Papers by Guido Vanden Berghe
Exponentially-fitted Stormer/Verlet methods are constructed taking into account a six-step flow c... more Exponentially-fitted Stormer/Verlet methods are constructed taking into account a six-step flow chart. It is shown that the thus constructed methods, when applied to strongly oscillating problems, are equivalent respectively to Cautschi and Deulfhard methods. As an illustration the constructed algorithms are used to solve bounded states as well as resonance states of Schrodinger equations. It is seen that all the versions are of order four but the way the error on the eigenvalues increases with the energy differs from one version to the other for the two problems considered.
Recent Advances in Computational and Applied Mathematics, 2011
The construction of symmetric and symplectic exponentially-fitted Runge-Kutta methods for the num... more The construction of symmetric and symplectic exponentially-fitted Runge-Kutta methods for the numerical integration of Hamiltonian systems with oscillatory solutions is reconsidered. In previous papers fourth-order and sixth-order symplectic exponentiallyfitted integrators of Gauss type, either with fixed or variable nodes, have been derived. In this paper new such integrators are constructed by making use of the six-step procedure of Ixaru and Vanden Berghe (Exponential fitting, Kluwer Academic Publishers, 2004). Numerical experiments for some oscillatory problems are presented and compared to the results obtained by previous methods.
International Journal of Computer Mathematics, 1991
ABSTRACT
Journal of Computational and Applied Mathematics, 1994
ABSTRACT
AIP Conference Proceedings, 2007
Tuned methods of the exponentially-fitted kind are derived and applied to fourth order ordinary d... more Tuned methods of the exponentially-fitted kind are derived and applied to fourth order ordinary differential equations subject to a special kind of boundary condtions. In this paper we analyse and construct several methods of order 2. A numerical experiment is performed to sustain the derived technique.
AIP Conference Proceedings, 2009
Second-order boundary value problems are solved by means of a new type of exponentially-fitted me... more Second-order boundary value problems are solved by means of a new type of exponentially-fitted methods that are modifications of the Numerov method. These methods depend upon a set of parameters which can be tuned to solve the problem at hand more accurately. Their values can be fixed over the entire integration interval, but they can also be determined locally from the local truncation error. A numerical example is given to illustrate the ideas.
Ijcm, 1996
To overcome the “order barrier” imposed by A-stability on linear multistep methods (LMMs) several... more To overcome the “order barrier” imposed by A-stability on linear multistep methods (LMMs) several authors have constructed A-stabilized extended one-step methods of order m by coupling m−1 LLMs for 2 ≤ m ≤ 5. In the present paper we discuss an alternative derivation technique for these methods based on Runge-Kutta methods (RKM). With this technique we are able to present examples of A-stable and L-stable methods for 2 ≤ m ≤ 7.
Journal of Computational and Applied Mathematics, 2011
J Comput Appl Math, 2003
... A general analysis of the operator O l k for arbitrary j values has been given by Hughes and ... more ... A general analysis of the operator O l k for arbitrary j values has been given by Hughes and Yadegar [5]. The following general definition is given: (1) where μ=0,
,j, k 0, (2) (3)Q ±μ =T(j, μ)l ± μ and where j 1 m 1 j 2 m 2 | jm represents the well-known ClebschGordan coefficient ...
We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods... more We consider the construction of P-stable, multi-parameter exponentially fitted Obrechkoff methods for second order differential equations. An earlier result for single-parameter exponential fitting is reexamined and extended to multiparameter, multi-order exponential fitting.
Exponential fitted algorithms for initial value and boundary value methods and for the calculatio... more Exponential fitted algorithms for initial value and boundary value methods and for the calculation of quadrature rules are introduced. Special attention is paid to the form of the leading order term of the error and to the global error term in particular. It is shown in which way a “best value” for the occurring frequency can be selected. Numerical experiments illustrate the proposed strategy.
Exponential Fitting, 2004
ABSTRACT
Exponential Fitting, 2004
Second-order boundary value problems are solved with exponentially-fltted Numerov methods. In ord... more Second-order boundary value problems are solved with exponentially-fltted Numerov methods. In order to attribute a value to the free parameter in such a method, we look at the leading term of the local truncation error. By solving the problem in two phases, a value for this parameter can be found such that the tuned method behaves like a sixth order method. Furthermore, guidelines to choose between multiple possible values for this parameter are given.
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Papers by Guido Vanden Berghe