scholarly journals A Class of Explicit Integrators with o-grid Interpolation for Solving Non-linear Systems of First Order ODEs

2020 ◽  
pp. 106-118
Author(s):  
U. W. Sirisena ◽  
S. I. Luka ◽  
S. Y. Yakubu
Keyword(s):  
Research Work ◽  
Stability And Convergence ◽  
Initial Value ◽  
First Order ◽  
Non Linear ◽  
Improved Stability ◽  
Linear Problems ◽  
The Stability ◽  
Stability And Accuracy ◽  
Grid Interpolation

This research work is aimed at constructing a class of explicit integrators with improved stability and accuracy by incorporating an off-gird interpolation point for the purpose of making them effcient for solving stiff initial value problems. Accordingly, continuous formulations of a class of hybrid explicit integrators are derived using multi-step collocation method through matrix inversion technique, for step numbers k = 2; 3; 4: The discrete schemes were deduced from their respective continuous formulations. The stability and convergence analysis were carried out and shown to be A(α)-stable and convergent respectively. The discrete schemes when implemented as block integrators to solve some non-linear problems, it was observed that the results obtained compete favorably with the MATLAB ode23 solver.

scholarly journals Numerical Study on Phase-Fitted and Amplification-Fitted Diagonally Implicit Two Derivative Runge-Kutta Method for Periodic IVPS

2021 ◽  
Vol 50 (6) ◽  
pp. 1799-1814
Author(s):  
Norazak Senu ◽  
Nur Amirah Ahmad ◽  
Zarina Bibi Ibrahim ◽  
Mohamed Othman
Keyword(s):  
Fourth Order ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Numerical Study ◽  
Kutta Method ◽  
Initial Value ◽  
First Order ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
The Stability

A fourth-order two stage Phase-fitted and Amplification-fitted Diagonally Implicit Two Derivative Runge-Kutta method (PFAFDITDRK) for the numerical integration of first-order Initial Value Problems (IVPs) which exhibits periodic solutions are constructed. The Phase-Fitted and Amplification-Fitted property are discussed thoroughly in this paper. The stability of the method proposed are also given herewith. Runge-Kutta (RK) methods of the similar property are chosen in the literature for the purpose of comparison by carrying out numerical experiments to justify the accuracy and the effectiveness of the derived method.

An Efficient Numerical Simulation for Solving Dynamical Systems With Uncertainty

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
Ali Ahmadian ◽  
Soheil Salahshour ◽  
Chee Seng Chan ◽  
Dumitur Baleanu
Keyword(s):  
Dynamical Systems ◽  
Stability And Convergence ◽  
Uncertain Parameters ◽  
Crucial Issue ◽  
Computational Costs ◽  
First Order ◽  
Wide Range ◽  
Linear And Nonlinear ◽  
The Stability ◽  
Fuzzy Dynamical Systems

In a wide range of real-world physical and dynamical systems, precise defining of the uncertain parameters in their mathematical models is a crucial issue. It is well known that the usage of fuzzy differential equations (FDEs) is a way to exhibit these possibilistic uncertainties. In this research, a fast and accurate type of Runge–Kutta (RK) methods is generalized that are for solving first-order fuzzy dynamical systems. An interesting feature of the structure of this technique is that the data from previous steps are exploited that reduce substantially the computational costs. The major novelty of this research is that we provide the conditions of the stability and convergence of the method in the fuzzy area, which significantly completes the previous findings in the literature. The experimental results demonstrate the robustness of our technique by solving linear and nonlinear uncertain dynamical systems.

scholarly journals Phase-Fitted and Amplification-Fitted Higher Order Two-Derivative Runge-Kutta Method for the Numerical Solution of Orbital and Related Periodical IVPs

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
N. A. Ahmad ◽  
N. Senu ◽  
F. Ismail
Keyword(s):  
Numerical Solution ◽  
Numerical Experiments ◽  
Initial Value ◽  
Oscillatory Solutions ◽  
First Order ◽  
Runge Kutta Method ◽  
Algebraic Order ◽  
Runge Kutta ◽  
The Stability ◽  
Order Four

A phase-fitted and amplification-fitted two-derivative Runge-Kutta (PFAFTDRK) method of high algebraic order for the numerical solution of first-order Initial Value Problems (IVPs) which possesses oscillatory solutions is derived. We present a sixth-order four-stage two-derivative Runge-Kutta (TDRK) method designed using the phase-fitted and amplification-fitted property. The stability of the new method is analyzed. The numerical experiments are carried out to show the efficiency of the derived methods in comparison with other existing Runge-Kutta (RK) methods.

scholarly journals A parameter uniform almost first order convergent numerical method for non-linear system of singularly perturbed differential equations

BIOMATH ◽  
2016 ◽  
Vol 5 (2) ◽  
pp. 1608111
Author(s):  
Ishwariya Raj ◽  
Princy Mercy Johnson ◽  
John J.H Miller ◽  
Valarmathi Sigamani
Keyword(s):  
Linear System ◽  
Differential Equations ◽  
Numerical Method ◽  
Shishkin Mesh ◽  
Singularly Perturbed ◽  
Initial Value ◽  
First Order ◽  
Non Linear ◽  
Perturbation Parameters ◽  
Non Linear System

In this paper an initial value problem for a non-linear system of two singularly perturbed first order differential equations is considered on the interval (0,1].The components of the solution of this system exhibit initial layers at 0. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be almost first order convergent in the maximum norm uniformly in the perturbation parameters.

scholarly journals Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2225
Author(s):  
Maria Jesus Moreta
Keyword(s):  
Computational Cost ◽  
Intermediate Stage ◽  
Second Order ◽  
First Order ◽  
New Class ◽  
Nyström Methods ◽  
Non Linear ◽  
Second Order In Time ◽  
Linear Problems ◽  
Runge Kutta

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.

