45,835 research outputs found
A One Step Method for the Solution of General Second Order Ordinary Differential Equations
In this paper, an implicit one step method for the numerical solution of second order initial value problems of ordinary differential equations has been developed by collocation and interpolation technique. The introduction of an o step point guaranteed the zero stability and consistency of the method. The implicit method developed was implemented as a block which gave simultaneous solutions, as well as their rst derivatives, at both o step and the step point. A comparison of our method to the predictor-corrector method after solving some sample problems reveals that our method performs better
Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations
Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation
Converting DAE models to ODE models: application to reactive Rayleigh distillation
This paper illustrates the application of an index reduction method to some differential algebraic equations
(DAE) modelling the reactive Rayleigh distillation. After two deflation steps, this DAE is converted to an
equivalent first-order explicit ordinary differential equation (ODE). This ODE involves a reduced number of
dependent variables, and some evaluations of implicit functions defined, either from the original algebraic
constraints, or from the hidden ones. Consistent initial conditions are no longer to be computed; at the
opposite of some other index reduction methods, which generate a drift-off effect, the algebraic constraints
remain satisfied at any time; and, finally, the computational effort to solve the ODE may be less than the
one associated to the original DAE
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
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