A. Sandu, F. A. Potra, G. R. Carmichael, and V. Damian. Efficient implementation of fully implicit methods for atmospheric chemical kinetics. J. Comput. Phys., 129:101--110, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....matrix with sparse diagonal blocks R k , k = 1; n z , which are reaction Jacobians in a cell k (cf. Fig. 1) The sparsity of the blocks can be efficiently exploited in the course of the LU factorization of I Gamma R. Using a special preprocessor tool KPP (Kinetic PreProcessor [17]) an optimal ordering of the trace gases can be found for which the L and U factors are as sparse as possible. In practice this means that the matrix L U usually has only few percent more fill in than I Gamma R. Preserving sparsity of the reaction Jacobian is of crucial importance and this, ....

A. Sandu, F. A. Potra, G. R. Carmichael, and V. Damian. Efficient implementation of fully implicit methods for atmospheric chemical kinetics. J. Comput. Phys., 129:101--110, 1996.

....implicit method is suitable for solving stiff atmospheric chemical kinetics problems (box models) provided the costs for the linear system solutions and the Jacobian updates are minimized by using optimal sparsity routines. Such routines are provided by the chemical kinetics preprocessor KPP [4, 5, 23]. ros2 remains order two consistent with an arbitrary matrix A, A 6= F 0 (wn ) However, stability of the scheme is corrupted when A is not the exact Jacobian. As is explained in Section 1, straightforward application of method (3.3) to the coupled verticaltransport chemistry problem is ....

A. Sandu, F. A. Potra, G. R. Carmichael, and V. Damian. Efficient implementation of fully implicit methods for atmospheric chemical kinetics. J. Comp. Phys., 129:101--110, 1996.

....in performance. See [52] for numerical comparisons. 3.3 General purpose methods of Rosenbrock type We next discuss an alternative for the special purpose methods, viz. low order Rosenbrock methods from the stiff ODE field which use sparse matrix routines to reduce numerical algebra overhead [76, 77, 78]. Earlier experiences reported in [35] have shown that for stiff ODEs from different applications Rosenbrock solvers are competitive with other solvers for low to modest accuracies. Because for atmospheric applications the greatest interest lies in high efficiency for very low accuracy (two digits ....

....A readily exceeds 50 by far, since in large models species react with only a limited number of others. This high level of sparsity can be exploited to significantly reduce the costs for the Jacobian update, LU decomposition and backsolves. For atmospheric problems sparsity has been considered in [45, 76, 77, 78, 101, 105, 95]. In particular a pivoting technique using a diagonal Markowitz criterion proves to minimize fill in and to reduce the numerical algebra costs considerably for large models [76] To relieve modellers from a lot of error prone coding, for a long time symbolic preprocessing is used to automatically ....

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A. Sandu, F.A. Potra, G.R. Carmichael and V. Damian. Efficient implementation of fully implicit methods for atmospheric chemical kinetics, J. Comput. Phys. 129, 101--110, 1996.

....chemistry kinetics. However the performance of implicit methods depends critically on the implementation of linear algebra (which appears in connection with Newton steps) and on their restart overhead (when used in an operator splitting environment) We have begun to explore these ideas [25]. We believe that it is possible to exploit sparsity in the Jacobian systematically and hence to develop more efficient implementations of fully implicit methods for solving atmospheric chemistry problems. 1. We evaluated reordering techniques that lead to minimal fill in during LU decomposition. ....

....algebra are presented in Table 1. In the usual practice, atmospheric modellers sometimes employ Gear type methods (see [19] Our results indicate that when linear algebra is treated efficiently, implicit methods from different families, like Singly Diagonal Implicit Runge Kutta or Rosembrock, [25] can be competitive. Figure 2 compares results for three sparse codes with the widely used QSSA and CHEMEQ. A restart is carried out every hour or every 15 minutes (Figure 2) This restart corresponds to an equal step size of the transport scheme, in an operator splitting code. Since ....

A. Sandu, F.A. Potra, V. Damian, and G.R. Carmichael. Efficient implementation of fully implicit methods for atmospheric chemistry. Technical report 79, The University of Iowa, Department of Mathematics, Iowa City, October 1995.

