An extension of A-stability to alternating direction implicit methods

An alternating direction implicit (ADI) scheme was constructed by the method of approximate factorization. An A-stable linear multistep method (LMM) was used to integrate a model two-dimensional hyperbolic-parabolic partial differential equation. Sufficient conditions for the A-stability of the LMM were determined by applying the theory of positive real functions to reduce the stability analysis of the partial differential equations to a simple algebraic test. A linear test equation for partial differential equations is defined and then used to analyze the stability of approximate factorization schemes. An ADI method for the three-dimensional heat equation is also presented

Alternating direction implicit methods for parabolic equations with a mixed derivative

Alternating direction implicit (ADI) schemes for two-dimensional parabolic equations with a mixed derivative are constructed by using the class of all A sub 0-stable linear two-step methods in conjunction with the method of approximation factorization. The mixed derivative is treated with an explicit two-step method which is compatible with an implicit A sub 0-stable method. The parameter space for which the resulting ADI schemes are second order accurate and unconditionally stable is determined. Some numerical examples are given

Numerical solution of a coupled pair of elliptic equations from solid state electronics

Iterative methods are considered for the solution of a coupled pair of second order elliptic partial differential equations which arise in the field of solid state electronics. A finite difference scheme is used which retains the conservative form of the differential equations. Numerical solutions are obtained in two ways, by multigrid and dynamic alternating direction implicit methods. Numerical results are presented which show the multigrid method to be an efficient way of solving this problem

The application of alternating-direction implicit methods to the space-dependent kinetics equations

Cover title"MIT-3903-3."Originally presented as the first author's thesis (Ph. D.), M.I.T. Dept. of Nuclear Engineering, 1969Includes bibliographical references (leaves 54-55)Prepared under U.S. Atomic Energy Commission AT(30-1) 390

Fast gain calibration in radio astronomy using alternating direction implicit methods: Analysis and applications

Context. Modern radio astronomical arrays have (or will have) more than one order of magnitude more receivers than classical synthesis arrays, such as the VLA and the WSRT. This makes gain calibration a computationally demanding task. Several alternating direction implicit (ADI) approaches have therefore been proposed that reduce numerical complexity for this task from $\mathcal{O}(P^3)$ to $\mathcal{O}(P^2)$, where $P$ is the number of receive paths to be calibrated. Aims. We present an ADI method, show that it converges to the optimal solution, and assess its numerical, computational and statistical performance. We also discuss its suitability for application in self-calibration and report on its successful application in LOFAR standard pipelines. Methods. Convergence is proved by rigorous mathematical analysis using a contraction mapping. Its numerical, algorithmic, and statistical performance, as well as its suitability for application in self-calibration, are assessed using simulations. Results. Our simulations confirm the $\mathcal{O}(P^2)$ complexity and excellent numerical and computational properties of the algorithm. They also confirm that the algorithm performs at or close to the Cramer-Rao bound (CRB, lower bound on the variance of estimated parameters). We find that the algorithm is suitable for application in self-calibration and discuss how it can be included. We demonstrate an order-of-magnitude speed improvement in calibration over traditional methods on actual LOFAR data. Conclusions. In this paper, we demonstrate that ADI methods are a valid and computationally more efficient alternative to traditional gain calibration method and we report on its successful application in a number of actual data reduction pipelines.Comment: accepted for publication in Astronomy & Astrophysic

The Numerical Experiments on the Alternating Direction Implicit Method Using Interlacing Scanning for Elliptic Partial Differential Equation

In this paper, the authors introduce a new variant of alternating direction implicit methods. The alternating direction implicit methods (the ADI methods) are proposed to solve elliptic and parabolic partial differential equations in the paper of Peaceman and Rachford, in 1955. The numerical solution of elliptic partial differential equations by finite difference generally leads to the problem of matrix equation Ax=K. The matrix A is split into two matrices H and V in which their nonzero entries appear at the position corresponding row and column directional mesh points, respectively. The ADI methods consist of the alternating computation implicitly about the row and column directional matrix equations. Our variant ADI method is such a method that each row and column directional computations proceed on every other line interlacingly, in the closed mesh region. We prove the convergence of this variant ADI method (the interlacing ADI method). The average rate of convergence of the interlacing ADI method is approximated almost twice that of the normal ADI method. Numerical experiments on model problems show less iteration times than ones of the normal ADI methods for model problems

Field simulation of axisymmetric plasma screw pinches by alternating-direction-implicit methods

An axisymmetric plasma screw pinch is an axisymmetric column of ionized gaseous plasma radially confined by forces from axial and azimuthal currents driven in the plasma and its surroundings. This dissertation is a contribution to detailed, high resolution computer simulation of dynamic plasma screw pinches in 2-d {ital rz}-coordinates. The simulation algorithm combines electron fluid and particle-in-cell (PIC) ion models to represent the plasma in a hybrid fashion. The plasma is assumed to be quasineutral; along with the Darwin approximation to the Maxwell equations, this implies application of Ampere`s law without displacement current. Electron inertia is assumed negligible so that advective terms in the electron momentum equation are ignored. Electrons and ions have separate scalar temperatures, and a scalar plasma electrical resistivity is assumed. Altemating-direction-implicit (ADI) methods are used to advance the electron fluid drift velocity and the magnetic fields in the simulation. The ADI methods allow time steps larger than allowed by explicit methods. Spatial regions where vacuum field equations have validity are determined by a cutoff density that invokes the quasineutral vacuum Maxwell equations (Darwin approximation). In this dissertation, the algorithm was first checked against ideal MM stability theory, and agreement was nicely demonstrated. However, such agreement is not a new contribution to the research field. Contributions to the research field include new treatments of the fields in vacuum regions of the pinch simulation. The new treatments predict a level of magnetohydrodynamic turbulence near the bulk plasma surface that is higher than predicted by other methods