Nonlinear Stiff Problems Research Articles

Development of a Stable Implicit Method Without Numerical Dissipation for Nonlinear Structural Dynamic Analysis

This paper develops a self-starting implicit time integration method for nonlinear dynamic analysis. The method is single-step and is based on a non-dissipative second-order accurate approximation scheme. The method employs simple expressions for velocity and acceleration while internal forces are calculated at a specified point. This point results from a weighted combination of the relevant displacements and velocities at the bounds of the time step. In the linear case, the scheme is unconditionally stable, there is no amplitude decay for any time step value and the period elongation is the same as in the Newmark scheme. In nonlinear case, the scheme maintains the same numerical characteristics of dissipation and dispersion of the Newmark scheme while demonstrating a substantial improvement in stability. Summarizing, the presented scheme (i) is a single-step scheme with self-starting attribute; (ii) does not involve additional variables or artificial parameters chosen by the user; (iii) has at least second-order accuracy; (iv) shows to have very wide stability ranges in nonlinear analyses. It also proves that the computational effort per step required by the presented scheme is the same as the effort required in the widely used implicit single-step methods and a substantial reduction in the number of Newton–Raphson iterations in the steps is obtained. For comparison, methods with similar characteristics such as Newmark, generalized-[Formula: see text] and Bathe schemes are tested. Stiff nonlinear problems regarding elastic behavior and finite element structural models are analyzed to compare the considered solution methods.

Issues with positivity-preserving Patankar-type schemes

Patankar-type schemes are linearly implicit time integration methods designed to be unconditionally positivity-preserving. However, there are only little results on their stability or robustness. We suggest two approaches to analyze the performance and robustness of these methods. In particular, we demonstrate problematic behaviors of these methods that, even on very simple linear problems, can lead to undesired oscillations and order reduction for vanishing initial condition. Finally, we demonstrate in numerical simulations that our theoretical results for linear problems apply analogously to nonlinear stiff problems.

ENHANCED 3-POINT FULLY IMPLICIT SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING STIFF INITIAL VALUE PROBLEMS

This paper modified an existing 3–point block method for solving stiff initial value problems. The modification leads to the derivation of another 3 – point block method which is suitable for solving stiff initial value problems. The method approximates three solutions values per step and its order is 5. Different sets of formula can be generated from it by varying a parameter ρ ϵ (-1, 1) in the formula. It has been shown that the method is both Zero stable and A–Stable. Some linear and nonlinear stiff problems are solved and the result shows that the method outperformed an existing method and competes with others in terms of accuracy

Formulation of a New Implicit Method for Group Implicit BBDF in Solving Related Stiff Ordinary Differential Equations

This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.

Canonical Euler splitting method for nonlinear composite stiff evolution equations

In this paper, a new splitting method, called canonical Euler splitting method (CES), is constructed and studied, which can be used for the efficient numerical solution of general nonlinear composite stiff problems in evolution equations of various type, such as ordinary differential equations (ODEs), semi-discrete unsteady partial differential equations (PDEs) and ordinary or partial Volterra functional differential equations (VFDEs), and can significantly improve the computing speed on the basis of ensuring the computing quality. Stability, consistency and convergence theories of this method are established. A series of numerical experiments are given which check the efficiency of CES method and confirm our theoretical results.

Behaviour of the Extrapolated Implicit IMR and ITR With and Without Compensated Summation

Extrapolation involves taking a certain linear combination of the numerical solutions of a base method applied with different stepsizes to obtain greater accuracy. This linear combination is done so as to eliminate the leading error term. The technique of extrapolation in accelerating convergence has been successfully in numerical solution of ordinary differential equations. In this study, symmetric Runge-Kutta methods for solving linear and nonlinear stiff problem are considered. Symmetric methods admit asymptotic error expansion in even powers of the stepsize and are therefore of special interest because successive extrapolations can increase the order by two at time. Although extrapolation can give greater accuracy, due to the stepsize chosen, the numerical approximations are often destroy due to the accumulated round off errors. Therefore, it is important to control the rounding errors especially when applying extrapolation. One way to minimize round off errors is by applying compensated summation. In this paper, the numerical results are given for the symmetric Runge-Kutta methods Implicit Midpoint and Implicit Trapezoidal Rule applied with and without compensated summation. The result shows that symmetric methods with higher level extrapolation using compensated summation gives much smaller errors. On the other hand, symmetric methods without compensated summation when applied with extrapolation, the errors are affected badly by rounding errors

