821 research outputs found
Fixed Coefficients Block Backward Differentiation Formulas for the Numerical Solution of Stiff Ordinary Differential Equations
This paper focuses on the derivation of implicit 2-point block method based on Backward Differentiation Formula (BDF) which will be called BBDF of variable step size
for solving first order stiff initial value problems (IVPs) for Ordinary Differential Equations (ODEs). The method presented is similar to the form of standard BDF. This
allows us to store the coefficients of the y values and thus avoiding calculating the differentiation coefficients at each step but robust enough to allow for step size variation.Plots of their regions of absolute stability for the method are also presented. The efficiency of the 2-point BBDF is compared with the conventional variable step variable order BDF(VSVOBDF) method. Numerical results indicate that the resulting 2-point BBDF method outperform the VSVOBDF method in both execution time and accuracy
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
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
Partitioning Techniques and Their Parallelization for Stiff System of Ordinary Differential Equations
A new code based on variable order and variable stepsize component wise
partitioning is introduced to solve a system of equations dynamically. In previous
partitioning technique researches, once an equation is identified as stiff, it will
remain in stiff subsystem until the integration is completed. In this current
technique, the system is treated as nonstiff and any equation that caused stiffness
will be treated as stiff equation. However, should the characteristics showed the
elements of nonstiffness, and then it will be treated again with Adam method. This
process will continue switching from stiff to nonstiff vice versa whenever it is
necessary until the interval of integration is completed.Next, a block method with R-points generate R new approximate solution values;is
a strategy for solving a system and also for parallelizing ODEs. Partitioning this
block method to solve stiff differential equations is a new strategy; it is more
efficient and takes less computational time compared to the sequential methods.
Two partitioning techniques are constructed, Intervalwise Block Partitioning (IBP)
and Componentwise Block Partitioning (CBP). Numerical results are compared as
validation of its effectiveness.
Intervalwise block partitioning will initially treat the systems of equations as
nonstiff and solve them using Adams method, by switching to the Backward
Differentiation formula when there is a step failure and indication of stiffness.
Componentwise block partitioning will place the necessary equations that cause
instability and stiffness into the stiff subsystem and solve using Backward
Differentiation Formula, while all other equations will still be treated as non-stiff
and solved using Adams formula.
Parallelizing the partitioning strategies using Message Passing Interface (MPI) is
the most appropriate method to solve large system of equations. Parallelizing the
right algorithm in the partitioning code will give a better perfonnance with shorter
execution times. The graphs of its performance and execution time, visualize the
advantages of parallelizing
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
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
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
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
Variable order block method for solving second order ordinary differential equations
This paper proposed 2-point block backward differentiation formulas (BBDF) of order 3, 4, and 5 for direct solution of second order ordinary differential equations. These methods were derived via backward difference interpolation polynomial with two solutions are produced simultaneously at each step. All the three different orders of 2-point BBDF is implemented in variable order scheme. The scheme utilizes the local truncation error, which is generated by the single order of 2-point BBDF method. Numerical results are presented to illustrate the validity of the proposed scheme
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