Stiff Initial Value Problems Research Articles

Step-by-step time discrete Physics-Informed Neural Networks with application to a sustainability PDE model

The use of Artificial Neural Networks (ANNs) has spread massively in several research fields. Among the various applications, ANNs have been exploited for the solution of Partial Differential Equations (PDEs). In this context, the so-called Physics-Informed Neural Networks (PINNs) are considered, i.e. neural networks generally constructed in such a way as to compute a continuous approximation in time and space of the exact solution of a PDE.In this manuscript, we propose a new step-by-step approach that allows to define PINNs capable of providing numerical solutions of PDEs that are discrete in time and continuous in space. This is done by establishing connections between the network outputs and the numerical approximations computed by a classical one-stage method for stiff Initial Value Problems (IVPs). Links are also highlighted between the step-by-step PINNs derived here, and the time discrete models based on Runge–Kutta (RK) methods proposed so far in literature. To evaluate the efficiency of the new approach, we build such PINNs to solve a nonlinear diffusion-reaction PDE model describing the process of production of renewable energy through dye-sensitized solar cells. The numerical experiments show that not only the new step-by-step PINNs are able to well reproduce the model solution, but also highlight that the proposed approach can constitute an improvement over existing continuous and time discrete models.

A Non-Standard Finite Difference Schemes for the Solution of Stiff Initial Value Problems

In this study, we introduce a novel non-standard finite difference (NSFD) scheme designed to address the challenges posed by stiff initial value problems. Stiffness in differential equations often leads to numerical instability and requires specialized methods for stable and accurate solutions. A novel set of numerical schemes for solving stiff ordinary differential equations caused by the decay of radioactive substances developed. This paper demonstrates the power of normalization in the discretization function. We employed non-local approximation and renormalization of the denominator function to create qualitatively stable schemes for a stiff ordinary differential equation. The schemes' stability properties were verified using numerical experiments. The schemes' performance is evaluated in comparison to other typical finite difference schemes

Hybrid block formulae whose eigenvalues of Jacobian matrices are close to the imaginary axis of the complex plane

Methods for integrating stiff initial value problems are required to be A-stable. Of great interest are A-stable methods whose Jacobian matrices have their eigenvalues close to the imaginary axis of the complex plane. This class of A-stable methods are very rare. This paper is on the development of a new family of A-stable hybrid block method whose Jacobian matrices possess eigenvalues on the imaginary of the complex plane via interpolation and collocation techniques. The family of methods developed herein are A-stable for order p ≤ 18. Numerical solutions generated by the new method are compared with existing methods in the literature. The numerical results show that the new class of methods are more efficient and accurate.

A New Fixed Coefficient Diagonally Implicit Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Stiff initial value problems in ordinary differential equations occur when solution components evolve at varying rates, posing challenges for traditional computational methods. Specialized techniques are crucial for maintaining accuracy and stability during rapid transitions, emphasizing their significance in developing reliable numerical algorithms across scientific and engineering applications. This study aims to develop a new fixed coefficient 3-point diagonally implicit block backward differentiation formula for the numerical solution of first order stiff initial value problems. The method is constructed by integrating a triangular matrix into the coefficient matrix of an existing extended 3-point super class of block BDF for solving stiff initial value problems. The selection of a fixed coefficient within the interval accompanies this integration to ensure optimal stability. The method is found to order five. Stability analysis indicates that the method is consistent, zero-stable, and almost A-stable, validating its applicability to stiff initial value problems. Implementation of the method involves Newton’s iteration, and a code in the C programming language is devised to demonstrate its effectiveness. Comparative examination of numerical outcomes with the existing 3BBDF and 3ESBBDF methods highlights the proposed method's enhanced accuracy and reduced computation time.

One‐Step Family of Three Optimized Second‐Derivative Hybrid Block Methods for Solving First‐Order Stiff Problems

This paper introduces a novel approach for solving first‐order stiff initial value problems through the development of a one‐step family of three optimized second‐derivative hybrid block methods. The optimization process was integrated into the derivation of the methods to achieve maximal accuracy. Through a rigorous analysis, it was determined that the methods exhibit properties of consistency, zero‐stability, convergence, and A‐stability. The proposed methods were implemented using the waveform relaxation technique, and the computed results demonstrated the superiority of these schemes over certain existing methods investigated in the study.

A Study on Convergence and Region of Absolute Stability of Implicit Two – Step Multiderivative Method for Solving Stiff Initial Value Problems of First Order Ordinary Differential Equations.

This study investigated and related the convergence and region of absolute stability of an implicit two - step multiderivative method, used in solving sampled stiff initial value problem of first order ordinary differential equation. It adopted the general implicit multiderivative linear multistep method at step - number (k) = 2 with derivative order (l) varied from 1 - 6 to develop six different variants of the method. Boundary locus method was adopted to determine the intervals of absolute stability which were plotted on the complex plane to show the regions of absolute stability. The variant methods were used to solve sampled stiff initial value problem of first order ordinary differential equation. The resulting numerical solutions were compared with the exact solution to determine accuracy and convergence of the methods. The study showed that two - step first derivative, two - step second derivative, two - step third derivative, two - step fifth derivative method and two - step sixth derivative methods yielded more accurate and convergent results with wider regions of absolute stability than two - step fourth derivative.