Nonlinear Stability and B-convergence of Additive Runge-Kutta Methods for Nonlinear Stiff Problems

AbstractIn this paper, we are devoted to nonlinear stability and B-convergence of additive Runge-Kutta (ARK) methods for nonlinear stiff problems with multiple stiffness. The concept of -algebraic stability of ARK methods for a class of stiff problems Kστ is introduced, and it is proven that this stability implies some contractive properties of the ARK methods. Some results on optimal B-convergence of ARK methods for Kσ,0 are given. These new results extend the existing ones of RK methods and ARK methods in the references. Numerical examples test the correctness of our theoretical analysis.

Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems

In this paper, meta-heuristic intelligent approaches are developed for handling nonlinear oscillatory problems with stiff and non-stiff conditions. The mathematical modeling of these oscillators is accomplished using feed-forward artificial neural networks (ANNs) in the form of an unsupervised manner. The accuracy as well as efficiency of the model is subject to the tuning of adaptive parameters for ANNs that are highly stochastic in nature. These optimal weights are carried out with swarm intelligence and pattern search methods hybridized with an efficient local search technique based on constraints minimization known as active set algorithm. The proposed schemes are validated on various stiff and non-stiff variants of the oscillator. The significance, applicability and reliability of the proposed scheme are well established based on comparison made with the results of standard numerical solver.

Two classes of implicit–explicit multistep methods for nonlinear stiff initial-value problems

The initial value problems of nonlinear ordinary differential equations which contain stiff and nonstiff terms often arise from many applications. In order to reduce the computation cost, implicit–explicit (IMEX) methods are often applied to these problems, i.e. the stiff and non-stiff terms are discretized by using implicit and explicit methods, respectively. In this paper, we mainly consider the nonlinear stiff initial-value problems satisfying the one-sided Lipschitz condition and a class of singularly perturbed initial-value problems, and present two classes of the IMEX multistep methods by combining implicit one-leg methods with explicit linear multistep methods and explicit one-leg methods, respectively. The order conditions and the convergence results of these methods are obtained. Some efficient methods are constructed. Some numerical examples are given to verify the validity of the obtained theoretical results.

Solution of diffusion–dispersion models using a computationally efficient technique of orthogonal collocation on finite elements with cubic Hermite as basis

In this paper, linear and non linear diffusion–dispersion models involving fluid flow through porous cylindrical particles are solved using orthogonal collocation on finite elements with cubic Hermite as basis. The technique involves partitioning of axial domain into equal elements and then orthogonal collocation method with cubic Hermite as basis function is applied within each element. Exit concentration profiles are drawn for Peclet numbers ranging from 0 (perfect mixing) to ∞ (perfect displacement). Proposed technique is computationally efficient, stable and yields accurate solution, even for nonlinear stiff problem. Correlation with industrial parameters is also presented.

A study on linear and nonlinear stiff problems using single-term Haar wavelet series technique

A study on linear and nonlinear stiff problems using single-term Haar wavelet series technique

Explicit Numerical Methods for Solving Stiff Dynamical Systems

The precise integration method can give precise numerical result for linear invariant dynamical system, and can be used to solve stiff linear invariant dynamical system. In this paper, precise integration is compounded with Runge–Kutta method and a new effective integration method is presented for solving nonlinear stiff problems. The numerical examples are given to demonstrate the validity and effectiveness of the proposed method.

A filtered flame approach for simulation of unsteady laminar premixed flames

The simulation of laminar flames consists of capturing the evolution of a very large number of species that may react within a broad range of time scales. It results in a highly non-linear stiff numerical problem that requires large computational resources. In the present paper, an alternative approach for the simulation of unsteady premixed flames is proposed. The approach focuses on describing the flame using a single scalar only (so-called progress variable) on relatively coarse grids. The unresolved details of the flame structure are analysed and described in term of spatial filtering. A model originally developed for large eddy simulation is discussed in the framework of unsteady laminar flames. The ability of the present approach to represent the filtered flame structure is validated against detailed numerical simulations of freely propagating laminar flames. Further, simulations of steady and unsteady laminar Bunsen flames are presented with a focus on the influence of the numerical grid and filter width upon the flame dynamics.

