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

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

A Two-Dimensional Mesh Refinement Method for Problems with One-Dimensional Singularities

This paper introduces a method for resolving internal layers that can occur in the solutions of time-dependent differential equations in two space dimensions. Singular features in these solutions that are essentially one-dimensional in nature but are not oriented with the computational mesh are resolved using one-dimensional mesh refinement techniques with a procedure that is similar to an ADI method. A careful interpolation procedure assures that the resolution obtained in each ADI step is not lost in the succeeding ADI step

An alternating direction implicit method for the Control Data STAR-100 vector computer

An implementation of the alternating direction implicit (ADI) method for the Control Data STAR-100 computer is presented and analyzed. Two parallel algorithms, both of which are most efficient when used to solve many independent tridiagonal systems of equations, are discussed relative to their usefulness in an ADI implementation on the STAR-100 computer. It is shown that it may be desirable to alternate between the parallel algorithms as the direction of implicitness is alternated in order to eliminate the data rearrangement which would otherwise be required. The applicability of the two parallel tridiagonal solvers to several other numerical algorithms is also discussed

SFADI: the Speckle-Free Angular Differential Imaging method

We present a new processing technique aimed at significantly improving the angular differential imaging method (ADI) in the context of high-contrast imaging of faint objects nearby bright stars in observations obtained with extreme adaptive optics (EXAO) systems. This technique, named "SFADI" for "Speckle-Free ADI", allows to improve the achievable contrast by means of speckles identification and suppression. This is possible in very high cadence data, which freeze the atmospheric evolution. Here we present simulations in which synthetic planets are injected into a real millisecond frame rate sequence, acquired at the LBT telescope at visible wavelength, and show that this technique can deliver low and uniform background, allowing unambiguous detection of $10^{-5}$ contrast planets, from $100$ to $300$ mas separations, under poor and highly variable seeing conditions ($0.8$ to $1.5$ arcsec FWHM) and in only $20$ min of acquisition. A comparison with a standard ADI approach shows that the contrast limit is improved by a factor of $5$. We extensively discuss the SFADI dependence on the various parameters like speckle identification threshold, frame integration time, and number of frames, as well as its ability to provide high-contrast imaging for extended sources, and also to work with fast acquisitions.Comment: Accepted for publication in Ap