E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. (1991), Berlin, Heidelberg, New York, 1991  Home/Search   Document Not in Database   Summary   Related Articles   Check

....based on the work of Lurmann et al. see [14] and Atkinson et al. see [1] is representative of those presently being used in the study of chemically perturbed environments. The results are tested against an exact solution (which was considered to be that obtained with the code RADAU5 [7] with very tight tolerances) In Figure 1 a work precision diagram is shown for QSSA, Extrapolated QSSA, and VODE (a BDF code, see [17] The number of significant digits is a measure of the error over all components. For the same computational effort Extrapolated QSSA is clearly more accurate ....

Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems; Springer Verlag, Berlin, 1991.

....direction of the flow of information, and it is therefore best suited for two point boundary value problems. Quite differently, for initial value problems it is sometimes desirable to have stiffly accurate schemes, that are able to damp the higher frequencies components from the computed response [11]. This is achieved in practice by forcing a lack of symmetry in the scheme, that incorporates the knowledge on the direction of flow of information within the element. It is possible to derivesuchschemes in the present framework, allowing one single jump discontinuityatx i while enforcing the ....

E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and Differential--Algebraic Problems, Springer-Verlag, 1991.

.... projection and pressure Poisson reformulations (e.g. 8, 12, 20, 24, 26] Another topic of great recent interest is the numerical solution of differentialalgebraic equations (DAEs) In their most popular special form, these are ordinary differential equations with some equality constraints (e.g. [6, 13]) Recall that an important concept for measuring the difficulty in solving DAEs is given by the (differential) index, which is defined by the minimal number of analytical constraint differentiations such that the DAE can be transformed by algebraic manipulations into an explicit first order ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, SpringerVerlag, 1991.

....satisfies 11 0( nn gty where 1 = nnn tth . For such RK methods, it is also natural to take 1 : nns zZ . For RK methods whose coefficients satisfy 1 0 = j a for 1 = j. s , it is natural to take 1 : nn Zz . Other alternatives based on RK coefficients are half explicit RK methods [21 23], partitioned RK methods [24,25] and SPARK methods [26] The RK method used in our numerical experiments is the 2 stage Lobatto IIIA method, simply speaking the trapezoidal rule. Convergence of order 2 for the y component is achieved, meaning that the global error of the y component on a ....

....meaning that the global error of the y component on a finite interval between the exact solution and the trapezoidal rule approximation is bounded by Consth where max : nn hh . Detailed convergence results for Lobatto IIIA methods can be found in Hairer et al. 22] Hairer and Wanner [23] and Jay [27] The Butcher tableau coefficients i c ij a j b of the 2 stage Lobatto IIIA RK method is given as follows 0 0 2 1 2 1 For this method we obtain the following system of nonlinear equations for 1 n y and 1 n z ( 1111 11 2 nnnnnnnn nn ....

Hairer, E. and Wanner, G., 1996, Solving Ordinary Differential Equations II. Stiff and Differential-algebraic Problems, Second Revised Edition, Springer, Berlin.

....stiffness AMS subject classification: 65F10, 65H10, 65L05, 65L06, 65L80, 70F20, 70F25, 70H03, 70H45 1. Introduction In this article a broad class of systems of possibly stiff and implicit differential algebraic equations (DAEs) is considered, including Hessenberg DAEs of index 1, 2, and 3 [1,5,6,8,9]. These equations encompass the formulation of mechanical systems with mixed constraints of holonomic, nonholonomic, scleronomic, and rheonomic types [7,16,17] Solutions to these DAEs can be approximated numerically by applying a class of super partitioned additive Runge Kutta (SPARK) methods, ....

....of implicit differential algebraic equations (DAEs) d q(t,y) v(t,y,z) d dt p t, y,z) f(t,y,z,u,#,#,#) d c(t, y, z, u) d(t,y,z,u,#,#,#) 1c) m(t, y, z, u, # ) 0, 1d) 0, 1e) which may present some stiffness. These equations encompass Hessenberg DAEs of index 1, 2, and 3 [1,5,6,8,9]. They also include the formulation of mechanical systems with mixed constraints of holonomic, nonholonomic, scleronomic, and rheonomic types [7,13,16,17] In mechanics the quantities q,v,p represent respectively gener alized coordinates, generalized velocities, and generalized momenta. The ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential--Algebraic Problems, Comput. Math., Vol. 14, 2nd revised ed. (Springer, Berlin, 1996).

