25 research outputs found
High-order convergent deferred correction schemes based on parameterized Runge-Kutta-Nyström methods for second-order boundary value problems
AbstractIterated deferred correction is a widely used approach to the numerical solution of first-order systems of nonlinear two-point boundary value problems. Normally, the orders of accuracy of the various methods used in a deferred correction scheme differ by 2 and, as a direct result, each time deferred correction is used the order of the overall scheme is increased by a maximum of 2. In [16], however, it has been shown that there exist schemes based on parameterized Runge–Kutta methods, which allow a higher increase of the overall order. A first example of such a high-order convergent scheme which allows an increase of 4 orders per deferred correction was based on two mono-implicit Runge–Kutta methods. In the present paper, we will investigate the possibility for high-order convergence of schemes for the numerical solution of second-order nonlinear two-point boundary value problems not containing the first derivative. Two examples of such high-order convergent schemes, based on parameterized Runge–Kutta-Nyström methods of orders 4 and 8, are analysed and discussed
A Sixth-Order Extension to the MATLAB Package bvp4c of J. Kierzenka and L. Shampine
A new two-point boundary value problem algorithm based upon the MATLAB bvp4c package of Kierzenka and Shampine is described. The algorithm, implemented in a new package bvp6c, uses the residual control framework of bvp4c (suitably modified for a more accurate finite difference approximation) to maintain a user specified accuracy. The new package is demonstrated to be as robust as the existing software, but more efficient for most problems, requiring fewer internal mesh points and evaluations to achieve the required accuracy
Superconvergant interpolants for the collocation solution of boundary value ordinary differential equations
Publisher's version/PDFA long-standing open question associated with the use of collocation methods for boundary value ordinary differential equations is concerned with the development of a high order continuous solution approximation to augment the high order discrete solution approximation, obtained at the mesh points which subdivide the problem interval. It is well known that the use of collocation at Gauss points leads to solution approximations at the mesh points for which the global error is O(h[superscript 2k]), where k is the number of collocation points used per subinterval and h is the subinterval size. This discrete solution is said to be superconvergent. The collocation solution also yields a C[superscript 0] continuous solution approximation that has a global error of O(h[supercript k+1]). In this paper, we show how to efficiently augment the superconvergent discrete collocation solution to obtain C[superscript 1] continuous "superconvergent" interpolants whose global errors are O(h[superscript 2k]). The key ideas are to use the theoretical
framework of continuous Runge-Kutta schemes and to augment the collocation solution with inexpensive monoimplicit Runge-Kutta stages. Specific schemes are derived for k = 1, 2, 3, and 4. Numerical results are provided to support the theoretical analysis
Fully well-balanced entropy controlled discontinuous Galerkin spectral element method for shallow water flows: global flux quadrature and cell entropy correction
We present a novel approach for solving the shallow water equations using a
discontinuous Galerkin spectral element method. The method we propose has three
main features. First, it enjoys a discrete well-balanced property, in a spirit
similar to the one of e.g. [20]. As in the reference, our scheme does not
require any a-priori knowledge of the steady equilibrium, moreover it does not
involve the explicit solution of any local auxiliary problem to approximate
such equilibrium. The scheme is also arbitrarily high order, and verifies a
continuous in time cell entropy equality. The latter becomes an inequality as
soon as additional dissipation is added to the method. The method is
constructed starting from a global flux approach in which an additional flux
term is constructed as the primitive of the source. We show that, in the
context of nodal spectral finite elements, this can be translated into a simple
modification of the integral of the source term. We prove that, when using
Gauss-Lobatto nodal finite elements this modified integration is equivalent at
steady state to a high order Gauss collocation method applied to an ODE for the
flux. This method is superconvergent at the collocation points, thus providing
a discrete well-balanced property very similar in spirit to the one proposed in
[20], albeit not needing the explicit computation of a local approximation of
the steady state. To control the entropy production, we introduce artificial
viscosity corrections at the cell level and incorporate them into the scheme.
