12 research outputs found
On numerical methods for the semi-nonrelativistic system of the nonlinear Dirac equation
Solving the nonlinear Dirac equation in the nonrelativistic limit regime numerically is difficult, because the solution oscillates in time with frequency of , where is inversely proportional to the speed of light. It was shown in [7], however, that such solutions can be approximated up to an error of by solving the semi-nonrelativistic limit system, which is a non-oscillatory problem. For this system, we construct a two-step method, called the exponential explicit midpoint rule, and prove second-order convergence of the semi-discretization in time. Furthermore, we construct a benchmark method based on standard techniques and compare the efficiency of both methods. Numerical experiments show that the new integrator reduces the computational costs per time step to 40% and within a given runtime improves the accuracy by a factor of six
Symmetric integrators with improved uniform error bounds and long-time conservations for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime
In this paper, we are concerned with symmetric integrators for the nonlinear
relativistic Klein--Gordon (NRKG) equation with a dimensionless parameter
, which is inversely proportional to the speed of light.
The highly oscillatory property in time of this model corresponds to the
parameter and the equation has strong nonlinearity when \eps is
small. There two aspects bring significantly numerical burdens in designing
numerical methods. We propose and analyze a novel class of symmetric
integrators which is based on some formulation approaches to the problem,
Fourier pseudo-spectral method and exponential integrators. Two practical
integrators up to order four are constructed by using the proposed symmetric
property and stiff order conditions of implicit exponential integrators. The
convergence of the obtained integrators is rigorously studied, and it is shown
that the accuracy in time is improved to be \mathcal{O}(\varepsilon^{3}
\hh^2) and \mathcal{O}(\varepsilon^{4} \hh^4) for the time stepsize \hh.
The near energy conservation over long times is established for the multi-stage
integrators by using modulated Fourier expansions. These theoretical results
are achievable even if large stepsizes are utilized in the schemes. Numerical
results on a NRKG equation show that the proposed integrators have improved
uniform error bounds, excellent long time energy conservation and competitive
efficiency
A fourth-order compact time-splitting method for the Dirac equation with time-dependent potentials
In this paper, we present an approach to deal with the dynamics of the Dirac
equation with time-dependent electromagnetic potentials using the fourth-order
compact time-splitting method (). To this purpose, the
time-ordering technique for time-dependent Hamiltonians is introduced, so that
the influence of the time-dependence could be limited to certain steps which
are easy to treat. Actually, in the case of the Dirac equation, it turns out
that only those steps involving potentials need to be amended, and the scheme
remains efficient, accurate, as well as easy to implement. Numerical examples
in 1D and 2D are given to validate the scheme.Comment: 24pages, 8 figure
Generalized averaged Gaussian quadrature and applications
A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal
MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications
Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described