Search results for "Linear equation"
showing 10 items of 102 documents
Nonresistive dissipative magnetohydrodynamics from the Boltzmann equation in the 14-moment approximation
2018
We derive the equations of motion of relativistic, non-resistive, second-order dissipative magnetohydrodynamics from the Boltzmann equation using the method of moments. We assume the fluid to be composed of a single type of point-like particles with vanishing dipole moment or spin, so that the fluid has vanishing magnetization and polarization. In a first approximation, we assume the fluid to be non-resistive, which allows to express the electric field in terms of the magnetic field. We derive equations of motion for the irreducible moments of the deviation of the single-particle distribution function from local thermodynamical equilibrium. We analyze the Navier-Stokes limit of these equati…
Efficient numerical methods for pricing American options under stochastic volatility
2007
Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two-dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M-matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved u…
A fast hierarchical dual boundary element method for three-dimensional elastodynamic crack problems
2010
In this work a fast solver for large-scale three-dimensional elastodynamic crack problems is presented, implemented, and tested. The dual boundary element method in the Laplace transform domain is used for the accurate dynamic analysis of cracked bodies. The fast solution procedure is based on the use of hierarchical matrices for the representation of the collocation matrix for each computed value of the Laplace parameter. An ACA (adaptive cross approximation) algorithm is used for the population of the low rank blocks and its performance at varying Laplace parameters is investigated. A preconditioned GMRES is used for the solution of the resulting algebraic system of equations. The precond…
Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems
2019
In this paper, a new technique is shown for deriving computable, guaranteed lower bounds of functional type (minorants) for two different cost functionals subject to a parabolic time-periodic boundary value problem. Together with previous results on upper bounds (majorants) for one of the cost functionals, both minorants and majorants lead to two-sided estimates of functional type for the optimal control problem. Both upper and lower bounds are derived for the second new cost functional subject to the same parabolic PDE-constraints, but where the target is a desired gradient. The time-periodic optimal control problems are discretized by the multiharmonic finite element method leading to lar…
Scheduled Relaxation Jacobi method: improvements and applications
2016
Elliptic partial differential equations (ePDEs) appear in a wide variety of areas of mathematics, physics and engineering. Typically, ePDEs must be solved numerically, which sets an ever growing demand for efficient and highly parallel algorithms to tackle their computational solution. The Scheduled Relaxation Jacobi (SRJ) is a promising class of methods, atypical for combining simplicity and efficiency, that has been recently introduced for solving linear Poisson-like ePDEs. The SRJ methodology relies on computing the appropriate parameters of a multilevel approach with the goal of minimizing the number of iterations needed to cut down the residuals below specified tolerances. The efficien…
DLPNO-MP2 second derivatives for the computation of polarizabilities and NMR shieldings
2021
We present a derivation and efficient implementation of the formally complete analytic second derivatives for the domain-based local pair natural orbital second order Møller–Plesset perturbation theory (MP2) method, applicable to electric or magnetic field-response properties but not yet to harmonic frequencies. We also discuss the occurrence and avoidance of numerical instability issues related to singular linear equation systems and near linear dependences in the projected atomic orbital domains. A series of benchmark calculations on medium-sized systems is performed to assess the effect of the local approximation on calculated nuclear magnetic resonance shieldings and the static dipole …
One pendulum to run them all
2013
The analytical solution for the three-dimensional linear pendulum in a rotating frame of reference is obtained, including Coriolis and centrifugal accelerations, and expressed in terms of initial conditions. This result offers the possibility of treating Foucault and Bravais pendula as trajectories of the same system of equations, each of them with particular initial conditions. We compare them with the common two-dimensional approximations in textbooks. A previously unnoticed pattern in the three-dimensional Foucault pendulum attractor is presented.
Physical model, theoretical aspects and applications of the flight of a ball in the atmosphere. Part I: Modelling of forces and torque, and theoretic…
1991
A model of the forces and the torque operating on a ball that is flying with rotation in the atmosphere of the Earth, and the resulting system of ordinary differential equations, are derived from mechanics and aerodynamics. The system of equations allows the theoretical aspects of the flight of a ball, such as the boundedness of its kinetic energy, the curvature of the orbit or the velocity function, to be investigated using certain transformations of the variables. The solutions of the corresponding ordinary or boundary value problems, computed numerically, are used to treat certain tasks in international ball games, for example, the maximum and minimum velocities of a volleyball service.
Numerical relativistic hydrodynamics: Local characteristic approach.
1991
We extend some recent Ishock capturing methodsR designed to solve nonlinear hyperbolic systems of conservation laws and which avoid the use of artifical viscosity for treating strong discontinuities to a relativistic hydrodynamics system of equations. Some standard shock-tube problems and radial accretion onto a Schwarzschild black hole are used to calibrate our code.
CFC+: Improved dynamics and gravitational waveforms from relativistic core collapse simulations
2004
Core collapse supernovae are a promising source of detectable gravitational waves. Most of the existing (multidimensional) numerical simulations of core collapse in general relativity have been done using approximations of the Einstein field equations. As recently shown by Dimmelmeier et al (2002a,b), one of the most interesting such approximation is the so-called conformal flatness condition (CFC) of Isenberg, Wilson and Mathews. Building on this previous work we present here new results from numerical simulations of relativistic rotational core collapse in axisymmetry, aiming at improving the dynamics and the gravitational waveforms. The computer code used for these simulations evolves th…