A sphere-equivalency approach for calculating the elastic property of an isotropic solid containing aligned inclusions: Modeling and experiment data verification

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. C239-C249
Author(s):  
Song Xu ◽  
Xiaoming Tang ◽  
Yuanda Su
Keyword(s):  
Elastic Properties ◽  
Elastic Wave Scattering ◽  
Fluid Saturation ◽  
Isotropic Solid ◽  
Improved Stability ◽  
High Crack ◽  
The Stability ◽  
Stability And Accuracy ◽  
Shallow Crust ◽  
Mean Square Errors

Accurate modeling of elastic properties of cracked rocks in the earth’s shallow crust has long been an important topic in the field of geophysics. A sphere-equivalency approach of elastic wave scattering was used to model the elastic moduli of an isotropic solid containing aligned cracks. The results were compared with those of the existing Eshelby-Cheng and Hudson’s theories for dry and fluid saturation conditions, showing remarkably improved stability and accuracy for high crack concentrations, especially for Hudson’s second-order model. The stability and accuracy of the new approach were determined for varying solid and crack parameters. Finally, the new and existing theories were applied to model the laboratory ultrasonic experimental data measured on artificially cracked samples with varying wave frequencies and crack concentrations. Compared to Hudson’s theory, the new model agrees significantly better with the data. Specifically, the root-mean-square errors of theoretical fitting to data from our model are generally smaller than those from the other two models. We have thus developed an effective tool for modeling elastic properties of cracked rocks.

Semi-Analytical Solution of a Non-Linear Heat Transfer Model for a Tubular Co-Current Moving-Bed Reactor with a First-Order Chemical Reaction in the Solid Phase

2021 ◽  
Author(s):  
Lucas Ivan de Souza Vereza Medeiros ◽  
Sávio Leandro Bertoli ◽  
Marcel Jefferson Gonçalves ◽  
Tuany Gabriela Hoffmann ◽  
Betina Louise Angioletti ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Chemical Reaction ◽  
Solid Phase ◽  
Transfer Model ◽  
Moving Bed ◽  
First Order ◽  
Order Chemical Reaction ◽  
Non Linear ◽  
Linear Problems ◽  
First Order Chemical Reaction

Abstract The development of mathematical models plays a fundamental role in the design, optimization and control of processes. Regarding heat transfer in moving bed reactors, the chemical reaction implies in the inclusion of a non-homogeneous and non-linear term in model equations, making the analytical integration a very difficult task. Up to date, there is not an analytic and/or a semi-analytic solution to a heat transfer model of a moving bed reactor (MBR) with isothermal walls to distributed parameter in the solid phase. Therefore, starting from analytical solutions of the associated homogeneous (linear) problems and through the spectral expansion of the non-homogeneous vector, this work presents strategies for determining semi-analytical solutions of non-homogeneous and non-linear problems. A MBR with a first-order chemical reaction in the solid phase - kaolinite dehydroxylation in the kaolinite flash calcination process - is selected as the case study; however, the strategies can easily be applied to other non-linear models. Results for conversion, and fluid and particle temperatures, are given for different parameter values. The solutions perform stable, fast and accurate. When compared with a hybrid Finite Difference and Finite Analytic (FD\&FA) numerical method, the solution showed a very good agreement.

TOPOLOGY DEPENDENCE IN LATTICE SIMULATIONS OF NON-LINEAR PDES ON A MIMD COMPUTER

1994 ◽  
Vol 05 (06) ◽  
pp. 957-971
Author(s):  
P. VALIN ◽  
B. GOULARD ◽  
M. SANIELEVICI
Keyword(s):  
Scalar Fields ◽  
Final States ◽  
Complex Scalar ◽  
Fortran Code ◽  
Non Linear ◽  
Execution Speed ◽  
Mimd Computer ◽  
The Stability ◽  
Stability And Accuracy ◽  
Pseudo Spectral

We tested the parallelization of explicit schemes for the solution of non-linear classical field theories of complex scalar fields which are capable of simulating hadronic collisions. Our attention focused on collisions in a fractional model with a particularly rich inelastic spectrum of final states. Relativistic collisions of all types were performed by computer on large lattices (64 to 256 sites per dimension). The stability and accuracy of the objects were tested by the use of two other methods of solutions: Pseudo-spectral and semi-implicit. Parallelization of the Fortran code on a 64-transputer MIMD Volvox machine revealed, for certain topologies, communication deadlock and less-than-optimum routing strategies when the number of transputers used was less than the maximum. The observed speedup, for N transputers in an appropriate topology, is shown to scale approximately as N, but the overall gain in execution speed, for physically interesting problems, is a modest 2–3 when compared to state-of-the-art workstations.

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