....high resolution on a wide variety of nonlinear circuits. It is also suitable for combining with matrix implicit Krylov subspace solvers in order to analyze large circuits with moderate computational cost. Another way of viewing our discretization scheme is as an implicit Runge Kutta (IRK) method [13]. Leveraging the theoretical framework available for analysis of IRK methods has given us an increased understanding of limitations of numerical techniques traditionally used in analog and RF circuit simulation, in particular the computational advantages associated with the superior stability ....

E. Hairer and G. Wanner, "Solving ordinary differential equations II", Springer-Verlag, 1991.

....intervals can be placed in a denser timestep spacing, and possibly the order of the method lowered. Third, the method has excellent numerical stability properties. It can be shown that our scheme is a member of a particular class of implicit Runge Kutta (IRK) methods, the collocation IRK methods [3]. Higher order implicit Runge Kutta methods are multi stage methods, that is, they are implicit in more than one point at a time. Because of the multiple implicit states, and unlike multi step methods such as the Gear methods, IRK methods can be constructed that are A stable even at high order. By ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, 1991.

....a constraint on div A C . Obviously, this forfeits uniqueness of the solution in parts of the domain where oe = 0, but the solution for B remains unique everywhere. Inside the conductor, where oe 0, we get a unique A. For the sake of stability, timestepping schemes for (1) have to be L stable [26]. This requirement can only be met by implicit schemes like SDIRK methods. In each timestep they entail the solution of a degenerate elliptic boundary value problem of the form curl curl u fiu = f u Theta n = 0 on (2) In this context, u denotes the new approximation to A to be computed ....

E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1991.

....uses finite elements and constraints without a reference frame. While it naturally leads to problems in the higher frequencies and even to instabilities, see also Shabana [10] stabilization techniques exist which add an artificial numerical damping, like the HHT method, see e.g. Hairer et al. [6, 7]. Alternatively, Gonzalez and Simo [5] derived stable energy momentum methods for Hamiltonian systems and it has been extended to contact and impact problems by Demkowicz and Bajer [3] The finite element method has been used for the modelling of multibody dynamics problems with contact, see e.g. ....

E. Hairer and G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems, Springer--Verlag Berlin Heidelberg (1991).

....system of ODEs, written in the form dw(t) F(w(t) t 0; 33) 16 where the vector w(t) contains the components w i (t) Our first observation is that this system is a stiff ODE. This means that the Jacobian matrix F= w has widely spread eigenvalues (for a discussion on stiffness we refer to [8]) Stiffness has a direct consequence on the choice of the time integration technique. An explicit method, which is simple and cheap per step, would be forced by the stiff system to take small time steps in order to avoid instabilities. This time step restriction is in our application so severe ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-algebraic Problems, 2nd ed., Springer Series in Computational Mathematics 14, Springer Verlag, Berlin (1996).

....approximation for c(tn) and Here J is an approximation of the Jacobian OF Oc at c = c n and is a free parameter. With the exact Jacobian for J, the stability function reads R(z) 1 (1 27)z ( 2 2)z (4. 1) from which it follows that the method is A stable if and only if 1 4 [4]. Furthermore the method is L stable if 1 4 v and the exact Jacobian matrix is used for J. The scheme is of second order in T regardless of the choice for J. In [4] one finds ample evidence that Rosenbrock methods are well suited to solve stiff ODEs in the low to moderate accuracy range. ....

....reads R(z) 1 (1 27)z ( 2 2)z (4.1) from which it follows that the method is A stable if and only if 1 4 [4] Furthermore the method is L stable if 1 4 v and the exact Jacobian matrix is used for J. The scheme is of second order in T regardless of the choice for J. In [4] one finds ample evidence that Rosenbrock methods are well suited to solve stiff ODEs in the low to moderate accuracy range. However, with the exact Jacobian c9F c9c the method cannot be efficiently applied since the linear system solutions are much too expensive. We therefore apply it with an ....

E. Hairer, G. Wanner, Solving ordinary differential equations II. Stiff and differential algebraic problems, 2nd ed., Springer-Verlag, Berlin, 1996.