A review of theoretical and numerical analysis for nonlinear stiff Volterra functional differential equations

In this review, we present the recent work of the author in comparison with various related results obtained by other authors in literature. We first recall the stability, contractivity and asymptotic stability results of the true solution to nonlinear stiff Volterra functional differential equations (VFDEs), then a series of stability, contractivity, asymptotic stability and B-convergence results of Runge-Kutta methods for VFDEs is presented in detail. This work provides a unified theoretical foundation for the theoretical and numerical analysis of nonlinear stiff problems in delay differential equations (DDEs), integro-differential equations (IDEs), delayintegro-differential equations (DIDEs) and VFDEs of other type which appear in practice.

Analysis of linear and nonlinear stiff problems using the RK‐Butcher algorithm

I present a numerical solution of linear and nonlinear stiff problems using the RK‐Butcher algorithm. The obtained discrete solutions using the RK‐Butcher algorithm are found to be very accurate and are compared with the exact solutions of the linear and nonlinear stiff problems and also with the Runge‐Kutta method based on arithmetic mean (RKAM). A topic of stability for the RK‐Butcher algorithm is discussed in detail. Error graphs for discrete and exact solutions are presented in a graphical form to show the efficiency of the RK‐Butcher algorithm. The results obtained show that RK‐Butcher algorithm is more useful for solving linear and nonlinear stiff problems and the solution can be obtained for any length of time.

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

Linear multistep methods applied to stiff initial value problems—A survey

The numerical approximation of solutions of differential equations has been and continues to be one of the principal concerns of numerical analysis. Linear multistep methods and, in particular, backward differentiation formulae (BDFs) are frequently used for the numerical integration of stiff initial value problems. Such stiff problems appear in a variety of applications. While the intuitive meaning of stiffness is clear to all specialists, there has been much controversy about its correct mathematical definition. We present a historical development of the concept of stiffness. A survey of convergence results for special classes of stiff problems based on these different concepts of stiffness is given, e.g., for linear, stiff systems, problems in singular perturbation form, nonautonomous stiff systems, and rather general nonlinear stiff problems. Different approaches proving convergence of linear multistep methods applied to stiff initial value problems are introduced. It is further indicated that the corresponding proofs for singular perturbation problems are compatible with a nonlinear transformation and thus convergence of a quite general class of nonlinear problems seems to be covered.

Numerical methods for evolutionary reaction–diffusion problems with nonlinear reaction terms

We have developed new methods to integrate efficiently reaction–diffusion parabolic problems with nonlinear reaction terms. In order to obtain uniform and unconditional convergence, such methods combine the advantages of alternating direction methods, the additive Runge–Kutta methods designed by Cooper and Sayfy for nonlinear Stiff problems as well as the use of Shishkin meshes in the singularly perturbed case. The resulting algorithms are only linearly implicit and they have the same order of computational complexity, per time step, that any explicit method. We show some numerical experiences which illustrate the good properties of our schemes predicted by the theoretical results.

Haar wavelet approach to nonlinear stiff systems

A simple and effective algorithm based on Haar wavelet is proposed to the solution of nonlinear stiff problems in this paper. The simulation result shows that the whole computation time can be reduced to one tenth of the well-known Runge–Kutta–Fehlberg approach, while the accuracy is nearly the same.

On the Starting Algorithms for Fully Implicit Runge-Kutta Methods

This paper is concerned with the behavior of starting algorithms to solve the algebraic equations of stages arising when fully implicit Runge-Kutta methods are applied to stiff initial value problems. The classical Lagrange extrapolation of the internal stages of the preceding step and some variants thereof that do not require any additional cost are analyzed. To study the order of the starting algorithms we consider three different approaches. First we analyze the classical order through the theory of Butcher's series, second we derive the order on the Prothero and Robinson model and finally we study the stiff order for a general class of dissipative problems. A detailed study of the orders of some starting algorithms for Gauss, Radau IA-IIA, Lobatto IIIA-C methods is also carried out. Finally, to compare the most relevant starting algorithms studied here, some numerical experiments on well known nonlinear stiff problems are presented.