A Computational Strategy of Variable Step, Variable Order for Solving Stiff Systems of Ordinary Differential Equations

This research study focuses on a computational strategy of variable step, variable order (CSVSVO) for solving stiff systems of ordinary differential equations. The idea of Newton’s interpolation formula combine with divided difference as the basis function approximation will be very useful to design the method. Analysis of the performance strategy of variable step, variable order of the method will be justified. Some examples of stiff systems of ordinary differential equations will be solved to demonstrate the efficiency and accuracy.

Real-Time Simulation of Fluid Power Systems

System-simulations involving fluid-power structures often result in numerically stiff model equations which may require prohibitively small simulation time steps when being tackled with a fixed-step solver. This poses a challenge in situations where real-time performance is required. This paper presents a practical rule-of-thumb to estimate the maximum permissible step-size for a given fluid power system and explains the influence of the relevant physical quantities on the step size requirement in simple terms. A categorization of methods suitable to relax the step-size requirement is proposed. Many research papers have been produced about methods and examples of how to improve real-time performance of fluid power systems, or stiff systems in general. The proposed categorization can be seen as a map for the simulation engineer to understand the basic point-of-attacks for the real-time simulation problem.

A Set of New Stable, Explicit, Second Order Schemes for the Non-Stationary Heat Conduction Equation

This paper introduces a set of new fully explicit numerical algorithms to solve the spatially discretized heat or diffusion equation. After discretizing the space and the time variables according to conventional finite difference methods, these new methods do not approximate the time derivatives by finite differences, but use a combined two-stage constant-neighbour approximation to decouple the ordinary differential equations and solve them analytically. In the final expression for the new values of the variable, the time step size appears not in polynomial or rational, but in exponential form with negative coefficients, which can guarantee stability. The two-stage scheme contains a free parameter p and we analytically prove that the convergence is second order in the time step size for all values of p and the algorithm is unconditionally stable if p is at least 0.5, not only for the linear heat equation, but for the nonlinear Fisher’s equation as well. We compare the performance of the new methods with analytical and numerical solutions. The results suggest that the new algorithms can be significantly faster than the widely used explicit or implicit methods, particularly in the case of extremely large stiff systems.

FiCoS: A fine-grained and coarse-grained GPU-powered deterministic simulator for biochemical networks

Mathematical models of biochemical networks can largely facilitate the comprehension of the mechanisms at the basis of cellular processes, as well as the formulation of hypotheses that can be tested by means of targeted laboratory experiments. However, two issues might hamper the achievement of fruitful outcomes. On the one hand, detailed mechanistic models can involve hundreds or thousands of molecular species and their intermediate complexes, as well as hundreds or thousands of chemical reactions, a situation generally occurring in rule-based modeling. On the other hand, the computational analysis of a model typically requires the execution of a large number of simulations for its calibration or to test the effect of perturbations. As a consequence, the computational capabilities of modern Central Processing Units can be easily overtaken, possibly making the modeling of biochemical networks a worthless or ineffective effort. To the aim of overcoming the limitations of the current state-of-the-art simulation approaches, we present in this paper FiCoS, a novel “black-box” deterministic simulator that effectively realizes both a fine-grained and a coarse-grained parallelization on Graphics Processing Units. In particular, FiCoS exploits two different integration methods, namely, the Dormand–Prince and the Radau IIA, to efficiently solve both non-stiff and stiff systems of coupled Ordinary Differential Equations. We tested the performance of FiCoS against different deterministic simulators, by considering models of increasing size and by running analyses with increasing computational demands. FiCoS was able to dramatically speedup the computations up to 855×, showing to be a promising solution for the simulation and analysis of large-scale models of complex biological processes.

A New L-Stable Third Derivative Hybrid Method for Solving First Order Ordinary Differential Equations

In this paper, an L-stable third derivative multistep method has been proposed for the solution of stiff systems of ordinary differential equations. The continuous hybrid method is derived using interpolation and collocation techniques of power series as the basis function for the approximate solution. The method consists of the main method and an additional method which are combined to form a block matrix and implemented simultaneously. The stability and convergence properties of the block were investigated and discussed. Numerical examples to show the efficiency and accuracy of the new method were presented.

Uniform Order Eleven of Eight-Step Hybrid Block Backward Differentiation Formulae for the Solution of Stiff Ordinary Differential Equations

In this research, we developed a uniform order eleven of eight step Second derivative hybrid block backward differentiation formula for integration of stiff systems in ordinary differential equations. The single continuous formulation developed is evaluated at some grid point of x=x_(n+j),j=0,1,2,3,4,5 and6 and its first derivative was also evaluated at off-grid point x=x_(n+j),j=15/2 and grid point x=x_(n+j),j=8. The method is suitable for the solution of stiff ordinary differential equations and the accuracy and stability properties of the newly constructed method are investigated and are shown to be A-stable. Our numerical results obtained are compared with the theoretical solutions as well as ODE23 solver.

Robust optimization of stiff delayed systems: application to a fluid catalytic cracking unit

We apply a robust steady state optimization method for stiff delay differential equations to the economic optimization of a fluidized catalytic cracking unit. Stiff systems of differential equations appear in this case due to the different time scales in the gas and fluid phase. Delays result from the catalyst hold-ups in the standpipes connecting riser and regenerator. We show that the proposed robust optimization method can cope with stiffness and delays. Moreover, the proposed method is capable of simultaneously optimizing the process parameters and tuning controller parameters.