....for multistep methods. The explicit Adams 2 step method is used to illustrate the techniques required. Key words. Multistep methods, stepsize selection, stability. AMS subject classifications. primary 65L05 1 Introduction The term SC stability (step control) was introduced in Hairer Wanner [1]. Loosely speaking an algorithm for solving ODEs is SC stable if the stepsizes chosen vary smoothly (usually slowly) when stability restricts the stepsize. If an algorithm is not SC stable, oscillating stepsizes and loss of smoothness in the solution will occur and in most cases there are frequent ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, Berlin, 1991.

....a high frequency of rejected steps. Both smoothness in the solutions and efficiency are affected adversely. In Hall [3] and Hall Higham [4] a theory was developed which gave conditions under which this non smooth type of behaviour will occur. This theory, called SC stability in Hairer Wanner [2], initiated some research to look for improved methods and improved stepsize controllers. In Higham Hall [5] new RK embedded formulae were derived which behaved smoothly with respect to the standard Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK ....

....may involve a slight loss of efficiency for the non stiff case. The purpose of this paper is describe an alternative controller, that can also be used with any RK embedded pair. The precise details will be worked out for the widely acknowledged method of [1] RK5(4)7FM, which is called DOPRI5 in [2]. The same development can be carried out for any RK embedded pair. The effect of the new strategy on DOPRI8, derived in [9] will also be shown. Estimates of the dominant eigenvalue of the Jacobian, rather than techniques from control theory will be used in formulating the new strategy. The new ....

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E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, SpringerVerlag, Berlin, 1991.

....in a different exploitation of the task parallelism. Figure 8 shows the resulting speedup values on 32 processors of an IBM SP2 and Intel Paragon for an application of a DIIRK method with 5 stages to the solution of a stiff ODE system that results from a discretization of the Brusselator equation [8]. For dense ODE systems, the attained speedup values are larger for both the task parallel and data parallel execution. The task parallel execution is still much faster than the data parallel execution, but the difference in percentages is smaller. 4.3 Extrapolation methods Extrapolation methods ....

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer, 1991.

....equations using the program package d 3 f [10] a simulator based on UG. The equations are discretized by means of mass conserving finite volumes using a consistent velocity approximation [11] The semidiscrete equations are solved by a 10 Bastian et al. diagonally implicit Runge Kutta method [14,18]. The discretization is second order consistent both in space and time. The solution of the discrete system is done by a fully coupled fully implicit solving strategy. After linearization the linear subproblems are solved by a multigrid V cyle iteration using a modified Vanca type SSOR smoother ....

E. Hairer and G. Wanner. Solving ordinary differential equations II. SpringerVerlag, Berlin, 1991.

....form c 1 a 11 . c s a s1 Delta Delta Delta a ss b 1 Delta Delta Delta b s where a ij = 0 for i j and a ii 6= 0. In order to obtain equilibrium in the final stage s we assume that the scheme is stiffly accurate, i.e. b i = a si . Then, the method is L stable ([21], Proposition III.3.8) For the algorithmic formulation we set c 0 = 0. Now, we define the following scheme: Given (un ; oe n ; q n ) set ( i ; p i ) oe n ; q n ) Deltat n i Gamma1 X j=1 a ij ( Deltaoe n;j ; Deltaq n;j ) find v i 2 V g(tn c i Deltat n ) Gammag(t n c i Gamma1 ....

.... n 1 ) k i = Deltaoe n;i ; Deltaq n;i ) Gamma ( Delta oe n;i ; Delta q n;i ) Y i = y 0 Deltat n i X j=1 a ij k j : Then, the identity jjj y 1 jjj 2 = jjj y 0 jjj 2 2 Deltat n s X i=1 b i hh k i ; Y i ii Gamma ( Deltat n ) 2 s X i;j=1 m ij hh k i ; k j ii holds ([21], in the proof of Theorem IV.12.4) where hh ; ii denotes the inner product corresponding to jjj jjj. From ii) follows P m ij hh k i ; k j ii 0. By definition of the Runge Kutta method we have R( v i ) i ; p i ) i a ii Deltat n Deltaoe n;i = oe n Deltat n i X j=1 a ij Deltaoe ....

