Implicit integration methods

Section 7.6 gave a number of explicit integration formulas, and in this section we present implicit methods, which are more stable than the explicit formula because they utilize the information about the unknown point, in the integration formula. 

To derive implicit integration methods, we start with the Newton backward difference interpolation formula starting from the point backward (Eq. 7.85) rather than from the point backward as used in the generation of the explicit methods. With a single equation of the type shown by Eq. 7.87, we have 

Equation 7.107 is the implicit Euler method, which is in a similar form to the explicit Euler method, except that the evaluation of the function / is done at the unknown point y +,. Hence, Eq. 7.107 is a nonlinear algebraic equation and must be solved by trial to find y +i. Example 7.1 demonstrated this iteration process in solving nonlinear algebraic equations. 

We proceed further with the generation of more implicit schemes. If the second term in the RHS of Eq. 7.105 is retained, we have 

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In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

Implicit Methods By using different interpolation formulas involving yn 1, it is possible to derive implicit integration methods. Implicit methods result in a nonlinear equation to be solved for yn 1 so that iterative methods must be used. The backward Euler method is a first-order method. 

The explicit methods considered in the previous section involved derivative evaluations, followed by explicit calculation of new values for variables at the next point in time. As the name implies, implicit integration methods use algorithms that result in implicit equations that must be solved for the new values at the next time step. A single-ODE example illustrates the idea. 

Although this equation is a solvable quadratic (owing to the second order reaction), we will solve it nonetheless by numerical means to demonstrate the general procedure of the implicit integration method. The iteration scheme for the Newton-Raphson procedure is (Appendix A)... 

Summary. Chemical reaction systems are described by many reactions which can widely differ in the reaction velocity and the number of reactants. If all reactions are slow explicite methods can be used for the numerical simulation. Fast reactions, however, require the use of implicit methods. Very often there are only few fast reactions in a very large reaction system. Because of these few fast reactions the simulation has to be done with implicit integration methods, where the dominating part of the computation very often is spend to solve large scale linear equation system. 

In this paper we introduce a method to reduce the dimension of these systems by identifying the feist reactions and apply a partitioned explicit implicit integration method so that the linear equation system is solved only in a low dimensional space of fast reactions. 

In spatially homogeneous simulations, the concentration—time curves (with resolution Af) can be obtained via a recursive evaluation of function G. If operator splitting is used in a reactive flow model (i.e. the solution of the flow and chemistry steps are separated), then this fitted function can be applied instead of typically using implicit integration methods to solve the chemical rate equations. Potentially large savings in computational effort can be achieved. 

Eq. 4 at time t. Eor this reason the integration procedure is called an explicit integration method. On the other hand, the Houbolt, Wilson, and Newmark methods, considered in the next sections, use the equilibrium conditions at time t -I- At and are called implicit integration methods. 

Because fix 0) generally is nonlinear, (4.115) often cannot be rearranged to provide a direct expression for x. Then, (4.115) is said to generate an implicit integration method that requires a nonlinear algebraic system to be solved at each time step. 

The LIN method ( Langevin/Implicit/Normal-Modes ) combines frequent solutions of the linearized equations of motions with anharmonic corrections implemented by implicit integration at a large timestep. Namely, we express the collective position vector of the system as X t) = Xh t) + Z t). (In LN, Z t) is zero). The first part of LIN solves the linearized Langevin equation for the harmonic reference component of the motion, Xh t)- The second part computes the residual component, Z(t), with a large timestep. 

The many approaches to the challenging timestep problem in biomolecular dynamics have achieved success with similar final schemes. However, the individual routes taken to produce these methods — via implicit integration, harmonic approximation, other separating frameworks, and/or force splitting into frequency classes — have been quite different. Each path has encountered different problems along the way which only increased our understanding of the numerical, computational, and accuracy issues involved. This contribution reported on our experiences in this quest. LN has its roots in LIN, which... 

The implicit Crank-Nicholson integration method was used to solve the equation. Radial temperature and concentrations were calculated using the Thomas algorithm (Lapidus 1962, Carnahan et al,1969). This program allowed the use of either ideal or non-ideal gas laws. For cases using real gas assumptions, heat capacity and heat of reactions were made temperature dependent. 

The implicit Euler integration method is examined here. We will use the set of two sequential reactions, rate constants, and initial conditions described in the previous problem. Note This problem uses results from tasks 1-3 in the previous problem.)... 

Using a stepsize of h = 10-2 s integrate the equation set with the implicit Euler method over the range t = 0 to 5 s. Plot the log of species concentration versus time for all three species. 

The second approach is a fractional-step method we call asymptotic timestep-splitting. It is developed by consideration of the specific physics of the problem being solved. Stiffness in the governing equations can be handled "asymptotically" as well as implicitly. The individual terms, including those which lead to the stiff behavior, are solved as independently and accurately as possible. Examples of such methods include the Selected Asymptotic Integration Method (4,5) for kinetics problems and the asymptotic slow flow algorithm for hydrodynamic problems where the sound speed is so fast that the pressure is essentially constant (6, 2). ... 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

The system of equations is discretized in space by a finite voJume approach, while for the time integration an implicit Euler method is used. Particle and flow model are solved consecutively, which implies that conditions in the bed change slowly compared to the integration step. To reduce the required computation time the flow model is solved here only for one dimension even if the software library TOSCA provides also classes for a two dimensional approach. 

Gonzalez and Schlegel [209] have also developed a series of third- and fourth-order methods. All of their higher order approaches use implicit integrators and are extensions of the GS2 algorithm. One of these, a fourth-order method, uses the tangent and curvature vectors at the initial and final points of each step. 

For predictor-corrector and implicit integrators, the accuracy of the path can often be enhanced by tightening convergence criteria. Of the methods discussed in this chapter, this approach is apphcable for MB, GS2, and HS. MB and GS2 both employ constrained optimizations to determine the tangent at the endpoint of each step. In the practical... 

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