70 research outputs found
Error analysis of variable stepsize RungeâKutta methods for a class of multiply-stiff singular perturbation problems
AbstractIn this paper, we present some results on the error behavior of variable stepsize stiffly-accurate RungeâKutta methods applied to a class of multiply-stiff initial value problems of ordinary differential equations in singular perturbation form, under some weak assumptions on the coefficients of the considered methods. It is shown that the obtained convergence results hold for stiffly-accurate RungeâKutta methods which are not algebraically stable or diagonally stable. Some results on the existence and uniqueness of the solution of RungeâKutta equations are also presented
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SciCADE 95: International conference on scientific computation and differential equations
This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included
Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control
It is common practice to apply gradient-based optimization algorithms to
numerically solve large-scale ODE constrained optimal control problems.
Gradients of the objective function are most efficiently computed by
approximate adjoint variables. High accuracy with moderate computing time can
be achieved by such time integration methods that satisfy a sufficiently large
number of adjoint order conditions and supply gradients with higher orders of
consistency. In this paper, we upgrade our former implicit two-step Peer
triplets constructed in [Algorithms, 15:310, 2022] to meet those new
requirements. Since Peer methods use several stages of the same high stage
order, a decisive advantage is their lack of order reduction as for
semi-discretized PDE problems with boundary control. Additional order
conditions for the control and certain positivity requirements now intensify
the demands on the Peer triplet. We discuss the construction of 4-stage methods
with order pairs (4,3) and (3,3) in detail and provide three Peer triplets of
practical interest. We prove convergence for s-stage methods, for instance,
order s for the state variables even if the adjoint method and the control
satisfy the conditions for order s-1, only. Numerical tests show the expected
order of convergence for the new Peer triplets.Comment: 47 pages, 5 figure
Robust computational methods for two-parameter singular perturbation problems
Magister Scientiae - MScThis thesis is concerned with singularly perturbed two-parameter problems. We study a tted nite difference method as applied on two different meshes namely a piecewise mesh (of Shishkin type) and a graded mesh (of Bakhvalov type) as well as a tted operator nite di erence method. We notice that results on Bakhvalov mesh are better than those on Shishkin mesh. However, piecewise uniform meshes provide a simpler platform for analysis and computations. Fitted operator methods are even simpler in these regards due to the ease of operating on uniform meshes. Richardson extrapolation is applied on one of the tted mesh nite di erence method (those based on Shishkin mesh) as well as on the tted operator nite di erence method in order to improve the accuracy and/or the order of convergence. This is our main contribution to this eld and in fact we have achieved very good results after extrapolation on the tted operator finitete difference method. Extensive numerical computations are carried out on to confirm the theoretical results.South Afric
Singular Perturbations and Time-Scale Methods in Control Theory: Survey 1976-1982
Coordinated Science Laboratory was formerly known as Control Systems LaboratoryJoint Services Electronics Program / N00014-79-C-0424U.S. Air Force / AFOSR 78-363
Gratings: Theory and Numeric Applications
International audienceThe book containes 11 chapters written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers
Gratings: Theory and Numeric Applications, Second Revisited Edition
International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11
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