E. Hairer and G. Wanner, Solving ordinary differential equations II, Springer, 1991.

....First of all, it represents a so called stiff ODE, which may be roughly described as a system whose solution components locally are strongly decreasing. In other words, the linearization of the ODE has eigenvalues with small real parts. The stability theory of numerical intergration (cf. [6]) shows that for systems of this kind implicit discretization schemes (like implicit Runge Kutta methods) are strongly preferable over explicit ones. Using an explicit method, the stepsize has to be chosen very small in order to reflect the asymtotic behavior of the solution. Thus, an explicit ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1991.

....differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. Steinebach [12] modified the well known solver RODAS of Hairer and Wanner [1] to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third order Rosenbrock solver for the nonlinear situation. ....

....multistep method usually must be restarted at lower order, whereas one 1 step methods can continue at higher order. In addition, Rosenbrock methods avoid the solution of nonlinear equations, working the exact Jacobian directly into the integration formula (Rosen brock [8] Hairer and Wanner [1]) Nowadays they are widely accepted to work satisfactorily for moderate accuracy requirements which are typical for the solution of PDEs. It is a known fact that one step methods such as Rosenbrock, Runge Kutta, and extrapolation methods suffer from order reduction when they are applied to ....

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E. Hairer and G. Wanner, Solving ordinary differential equations II, stiff and differentialalgebraic problems, second edition, Springer--Verlag, Berlin, Heidelberg, New York (1996)

....the splitting error may spoil the solution. This is especially pronounced for the fast varying trace gases (the so called radicals) 4, 2, 3, 18] The most straightforward way to avoid splitting while still treating advection explicitly is to apply an implicit scheme, say a Rosenbrock scheme [5, 8], with a Jacobian containing incomes of only reactions and vertical mixing terms. Another alternative is to use the so called source splitting [12, 11] where the advection step is performed first and added as the source during the implicit vertical mixing reaction substep. In both cases ....

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-- Algebraic Problems. Springer Series in Computational Mathematics 14. Springer--Verlag, 1991.

....1 and 2 space dimensions, but our assumptions lead to some problems in 3 space dimensions. Finally, Section 5 shows that the results of Sections 2 to 4 extend to variable step sizes under mild restrictions on the time step sequence. We conclude this section by recalling some terminology (cf. Bu] [HaW]) A Runge Kutta method applied to an initial value problem u 0 = F (t; u) u(0) u 0 with a step size h 0 yields at t n = nh an approximation un , given recursively by un 1 = un h m X j=1 b j U 0 nj ; U ni = un h m X j=1 a ij U 0 nj ; U 0 ni = F (t n c i h; U ni ) i = ....

....solution of the nonlinear Runge Kutta equations can be bounded similarly. d) In finite dimension, the existence of a numerical solution can be shown under the method assumption of [CrR] Thm.II.5. 4: There exists a positive diagonal matrix D such that DO O T D is positive definite (cf. also [HaW], Ch. IV.14) Using condition (1.2) one shows that the iteration U 0(1) ni A(U (0) ni )U (1) ni = f(t n c i h) maps some ball into itself and applies Brouwer s fixed point theorem. Uniqueness is obtained from condition (1.4) only if there exists a numerical solution with internal stages ....

E. Hairer, G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential -Algebraic Problems. Springer-Verlag, 1991.

....div A outside Omega C . Obviously, this forfeits uniqueness of the solution in parts of the domain where oe = 0, but the solution for B remains unique everywhere. Inside the conductor, where oe 0, we get a unique A. For the sake of stability, timestepping schemes for (1) have to be L stable [26]. This requirement can only be met by implicit schemes like SDIRK methods. In each timestep they entail the solution of a degenerate elliptic boundary value problem of the form curl curl u fiu = f in Omega u Theta n = 0 on Omega : 2) In this context, u denotes the new approximation to A ....

E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1991.

....a subincrementation method, and it requires an assumption about the time dependence of strain during Dt n 1 . It has been taken to vary linearly from ee n to ee n 1 . An unconditionally stable and very efficient code for moderate accuracy requirements is the Radau5 routine by Hairer and Wanner [8]. We have used it previously for the integration of the SMA equations in the context of the torsional vibration of an SMA wire [15] and it showed excellent performance. It is based on an implicit Runge Kutta scheme (Radau IIa) and is particularly suited for stiff systems. 4 Simulation of an ....

Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential -algebraic Problems, Springer Series in Computational Mathematics, Springer Verlag, 1991

....equation (ODE) model of the form: x = f(x; u; t) 6) that can be solved with regular ODE solvers. The tendency has been to solve the resulting DAE set directly using a numerical DAE solver. Several powerful DAE solvers have been made available meanwhile, including DASSL (Petzold, 1982) and Radau (Hairer and Wanner, 1991). The problems with this approach are twofold. On the one hand, these numerical DAE solvers are not suited for solving higher index DAE systems. A symbolic algorithm is known (Pantelides, 1988) that makes it possible to automatically reduce the index of a DAE system down to index 1. This ....

Hairer, E. and G. Wanner (1991), Solving Ordinary Differential Equations II. Stiff and Differential--Algebraic Problems, Springer--Verlag, Berlin.

....equations. In contrast to fully implicit Runge Kutta methods, these methods require only the solution of linear systems of equations in each time step. They include Rosenbrock methods, W methods, and as an important particular case the extrapolated linearly implicit Euler method, see the books of Hairer Wanner (1991), Strehmel Weiner (1992) Deuflhard Bornemann (1994) and references therein. The book of Strehmel Weiner (1992) contains most of what is presently known in the literature about the error of linearly implicit one step methods for parabolic equations. In particular, it gives convergence ....

....the purpose of this paper to provide value judgements on the advantages of different methods. These would depend on such diverse topics as the availability of inexpensive Jacobians, the cost of numerical linear algebra, accuracy requirements, type and frequency of space grid adaptations, etc. See Hairer Wanner (1991) for some numerical comparisons between implementations of methods treated in this article and also fully implicit Runge Kutta and BDF codes. Throughout the paper, C will denote a generic constant which takes on different values on different occurences. 2. The equations considered in this paper ....

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Hairer, E., & Wanner, G. 1991 Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer, Berlin.

....used in the numerical experiments in both original and modified (sparse) versions. Each code is based on a different numerical scheme. ffl vode, the Variable coefficient ODE solver of Hindmarsch, Brown and Byrne, a BDF code. For details see [3] ffl sdirk4 written by Hairer and Wanner, part of [10]. Is based on a stiffly accurate, five stages, order four, singly diagonally implicit Runge Kutta method; Efficient implementation of fully implicit methods 18 Model A ROUTINE DEC SOL 1D 7S LINPACK U 714 73 1225 LINPACK O 411 73 922 LAPACK U 694 102 1408 LAPACK O 341 102 1055 HARWELL 393 39 ....

....20 Species Initial Species Initial name (ppb) name (ppb) O 8.15 O 3 656 NO 10.7 NO 2 2.75 HNO 3 0.35 H 2 O 6100 OH 0.2 HO 2 0.14 H 2 370 CH 4 490 CO 20 C O 1 HC 2.15 HOC 0.22 Table IV: Initial concentrations for stratospheric model A. ffl rodas written by Hairer and Wanner, part of [10]. Based on stiffly accurate Rosenbrock method of order four with six stages. 4.6 Test problems Test problem A corresponds to a stratospheric (altitude 40 Km) box model. It is available at NASA ftp site, contact Douglas E. Kinnison, kinnison1 llnl.gov. There are 38 species involved in 84 thermal ....

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1991.

....the time step = 30 minutes one normally has kf 0 k 2 O(10 6 ) kTk 2 O(10) 3.1) whereas the smallest in modulus eigenvalues are of order O(10 Gamma5 ) All three methods for solving (1. 2) are derived from a particular Rosenbrock method (which we call ros2) from the stiff ODE field [6, 11]. Let fl = 1 1 2 p 2 and denote A = F 0 (wn ) The Rosenbrock method reads wn 1 = wn 1 2 k 1 1 2 k 2 ; 3.2) I Gamma fl A) k 1 = F (wn ) I Gamma fl A) k 2 = F (wn k 1 ) Gamma 2flAk 1 ; 3. Integration methods 6 where wn w(tn ) and = t n 1 Gamma t n denotes the step ....

E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems. Springer-Verlag, Berlin, 2nd edition, 1996.

