Efficient Parallel Algorithms for Parabolic Problems
TL;DR: Domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation, showing that the resulting schemes are of second order global accuracy in space, and stable in the sense of Osher or in $L_{\infty }$.
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Abstract: Domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation. Implicit schemes are used in order to march with large time steps. Parallelization is realized by approximating interface values using explicit computation. Various techniques are examined, including a multistep second order explicit scheme and a one-step high-order scheme. We show that the resulting schemes are of second order global accuracy in space, and stable in the sense of Osher or in $L_{\infty }$. They are optimized with respect to the parallel efficiency.
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Citations
Non-iterative domain decomposition methods for a non-stationary Stokes–Darcy model with Beavers–Joseph interface condition☆
TL;DR: A three-step backward differentiation is used in the second method to achieve an accuracy order O ( h 3 + Δ t 3 ) , which is illustrated by a numerical example.
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Unconditional Stability of Corrected Explicit-Implicit Domain Decomposition Algorithms for Parallel Approximation of Heat Equations
Han-Sheng Shi,Hong-lin Liao +1 more
TL;DR: By adding an interface correction step to Kuznetsov's EIDs, the one-dimensional CEIDD procedure achieves unconditional stability without discarding the time-stepwise efficiency of the EIDD method.
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An efficient S-DDM iterative approach for compressible contamination fluid flows in porous media
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TL;DR: The developed method takes the excellent attractive advantages of both the non-overlapping domain decomposition and the splitting technique, and reduces computational complexities, large memory requirements and long computational durations.
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Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation
TL;DR: The unconditional stability of these schemes is proved, and the convergence rate of second order is also obtained, and numerical results are presented to examine the accuracy, stability and parallelism of the parallel schemes.
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Conservative domain decomposition schemes for solving two-dimensional heat equations
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TL;DR: In this paper, a new mass-conserved domain decomposition method for two-dimensional heat equations is proposed by combining the operator splitting technique and the C-N implicit scheme.
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Abstract: Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this book, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents Preface to the first edition page viii Preface to the second edition xi 1 Introduction 1 2 Parabolic equations in one space variable 7 2.1 Introduction 7 2.2 A model problem 7 2.3 Series approximation 9 2.4 An explicit scheme for the model problem 10 2.5 Difference notation and truncation error 12 2.6 Convergence of the explicit scheme 16 2.7 Fourier analysis of the error 19 2.8 An implicit method 22 2.9 The Thomas algorithm 24 2.10 The weighted average or θ-method 26 2.11 A maximum principle and convergence for µ(1 − θ) ≤ 1 2 33 2.12 A three-time-level scheme 38 2.13 More general boundary conditions 39 2.14 Heat conservation properties 44 2.15 More general linear problems 46 2.16 Polar coordinates 52 2.17 Nonlinear problems 54 Bibliographic notes 56 Exercises 56 v
A Finite Difference Domain Decomposition Algorithm for Numerical Solution of the Heat Equation
TL;DR: In this article, a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented, where interface values between subdomains are found by an explicit finite difference formula, and interior values are determined by backward differencing in time.
Additive Schwarz algorithms for parabolic convection-diffusion equations
TL;DR: Three additive Schwarz type domain decomposition methods for general, not necessarily selfadjoint, linear, second order, parabolic partial differential equations are introduced and the convergence rates of these algorithms are studied.
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Numerical solution of partial differential equations
Theodor Meis,Ulrich Marcowitz +1 more
TL;DR: In this paper, the Schroder-Trottenberg reduction method was used for band matrices and the Buneman algorithm for solving the Poisson equation on nonrectangular regions.
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The convergence rate for difference approximations to general mixed initial boundary value problems
TL;DR: In this article, the convergence rate for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations was investigated, and it was shown that if an energy estimate holds, then the extra boundary conditions can be of one order lower accuracy without destroying the convergence expected from the approximation at inner points.
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