Jacobian-free implicit MDRK methods for stiff systems of ODEs

In this work, an approximate family of implicit multiderivative Runge-Kutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al., 2020 [7]). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on top of the stage equation itself. From linearizing through Newton's method, it turns out that the conditioning of the Newton matrix behaves significantly different for both cases. We show that direct computation results in a matrix with a conditioning that is highly dependent on the stiffness, increasing exponentially in the stiffness parameter with the amount of derivatives. Adding separate formulas has a more favorable behavior, the matrix conditioning being linearly dependent on the stiffness, regardless of the amount of derivatives. Despite increasing the Newton system significantly in size, through several numerical results it is demonstrated that doing so can be considerably beneficial.

MAPLE SIMULATION CODES FOR STABILITY ANALYSIS OF VARIABLE STEP SUPER CLASS OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR INTEGRATING A SYSTEM OF FIRST ORDER STIFF IVPS

Strength of numerical scheme is rated by the properties it possessed and in turn the kind of problems it can handle. Zero stable method can effectively handle ODEs problem. While, an A – stable method can solve stiff ODEs problem. Analyzing stability of block methods are been carried out using various software. This work aimed at using simplified Maple simulation code to critically analyze avariable step size multi-block backward differentiation formula for the solution stiff initial value problems of ordinary differential equations. The Graphical comparisons of the simulated result obtained is made using Matlab to depict the performing schemes.

On the Stability of a High Order Stiffly Stable Parameter Dependent Nested Hybrid Linear Multistep Methods for Stiff ODEs

This paper is on the stability of a high order stiffly stable parameter dependent nested hybrid multistep method for the numerical integration of stiff initial value problems (IVPs) in ordinary differential equations (ODEs). The method incorporates one or more off-step points for better stability properties. The stability properties of the methods were investigated and the intervals of absolute stability of the methods with step number $k \leq 6$ are presented using the boundary locus techniques. The method is $A$-stable and $A(\alpha)$-stable which makes the methods more suitable for stiff initial value problems.

A new adaptive nonlinear numerical method for singular and stiff differential problems

In this work, a new adaptive numerical method is proposed for solving nonlinear, singular, and stiff initial value problems often encountered in real life. Starting with a fixed step size, the new method’s performance can be significantly enhanced by introducing an adaptive step-size approach. The qualitative properties of the proposed method have been investigated to determine the efficiency and reliability of the method. The proposed method is of fifth-order accuracy, zero stable, L-stable, and consistent. In addition, the proposed method is convergent, and its stability properties are also shown through its Order Stars. Finally, numerical experiments are conducted to illustrate the performance of the method. The results obtained show that the proposed method compares favourably with existing methods.

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

Estimating the Domain of Absolute Stability of a Numerical Scheme Based on the Method of Solution Continuation with Respect to a Parameter for Solving Stiff Initial Value Problems

The modeling of physical and technological processes often involves solving stiff initial value problems. In most cases, their exact solution is difficult to find, while numerical schemes sometimes fail to produce a sufficiently accurate solution in acceptable computation time. Moreover, for some classes of problems, numerical solution schemes are unsuitable because of their insufficient stability. This paper deals with numerical methods based on solution continuation with respect to arguments of various types that make it possible to enhance the stability of explicit numerical schemes. Most frequently, the used best argument is hardly applicable to problems in which the integral curves grow at a superpower or nearly exponential rate. Previously, the authors proposed a modification of the best argument that alleviates these disadvantages. In the present paper, we estimate the domain of absolute stability of the explicit Euler scheme as applied to problems transformed to a modified best argument of special form and refine the proof of a similar estimate for initial value problems transformed to the best argument. A test initial value problem is used to verify the resulting theoretical estimates and to analyze the application of the modified best argument of solution continuation.

Techniques for Mixed Multiple Shooting for solve Stiff Initial Value Problems

The object of the research is to develop a number of techniques on the subject of multiple shooting for solving stiff initial value problems, and these techniques increase the accuracy of numerical solutions of stiff problems, as well as reduce the accumulation of errors, In addition to reducing the algebraic equations to get to the solution. We used mixed-multiple shooting in the development of these techniques.

Development of technique of Backward integration step-by-step for solve stiff initial value problems

Our purpose in this paper is the development of the technique of backward integration step-by-step, In order to facilitating the use of this technique for solving the Stiff Problems. When using backward integration step-by-step for solving one differential equation of first order we need to solve () algebraic equations while using the new developed technique we need solve one algebraic equation. In general if the order of the differential equation is (), the solution needs solving ()of algebraic equation by using backward integration step-by-step. While the developed technique needs only solving () of algebraic equations.

