Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables

Deng, Dingwen; Zhang, Chengjian

doi:10.1007/s11075-012-9604-8

Cited by 12 publications

(11 citation statements)

References 25 publications

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“…Also, we refer the interested reader to [28,[31][32][33][34][35][36] for useful research works on ADI methods. More works can be seen in [37][38][39][40]24,[41][42][43].…”

Section: An Introduction About Hoc-adi Methodsmentioning

confidence: 99%

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

Shokri

Afshari

2015

Computer Physics Communications

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“…Also, we refer the interested reader to [28,[31][32][33][34][35][36] for useful research works on ADI methods. More works can be seen in [37][38][39][40]24,[41][42][43].…”

Section: An Introduction About Hoc-adi Methodsmentioning

confidence: 99%

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

Shokri

Afshari

2015

Computer Physics Communications

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“…The semi-discrete solution (6) is therefore defined in terms of a convolution integral, and as mentioned, traditional methods of discretization in space will bear a cost of O(N 2 ) operations per time step to evaluate u n+1 (x) at N spatial points. However, we have developed a fast convolution algorithm for (6), so that the numerical solution is obtained in O(N ) operations per time step. This is accomplished by first performing a "characteristic" decomposition…”

Section: Background and Notationmentioning

confidence: 99%

“…Our next goal is to approximate the differential operators (∂ xx ) m using the compositions of the convolution operator D from (6). We begin this process by observing…”

Section: An A-stable Family Of Schemes Of Order 2pmentioning

confidence: 99%

“…The application of the operator D p will be computed recursively, and so a scheme of order P will have a cost of O(P N ) per time step for N spatial discretization points. Notice that equation (10) is in fact identical to the original second order scheme (6), while the schemes (11) and (12) are fourth and sixth order, respectively. The local truncation errors will be O((c∆t/β) 2P +2 ), which means that the error constant will decrease with increasing β.…”

Section: An A-stable Family Of Schemes Of Order 2pmentioning

confidence: 99%

“…But the Fairweather's approach did not remain A-stable, as was the case for the second order scheme of Lees. The work on higher order ADI implementations for second order hyperbolic PDEs continues to this day, with recent emphasis placed on the use of compact finite differences [6]. But what of higher order schemes which are also Astable?…”

Section: Introductionmentioning

confidence: 99%

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Higher Order A-Stable Schemes for the Wave Equation Using a Successive Convolution Approach

Causley¹,

Christlieb²

2014

SIAM J. Numer. Anal.

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In several recent works, we developed a new second order, A-stable approach to wave propagation problems based on the method of lines transpose (MOL T ) formulation combined with alternating direction implicit (ADI) schemes. Because our method is based on an integral solution of the ADI splitting of the MOL T formulation, we are able to easily embed non-Cartesian boundaries and include point sources with exact spatial resolution. Further, we developed an efficient O(N ) convolution algorithm for rapid evaluation of the solution, which makes our method competitive with explicit finite difference (e.g., finite difference time domain) solvers, in terms of both accuracy and time to solution, even for Courant numbers slightly larger than 1. We have demonstrated the utility of this method by applying it to a range of problems with complex geometry, including cavities with cusps. In this work, we present several important modifications to our recently developed wave solver. We obtain a family of wave solvers which are unconditionally stable, accurate of order 2P , and require O(P d N ) operations per time step, where N is the number of spatial points and d the number of spatial dimensions. We obtain these schemes by including higher derivatives of the solution, rather than increasing the number of time levels. The novel aspect of our approach is that the higher derivatives are constructed using successive applications of the convolution operator. We develop these schemes in one spatial dimension, and then extend the results to higher dimensions, by reformulating the ADI scheme to include recursive convolution. Thus, we retain a fast, unconditionally stable scheme, which does not suffer from the large dispersion errors characteristic to the ADI method. We demonstrate the utility of the method by applying it to a host of wave propagation problems. This method holds great promise for developing higher order, parallelizable algorithms for solving hyperbolic PDEs and can also be extended to parabolic PDEs. Introduction.In recent works [2, 3], the method of lines transpose (MOL T ) has been utilized to solve the wave equation, resulting in a second order accurate, A-stable numerical scheme. The solution is constructed using a boundary-corrected integral equation, which is derived in a semidiscrete setting. Thus, the solution at time t n+1 is found by convolving the solution at previous time levels against the semidiscrete Green's function. Upon spatial discretization, traditional convolution requires O(N 2 ) operations for a total of N spatial points per time step. However, we have developed a fast convolution algorithm for the one-dimensional (1d) problem, which reduces the computational cost to O(N ) operations [2]. This efficiency was additionally extended to the wave equation in higher dimensions by applying alternate direction implicit (ADI) splitting to the semidiscrete elliptic differential operator.

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Improved ADI Scheme for Linear Hyperbolic Equations: Extension to Nonlinear Cases and Compact ADI Schemes

Zhang

Cai

2017

J Sci Comput

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Analysis of a fourth-order compact ADI method for a linear hyperbolic equation with three spatial variables

Cited by 12 publications

References 25 publications

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

High-order compact ADI method using predictor–corrector scheme for 2D complex Ginzburg–Landau equation

Higher Order A-Stable Schemes for the Wave Equation Using a Successive Convolution Approach

Improved ADI Scheme for Linear Hyperbolic Equations: Extension to Nonlinear Cases and Compact ADI Schemes

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