2,174 research outputs found
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Dynamical approach study of spurious steady-state numerical solutions of nonlinear differential equations. Part 1: The ODE connection and its implications for algorithm development in computational fluid dynamics
Spurious stable as well as unstable steady state numerical solutions, spurious asymptotic numerical solutions of higher period, and even stable chaotic behavior can occur when finite difference methods are used to solve nonlinear differential equations (DE) numerically. The occurrence of spurious asymptotes is independent of whether the DE possesses a unique steady state or has additional periodic solutions and/or exhibits chaotic phenomena. The form of the nonlinear DEs and the type of numerical schemes are the determining factor. In addition, the occurrence of spurious steady states is not restricted to the time steps that are beyond the linearized stability limit of the scheme. In many instances, it can occur below the linearized stability limit. Therefore, it is essential for practitioners in computational sciences to be knowledgeable about the dynamical behavior of finite difference methods for nonlinear scalar DEs before the actual application of these methods to practical computations. It is also important to change the traditional way of thinking and practices when dealing with genuinely nonlinear problems. In the past, spurious asymptotes were observed in numerical computations but tended to be ignored because they all were assumed to lie beyond the linearized stability limits of the time step parameter delta t. As can be seen from the study, bifurcations to and from spurious asymptotic solutions and transitions to computational instability not only are highly scheme dependent and problem dependent, but also initial data and boundary condition dependent, and not limited to time steps that are beyond the linearized stability limit
Pegasus: A New Hybrid-Kinetic Particle-in-Cell Code for Astrophysical Plasma Dynamics
We describe Pegasus, a new hybrid-kinetic particle-in-cell code tailored for
the study of astrophysical plasma dynamics. The code incorporates an
energy-conserving particle integrator into a stable, second-order--accurate,
three-stage predictor-predictor-corrector integration algorithm. The
constrained transport method is used to enforce the divergence-free constraint
on the magnetic field. A delta-f scheme is included to facilitate a
reduced-noise study of systems in which only small departures from an initial
distribution function are anticipated. The effects of rotation and shear are
implemented through the shearing-sheet formalism with orbital advection. These
algorithms are embedded within an architecture similar to that used in the
popular astrophysical magnetohydrodynamics code Athena, one that is modular,
well-documented, easy to use, and efficiently parallelized for use on thousands
of processors. We present a series of tests in one, two, and three spatial
dimensions that demonstrate the fidelity and versatility of the code.Comment: 27 pages, 12 figures, accepted for publication in Journal of
Computational Physic
Finite-difference methods for simulation models incorporating non-conservative forces
We discuss algorithms applicable to the numerical solution of second-order
ordinary differential equations by finite-differences. We make particular
reference to the solution of the dissipative particle dynamics fluid model, and
present extensive results comparing one of the algorithms discussed with the
standard method of solution. These results show the successful modeling of
phase separation and surface tension in a binary immiscible fluid mixture.Comment: 27 pages RevTeX, 9 figures, J. Chem. Phys. (in press
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