784 research outputs found
Numerical solution for stiff initial value problems using 2-point block multistep method
This paper focuses on the derivation of an improved 2-point Block Backward Differentiation Formula of order five (I2BBDF(5)) for solving stiff first order Ordinary Differential Equations (ODEs). The I2BBDF(5) method is derived by using Taylor's series expansion to obtain the coefficients of the formula. To verify the efficiency of the I2BBDF(5) method, stiff problems from the literature are tested and compared with the existing solver for stiff ODEs. From the numerical results, we conclude that the I2BBDF(5) method can be an alternative solver for solving stiff ODE
A new fifth order implicit block method for solving first order stiff ordinary differential equations
A new implicit block backward differentiation formula that computes 3–points simultaneously is derived. The method is of order 5 and solves system of stiff ordinary differential equations (ODEs). The stability analysis indicates that the
method is A–stable. Numerical results show that the method outperformed some existing block and non-block methodsfor solving stiff ODEs
Solving delay differential equations by using implicit 2-point block backward differentiation formula
In this paper, an implicit 2-point Block Backward Differentiation formula (BBDF) method was
considered for solving Delay Differential Equations (DDEs). The method was implemented by using a
constant stepsize via Newton Iteration. This implicit block method was expected to produce two points
simultaneously. The efficiency of the method was compared with the existing classical 1-point Backward
Differentiation Formula (BDF) in terms of execution time and accurac
Block Milne’s Implementation For Solving Fourth Order Ordinary Differential Equations
Block predictor-corrector method for solving non-stiff
ordinary differential equations (ODEs) started with Milne’s
device. Milne’s device is an extension of the block predictor corrector method providing further benefits and better results. This study considers Milne’s devise for solving fourth order ODEs. A combination of Newton’s backward difference interpolation polynomial and numerical integration method are applied and integrated at some selected grid points to formulate the block predictor-corrector method. Moreover, Milne’s devise advances the computational efficiency by applying the chief local
truncation error] (CLTE) of the block predictor-corrector
method after establishing the order. The numerical results were exhibited to attest the functioning of Milne’s devise in solving fourth order ODEs. The complete results were obtained with the aid of Mathematica 9 kernel for Microsoft Windows. Numerical results showcase that Milne’s device is more effective than existent methods in terms of design new step size, determining the convergence criteria and maximizing errors at all examined convergence levels
An implicit 2-point block extended backward differentiation formula for integration of stiff initial value problems
A class of implicit 2-point block extended backward differentiation formula (BEBDF) of order 4 is presented. The stability region of the method is constructed and shown to be A – stable. Results obtained are compared with an existing block backward differentiation formula (BBDF). The comparison shows that using constant step size and the same number of integration steps, our method achieves greater accuracy than
the 2-point BBDF and is suitable for solving stiff initial value problems
Blended General Linear Methods based on Boundary Value Methods in the GBDF family
Among the methods for solving ODE-IVPs, the class of General Linear Methods
(GLMs) is able to encompass most of them, ranging from Linear Multistep
Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able
to overcome typical drawbacks of the previous classes of methods. For example,
order barriers for stable LMF and the problem of order reduction for RK
methods. Nevertheless, these goals are usually achieved at the price of a
higher computational cost. Consequently, many efforts have been made in order
to derive GLMs with particular features, to be exploited for their efficient
implementation. In recent years, the derivation of GLMs from particular
Boundary Value Methods (BVMs), namely the family of Generalized BDF (GBDF), has
been proposed for the numerical solution of stiff ODE-IVPs. In particular, this
approach has been recently developed, resulting in a new family of L-stable
GLMs of arbitrarily high order, whose theory is here completed and fully
worked-out. Moreover, for each one of such methods, it is possible to define a
corresponding Blended GLM which is equivalent to it from the point of view of
the stability and order properties. These blended methods, in turn, allow the
definition of efficient nonlinear splittings for solving the generated discrete
problems. A few numerical tests, confirming the excellent potential of such
blended methods, are also reported.Comment: 22 pages, 8 figure
Weighted block Runge-Kutta method for solving stiff ordinary differential equations
In this paper, weighted block Runge-Kutta (WBRK) method is derived for solving stiff ordinary differential equations (ODEs). Implementation of weights on the method and its stability region are shown. Numerical results of the WBRK method are presented and compared with the existing methods to prove the ability of the proposed method to solve stiff ODEs. The results show that the WBRK method has better accuracy than the comparing methods
Programming codes of block-Milne's device for solving fourth-order ODEs
Block-Milne’s device is an extension of block-predictor-corrector method and
specifically developed to design a worthy step size, resolve the convergence
criteria and maximize error. In this study, programming codes of block-
Milne’s device (P-CB-MD) for solving fourth order ODEs are considered.
Collocation and interpolation with power series as the basic solution are
used to devise P-CB-MD. Analysing the P-CB-MD will give rise to the principal
local truncation error (PLTE) after determining the order. The P-CB-MD for
solving fourth order ODEs is written using Mathematica which can be utilized
to evaluate and produce the mathematical results. The P-CB-MD is very
useful to demonstrate speed, efficiency and accuracy compare to manual
computation applied. Some selected problems were solved and compared
with existing methods. This was made realizable with the support of the
named computational benefit
Fourth-order time-stepping for stiff PDEs on the sphere
We present in this paper algorithms for solving stiff PDEs on the unit sphere
with spectral accuracy in space and fourth-order accuracy in time. These are
based on a variant of the double Fourier sphere method in coefficient space
with multiplication matrices that differ from the usual ones, and
implicit-explicit time-stepping schemes. Operating in coefficient space with
these new matrices allows one to use a sparse direct solver, avoids the
coordinate singularity and maintains smoothness at the poles, while
implicit-explicit schemes circumvent severe restrictions on the time-steps due
to stiffness. A comparison is made against exponential integrators and it is
found that implicit-explicit schemes perform best. Implementations in MATLAB
and Chebfun make it possible to compute the solution of many PDEs to high
accuracy in a very convenient fashion
Numerical treatment of Kap's equation using a class of fourth order method
Kap's equation is a stiff initial value problem. This paper deals with the treatment of Kap's equation using a class of 4th order explicit Runge-Kutta method. Numerical computation was carried out using Microsoft Visual C++. The results of the computation were found to be highly accurate and consistent with minima errors. A comparison of the results generated from the scheme was also carried out vis-a-vis some other conventional explicit Runge-Kutta formulae. The proposed class of method was found to compare favourably well
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