....sparsity of the Jacobian matrix. This reduces the costs of solving the linear algebraic systems arising in the modified Newton iteration. In [21] and [8] it has been shown that this is very profitable for atmospheric chemistry problems. We used the sparse linear algebra implementation described in [22]. The necessary routines are automatically generated by the symbolic chemical preprocessor kpp [23] which transparently: ffl determines the sparse analytical Jacobian, ffl reorders the species using a diagonal Markowitz criterion, in order to minimize 72 the fill in resulting from the lu ....

A. Sandu, F.A. Potra, V. Damian, and G.R. Carmichael. Efficient implementation of fully implicit methods for atmospheric chemistry. Technical report 79, The University of Iowa, Department of Mathematics, Iowa City, October 1995.

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A. Sandu, F.A. Potra, V. Damian, and G.R. Carmichael. Efficient implementation of fully implicit methods for atmospheric chemical kinetics. Journal of Computational Physics: 129, 101--110, 1996.

....The first five of these are special purpose and compute the solution in an explicit way, as opposed to the last four which are all implicit and state of the art in the numerical stiff ode field. For our purpose, the solvers vode, sdirk4 and rodas have been provided with a sparse matrix technique [20] to economize on the numerical algebra overhead in the modified Newton solution. lsodes has already built in a sparsematrix technique and is a successor of lsode [14] which is popular amongst atmospheric chemists as a reference code. By providing the implicit solvers with sparse matrix techniques, ....

....the chemical conditions are identical and the calculated results very close, the performances of the numerical solvers depend on the chemical mechanism used. We will make this point later in the paper. ffl An important issue in our numerical comparison is the use of a sparse matrix technique [20] to economize on the linear algebra costs which the stiff solvers spend in the modified Newton iteration. As a measure of these costs, we give in Table 1 the number of nonzero elements in the Jacobian matrix, as well as the number of nonzero entries in the Newton matrix after the lu factorization. ....

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A. Sandu, F.A. Potra, V. Damian, and G.R. Carmichael. Efficient implementation of fully implicit methods for atmospheric chemistry. Technical report 79, University of Iowa, Department of Mathematics, 1995.

....see [17] widely used by atmospheric modellers. VODE is considered to have several advantages over LSODE when used to integrate systems of ODE arising from chemical kinetics (see [4] In order to take full advantage of the sparsity pattern of the Jacobian, VODE has been modified as described in [23] by replacing the general factorization and substitution routines dgefa and dgesl with specialized sparse routines. Results for both the standard and the modified VODE are presented. All integrators have been used with a lower bound of 0:01 [seconds] imposed on the chosen step size. The emissions ....

....its performance is affected by frequent restarts; this is one of the reasons why atmospheric scientists have chosen to develop their own integrators rather than using general solvers. However, if the linear algebra is done such that full advantage of the structure of the problem is taken (see [23]) the computational time of VODE is greatly reduced 4 and the code becomes competitive. Extrapolated QSSA and Symmetric QSSA perform well compared to DAE QSSA, Iterated QSSA or CHEMEQ (especially when a NAD higher than 1 is required) but not better than Sparse VODE or TWOSTEP. In three ....

A. Sandu, F.A. Potra, V. Damian, and G.R. Carmichael. Efficient implementation of fully implicit methods for atmospheric chemistry. Technical report 79, University of Iowa, Department of Mathematics, 1995.

....This research effort lead to the reduction of the chemistry integration time by an order of magnitude, and contributed to a better theoretical understanding of the behavior of numerical integrators for chemical rate equations. The results of our research are contained in several papers ([12, 13, 14, 16, 42, 43, 44, 45]) some of which have been published in prestigious professional journals such as SIAM Journal on Scientific Computing, Journal of Computational Physics, and Atmospheric Environment. In order to easily test different chemistry models and different integration schemes our group wrote a symbolic ....

A. Sandu, F. A. Potra, V. Damian-Iordache, and G. R. Carmichael. Efficient implementation of fully implicit methods for atmospheric chemistry. Journal of Computational Physics, 129:101--110, 1996.

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