35]  Zahari Zlatev,et al.  Computer Treatment of Large Air Pollution Models , 1995 .

[36]  Peter Deuflhard,et al.  Efficient numerical simulation and identification of large chemical reaction systems , 1986 .

[37]  I. Dimov,et al.  Numerical Analysis and Its Applications , 1996, Lecture Notes in Computer Science.

[38]  Kevin Burrage,et al.  Parallel and sequential methods for ordinary differential equations , 1995, Numerical analysis and scientific computation.

[39]  M. V. Belyi,et al.  Discrete analogs of boundary integral equations for elliptic boundary value problems , 1996 .