....or the memory requirements of the code. These attributes are relevant if the user has some preferences regarding time and accuracy [2] An IVP is said to be stiff if its solution has components with widely different time scales and the solution is dominated by the slowly varying components [3]. A measure of this property which is often recommended is the stiffness ratio, i.e. the ratio of the real parts of the maximum and minimum eigenvalues of the Jacobian of f in (1) The difficulty here is that this ratio does not inform us whether the components of interest are the slowly varying ....

....etc. The most important property of a difference method is the convergence to the exact solution if the step size goes to zero. This supposes a method order greater than one and method stability. Many formulae have been developed to mathematically describe the order of a formula (see for example [3]) but these formulae are method class dependent. A general tool for evaluating difference method must identify each case and apply the known order formulae or can interpret the definition of the method order, i.e. to find the value of p from the following equality: l n (hn ) y(t n 1 ) Gamma ....

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E. Hairer, and G. Wanner, Solving ordinary differential equations II. Stiff and differentialalgebraic problems, Springer, Berlin, 1991.

....a change of notation. For the Euler Lagrange equations, where f(x; y) f(x) Gamma g T x (x)y, the usual multistep discretization is h Gamma1 aex n = oe Gamma f(xn ) Gamma g T x (x n )y n Delta (1.2a) 0 = g(xn ) 1. 2b) This is stable only if oe(i) satisfies the root condition [1, 9]. Unfortunately, this prohibits the use of nonstiff methods such as the Adams Moulton (AM) or difference corrected BDF (dcBDF, 13, 3] methods which have order p = k 1. The source of the instability, however, is only in the y space but contaminates x as well. Exploiting this structure, ....

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems, Springer series in Computational Mathematics, 14, Springer--Verlag, New York, 1991.

....the thermodynamic properties, and the routines for the transport properties were adapted from the Sandia 1D flame code [9] Equations (1) 5) are a system of partial differential algebraic equations that have index 2. Equations of index greater that one are difficult to solve numerically [1] [8]. Therefore we derive an explicit equation for the velocity v. and thus reduce the index of system of the PDAEs to 1. For convenience, define: s 1 = Ns X i=1 Y i e i T (T ) s 2 = Ns X i=1 Y i ; s 3 = Ns X i=1 [ i ae Gamma 1 ae (ae Y i V i ) x ] s 4 = Ns X i=1 ....

....and thus is cheap to invert. Since fourth order discretization is used in the spatial variable, we would also like to use higher order discretization for the time integration. There are two distinct approaches that produce higher order methods based on a lower order method: extrapolation [8] and iterated deferred correction [2] Here we use the second approach. Because the algebraic equations in our PDAEs are solvable, for t 2 [a; b] we can write our problem in the form y 0 (t) F (y(t) 10) y(a) y 0 : This ODE can also be written as an integral equation, y(t) y(a) Z ....

E. Hairer & G. Wanner, Solving ordinary differential equations II, stiff and differential-algebraic problems, Springer-Verlag, New York, 1996

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E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. (1991), Berlin, Heidelberg, New York, 1991

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E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, 1991.

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E. HAIRER AND G. WANNER, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner. Solving ordinary differential equations II. Stiff and differential--algebraic problems., volume 14 of Springer Series in Comput. Mathematics. Springer--Verlag, second revised edition, 1996.

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Hairer, E. and Wanner, G. (1991). Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, volume 14 of SCM. Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer-Verlag, Berlin, 1996.

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E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Springer Series in Computational Mathematics vol. 14, Springer-Verlag 1991.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems. Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. SpringerVerlag, Berlin, 1996.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer-Verlag, Berlin, 1996.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer-Verlag, Berlin, 1996.

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E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, SpringerVerlag (Berlin), 1991.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, Stiff and Differential Algebraic problems. Second edition, Springer, 1996.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II, volume 14 of SCM. Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Stiff and DifferentialAlgebraic Problems. Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner, Solving ordinary differential equations II. Stiff and differentialalgebraic problems, Springer-Verlag, Berlin, 1991.

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E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential Algebraic Problems. Berlin, Germany: Springer-Verlag, 1996. 2 rev. ed.

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E. Hairer and G. Wanner (1991), Solving Ordinary Differential Equations II, Springer-Verlag, Berlin.

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Hairer, E. and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential--Algebraic Problems. Springer Verlag, Berlin, Heidelberg, New York, 1991.

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