A Variable Step Size Multi-Block Backward Differentiation Formula for Solving Stiff Initial Value Problem of Ordinary Differential Equations

A variable step size multi-block backward differentiation formula for solving stiff initial value problems of ordinary differential equations with a variable step size strategy was derived. The proposed method (VSSMBBDF) computes two approximate solution values at a time per integration step. The stability properties are achieved by varying the step size ratio in the formula to generate more zero stable schemes. The proposed method is also found to be an A-Stable scheme across different choices of the step size. The method is capable of solving stiff IVPs of ODEs. Approximates result from the system of stiff ODE problems considered are found to favorably validate the performance of the new method in terms of accuracy of the scale error and less executional time in respect to the two methods compared in the study. Hence, the proposed method can be an alternative solver for stiff IVPs of ODEs.

Two–Point Diagonally Implicit Extended Super Class of Block Backward Differentiation Formula for Stiff Initial Value Problems

In this paper, a diagonal form of the extended super class of block backward differentiation formula is derived. The method is fully implicit and approximates two solution values at a time. By varying a parameter 𝜌𝜖 (–1,1) in the formula, different sets of formulae can be generated. Analysis of the method indicated that the method is both zero and A–stable, hence, suitable for solving stiff initial value problems. Comparison of the method with some existing algorithms showed its advantage in terms of accuracy over some methods.

Convergence and Order of the 2-Point Diagonally Implicit Block Backward Differentiation Formula with Two Off-Step Points

The development and formulation of a most reliable and efficient numerical schemes for the integration of stiff systems of ordinary differential equations in terms of order, convergence, stability requirements, accuracy, and computational expense has been a major challenged in the study of modern numerical analysis. In this paper, the order and convergence properties of the 2-point diagonally implicit block backward differentiation formula with two off-step points for solving first order stiff initial value problems have been studied, the method was derived and found to be of order five. The necessary and sufficient conditions for the convergence of the method have also been established. It has shown that the 2-point diagonally implicit block backward differentiation formula with two off-step points is both consistent and zero stable, having satisfied these two conditions of consistency and that of zero stability, it is therefore concluded that the method converges and suitable for the numerical integration of stiff systems.

A modified Susceptible-Infected-Recovered epidemiological model

"Objectives This paper proposes an infectious disease model incorporating two new model compartments, hospitalization, and intensive care unit. Methods The model dynamics are analyzed using the local and global stability theory of nonlinear systems of ordinary differential equations. For the numerical simulations, we used the Rosenbrock method for stiff initial value problems. We obtained numerical simulations using MAPLE software. The returned MAPLE procedure was called only for points inside the range on which the method evaluated the numerical solution of the system with specified initial conditions. Results We proposed a new model to describe the dynamics of microparasitic infections. Numerical simulations revealed that the proposed model fitted with the expected behaviour of mi- croparasitic infections with ”acute epidemicity.” The numerical simulations showed consistency in the behaviour of the system. Conclusions The model proposed has ”robust” dynamics, supported by the global stability of its endemic state and the consistency of the numerical simulations regarding the model’s timeevolution behaviour. The introduction of the hospitalization and intensive care unit compartments in the proposed model revealed that it is essential to consider such policies in the case of ”acuteepidemicity” of microparasitic infections."

Convergence of the Fourth Order Variable Step Size Super Class of Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

In many fields of study such as science and engineering, various real life problems are created as mathematical models before they are solved. These models often lead to special class of ordinary differential equations known as stiff ODEs. A system is regarded as ‘stiff’; if the existing explicit numerical methods fail to efficiently integrate it, or when the step size is determined by the requirements of its stability, rather than the accurateness. The solution of stiff ODEs contains a component with both slowly and rapidly decaying rates due to a large difference in the time scale exhibited by the system. The stiffness property prevents the conventional explicit method from handling the problem efficiently. This nature of stiff ODEs has led to considerable research efforts in developing many implicit mathematical methods. This paper discussed the convergence and order of the current variable step size super-class of block backward differentiation formula (BBDF) for solving stiff initial value problems. The necessary conditions for the convergence of the fourth order variable step size super class of BBDF for solving stiff initial value problems, has been established in this work. It has been shown that the new method is both zero-stable and consistent, which are the requirements for the convergence of any numerical method. The order of the method is also derived to be four. It is therefore concluded that the method is convergent and has significance in solving more complex stiff initial value problems, and could be robustly applied in many fields of study.

A family of r-points 1-block implicit methods with optimized region of stability for stiff initial value problems in ordinary differential equations

In this paper, we proposed a family of r-points 1-block implicit methods with optimized region of stability. This family of methods is derived with Mathematical 10.4 software and the stability is investigated using boundary locus techniques. The block methods are consistence, zero stable, and A-stable and satisfy other stability requirements which finds them suitable for stiff problems in ODEs. Numerical experiments are presented and results are compared with other block methods and exact solutions of some stiff ordinary differential equations. The methods have been found to show competitiveness with other numerical methods.