The sequential-clustered method for dynamic chemical plant simulation

We describe the design, development and testing of a prototype simulator to study problems associated with robust and efficient solution of dynamic process problems, particularly for systems with models containing moderately to very stiff ordinary differential equations and associated algebraic equations.A new predictor--corrector integration strategy and modular dynamic simulator architecture allow for simultaneous treatment of equations arising from individual modules (equipment units), clusters of modules, or in the limit, all modules associated with a process. This "sequential-clustered" method allows for sequential and simultaneous modular integration as extreme cases.Testing of the simulator using simple but nontrivial plant models indicates that the clustered integration strategy is often the best choice, with good accuracy, reasonable execution time and moderate storage requirements.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28737/1/0000564.pd

SARK: a type-insensitive Runge-Kutta code

A novel solution method based on Mono-implicit Runge-Kutta methods has been fully developed and analysed for the numerical solution of stiff systems of ordinary differential equations (ODE). These Backward Runge-Kutta (BRK) methods have very desirable stability properties which make them efficient for solving a certain class of ODE which are not solved adequately by current methods. These stability properties arise from applying a numerical method to the standard test problem and analysing the resulting stability function. This technique, however, fails to show the full potential of a method. With this in mind a new graphical technique has been derived that examines the methods performance on the standard test case in much greater detail. This technique allows a detailed investigation of the characteristics required for a numerical integration of highly oscillatory problems. Numerical ODE solvers are used extensively in engineering applications, where both stiff and non-stiff systems are encountered, hence a single code capable of integrating the two categories, undetected by the user, would be invaluable. The BRK methods, combined with explicit Runge-Kutta (ERK) methods, are incorporated into such a code. The code automatically determines which integrator can currently solve the problem most efficiently. A switch to the most efficient method is then made. Both methods are closely linked to ensure that overheads expended in the switching are minimal. Switching from ERK to BRK is performed by an existing stiffness detection scheme whereas switching from BRK to ERK requires a new numerical method to be devised. The new methods, called extended BRK (EBRK) methods, are based on the BRK methods but are chosen so as to possess stability properties akin to the ERK methods. To make the code more flexible the switching of order is also incorporated. Numerical results from the type-insensitive code, SARK, indicate that it performs better than the most widely used non-stiff solver and is often more efficient than a specialized stiff solver

Multi-Adaptive Time-Integration

Time integration of ODEs or time-dependent PDEs with required resolution of the fastest time scales of the system, can be very costly if the system exhibits multiple time scales of different magnitudes. If the different time scales are localised to different components, corresponding to localisation in space for a PDE, efficient time integration thus requires that we use different time steps for different components. We present an overview of the multi-adaptive Galerkin methods mcG(q) and mdG(q) recently introduced in a series of papers by the author. In these methods, the time step sequence is selected individually and adaptively for each component, based on an a posteriori error estimate of the global error. The multi-adaptive methods require the solution of large systems of nonlinear algebraic equations which are solved using explicit-type iterative solvers (fixed point iteration). If the system is stiff, these iterations may fail to converge, corresponding to the well-known fact that standard explicit methods are inefficient for stiff systems. To resolve this problem, we present an adaptive strategy for explicit time integration of stiff ODEs, in which the explicit method is adaptively stabilised by a small number of small, stabilising time steps

Extrapolation-based implicit-explicit general linear methods

For many systems of differential equations modeling problems in science and engineering, there are natural splittings of the right hand side into two parts, one non-stiff or mildly stiff, and the other one stiff. For such systems implicit-explicit (IMEX) integration combines an explicit scheme for the non-stiff part with an implicit scheme for the stiff part. In a recent series of papers two of the authors (Sandu and Zhang) have developed IMEX GLMs, a family of implicit-explicit schemes based on general linear methods. It has been shown that, due to their high stage order, IMEX GLMs require no additional coupling order conditions, and are not marred by order reduction. This work develops a new extrapolation-based approach to construct practical IMEX GLM pairs of high order. We look for methods with large absolute stability region, assuming that the implicit part of the method is A- or L-stable. We provide examples of IMEX GLMs with optimal stability properties. Their application to a two dimensional test problem confirms the theoretical findings

A class of implicit-explicit two-step Runge-Kutta methods

This work develops implicit-explicit time integrators based on two-step Runge-Kutta methods. The class of schemes of interest is characterized by linear invariant preservation and high stage orders. Theoretical consistency and stability analyses are performed to reveal the properties of these methods. The new framework offers extreme flexibility in the construction of partitioned integrators, since no coupling conditions are necessary. Moreover, the methods are not plagued by severe order reduction, due to their high stage orders. Two practical schemes of orders four and six are constructed, and are used to solve several test problems. Numerical results confirm the theoretical findings