We provide theoretical and numerical characterizations of the accuracy and
equilibrium preservation of these corrections. Through extensive numerical
benchmarking, we validate our theoretical predictions, with considerable
improvements in accuracy for steady states, as well as enhanced robustness for
more complex scenario
Relaxation deferred correction methods and their applications to residual distribution schemes
The Deferred Correction (DeC) methods combined with the residual distribution (RD) approach allow the construction of high order continuous Galerkin (cG) schemes avoiding the inversion of the mass matrix. With the application of entropy correction functions we can even obtain entropy conservative/dissipative spatial discretizations in this context. To handle entropy production in time, a relaxation approach has been suggested by Ketcheson. The main idea is to slightly modify the time-step size such that the approximated solution fulfills the underlying entropy conservation/dissipation constraint. In this paper, we first study the relaxation technique applied to the DeC approach as an ODE solver, then we extend this combination to the residual distribution method, requiring more technical steps. The outcome is a class of cG methods that is fully entropy conservative/dissipative and where we can still avoid the inversion of a mass matrix
Discontinuous Galerkin methods for nonlinear scalar hyperbolic conservation laws: divided difference estimates and accuracy enhancement
In this paper, an analysis of the accuracy-enhancement for the discontinuous Galerkin (DG) method applied to one-dimensional scalar nonlinear hyperbolic conservation laws is carried out. This requires analyzing the divided difference of the errors for the DG solution. We therefore first prove that the alpha-th order (1 <= \alpha <= k+1) divided difference of the DG error in the L2-norm is of order k+(3-alpha)/2 when upwind fluxes are used, under the condition that |f'(u)| possesses a uniform positive lower bound. By the duality argument, we then derive superconvergence results of order k+(3-alpha)/2 in the negative-order norm, demonstrating that it is possible to extend the Smoothness-Increasing Accuracy-Conserving filter to nonlinear conservation laws to obtain at least (3k/2+1)th order superconvergence for post-processed solutions. As a by-product, for variable coefficient hyperbolic equations, we provide an explicit proof for optimal convergence results of order k+1 in the L2-norm for the divided differences of DG errors and thus (2k+1)th order superconvergence in negative-order norm holds. Numerical experiments are given that confirm the theoretical results
Application of exponential fitting techniques to numerical methods for solving differential equations
Ever since the work of Isaac Newton and Gottfried Leibniz in the late 17th century, differential equations (DEs) have been an important concept in many branches of science. Differential equations arise spontaneously in i.a. physics, engineering, chemistry, biology, economics and a lot of fields in between. From the motion of a pendulum, studied by high-school students, to the wave functions of a quantum system, studied by brave scientists: differential equations are common and unavoidable. It is therefore no surprise that a large number of mathematicians have studied, and still study these equations. The better the techniques for solving DEs, the faster the fields where they appear, can advance.
Sadly, however, mathematicians have yet to find a technique (or a combination of techniques) that can solve all DEs analytically. Luckily, in the meanwhile, for a lot of applications, approximate solutions are also sufficient. The numerical methods studied in this work compute such approximations. Instead of providing the hypothetical scientist with an explicit, continuous recipe for the solution to their problem, these methods give them an approximation of the solution at a number of discrete points. Numerical methods of this type have been the topic of research since the days of Leonhard Euler, and still are. Nowadays, however, the computations are performed by digital processors, which are well-suited for these methods, even though many of the ideas predate the modern digital computer by almost a few centuries. The ever increasing power of even the smallest processor allows us to devise newer and more elaborate methods.
In this work, we will look at a few well-known numerical methods for the solution of differential equations. These methods are combined with a technique called exponential fitting, which produces exponentially fitted methods: classical methods with modified coefficients. The original idea behind this technique is to improve the performance on problems with oscillatory solutions
Galerkin projection of discrete fields via supermesh construction
Interpolation of discrete FIelds arises frequently in computational physics.
This thesis focuses on the novel implementation and analysis of Galerkin
projection, an interpolation technique with three principal advantages over
its competitors: it is optimally accurate in the L2 norm, it is conservative,
and it is well-defined in the case of spaces of discontinuous functions.
While these desirable properties have been known for some time, the implementation
of Galerkin projection is challenging; this thesis reports the first
successful general implementation.
A thorough review of the history, development and current frontiers of
adaptive remeshing is given. Adaptive remeshing is the primary motivation
for the development of Galerkin projection, as its use necessitates the interpolation
of discrete fields. The Galerkin projection is discussed and the
geometric concept necessary for its implementation, the supermesh, is introduced.
The efficient local construction of the supermesh of two meshes
by the intersection of the elements of the input meshes is then described.
Next, the element-element association problem of identifying which elements
from the input meshes intersect is analysed. With efficient algorithms for
its construction in hand, applications of supermeshing other than Galerkin
projections are discussed, focusing on the computation of diagnostics of simulations
which employ adaptive remeshing. Examples demonstrating the effectiveness
and efficiency of the presented algorithms are given throughout.
The thesis closes with some conclusions and possibilities for future work
Mathematical Properties of a New Levin-Type Sequence Transformation Introduced by \v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. I. Algebraic Theory
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la [J. Math. Phys. \textbf{44}, 962
- 968 (2003)] introduced in connection with the summation of the divergent
perturbation expansion of the hydrogen atom in an external magnetic field a new
sequence transformation which uses as input data not only the elements of a
sequence of partial sums, but also explicit estimates
for the truncation errors. The explicit
incorporation of the information contained in the truncation error estimates
makes this and related transformations potentially much more powerful than for
instance Pad\'{e} approximants. Special cases of the new transformation are
sequence transformations introduced by Levin [Int. J. Comput. Math. B
\textbf{3}, 371 - 388 (1973)] and Weniger [Comput. Phys. Rep. \textbf{10}, 189
- 371 (1989), Sections 7 -9; Numer. Algor. \textbf{3}, 477 - 486 (1992)] and
also a variant of Richardson extrapolation [Phil. Trans. Roy. Soc. London A
\textbf{226}, 299 - 349 (1927)]. The algebraic theory of these transformations
- explicit expressions, recurrence formulas, explicit expressions in the case
of special remainder estimates, and asymptotic order estimates satisfied by
rational approximants to power series - is formulated in terms of hitherto
unknown mathematical properties of the new transformation introduced by
\v{C}\'{\i}\v{z}ek, Zamastil, and Sk\'{a}la. This leads to a considerable
formal simplification and unification.Comment: 41 + ii pages, LaTeX2e, 0 figures. Submitted to Journal of
Mathematical Physic