Hopscotch: a Fast Second-order Partial Differential Equation Solver

Gourlay, A. R.

doi:10.1093/imamat/6.4.375

Cited by 212 publications

(66 citation statements)

References 3 publications

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“…That is, after reconstructing the image with inpainting, one regards the reconstructed points in Ω \ K as known and the specified points in K as unknown. Then one inpaints the data in K using the information from Ω \ K. This leads to an overall solution that is smoother and propagates the high quality solution from Ω \ K to the more erroneous data in K. While this strategy may appear ad hoc at first glance, it can be justified as a numerically consistent approximation of the inpainting PDE in the sense of so-called Hopscotch schemes [34].…”

Section: Interpolation Swappingmentioning

confidence: 99%

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

et al. 2014

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Galić et al. [33] have shown that compression based on edge-enhancing anisotropic diffusion (EED) can outperform the quality of JPEG for medium to high compression ratios when the interpolation points are chosen as vertices of an adaptive triangulation. However, the reasons for the good performance of EED remained unclear, and they could not outperform the more advanced JPEG 2000. The goals of the present paper are threefold: Firstly, we investigate the compression qualities of various partial differential equations. This sheds light on the favourable properties of EED in the context of image compression. Secondly, we demonstrate that it is even possible to beat the quality of JPEG 2000 with EED if one uses specific subdivisions on rectangles and several important optimisations. These amendments include improved entropy coding, brightness and diffusivity optimisation, and interpolation swapping. Thirdly, we demonstrate how to extend our approach to 3-D and shape data. Experiments on classical test images and 3-D medical data illustrate the high potential of our approach.

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Section: Interpolation Swappingmentioning

confidence: 99%

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

et al. 2014

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“…Using the notation uj = u t jAx,nOt), the 1lopscotch method (see ( 3) , [ 4) ) is defined by given, the equation (4.10) defines un assuming X 1.…”

Section: The Hopscotch Methodsmentioning

confidence: 99%

On The Navier‐Stokes Equations with Constant Total Temperature

Gottlieb

Gustafsson

1976

Stud Appl Math

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For various applications in fluid dynamics, one can assume that the total temperature is constant. Therefore, the energy equations can be replaced by an algebraic relation. The resulting set of equations in the inviscid case is analyzed in this paper. It is shown that the system is strictly hyperbolic and well posed for the initial‐value problem. Boundary conditions are described such that the linearized system is well posed. The hopscotch method is investigated and numerical results are presented.

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“…In Section 2.1 this will be made more precise, and we present sufficient conditions for (1.5) which are applicable for parabolic problems. Some related results of Gourlay [3] are generalized and improved. For the odd-even hopscotch scheme applied to the convection-diffusion equation, using either central or one-sided differences in space, we shall give in Section 2.3 conditions on r and h which are necessary and sufficient for (1.5).…”

Section: Introductionmentioning

confidence: 99%

“…This paper is concerned with stability properties of the well-known hopscotch scheme, an efficient numerical integration method for time-dependent partial differential equations [2][3][4][5][6]. We shall examine the scheme for linear, homogeneous initial-boundary value problems.…”

Section: Introductionmentioning

confidence: 99%

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Linear stability of the hopscotch scheme

Hundsdorfer¹,

Verwer²

1989

Applied Numerical Mathematics

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423This paper is devoted to the hopscotch scheme, which is a numerical integration technique for time-dependent partial differential equations. We examine its linear stability properties. A general theorem is presented which provides sufficient conditions for boundedness of the numerical solution during time stepping on a fixed space-time mesh. This theorem has applications in the field of parabolic problems. For the one-space-dimensional convection-diffusion equation we present a detailed stability analysis of the odd-even scheme combined with central and one-sided finite differences. We compare stability based on the spectral condition with von Neumann stability.

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Hopscotch: a Fast Second-order Partial Differential Equation Solver

Cited by 212 publications

References 3 publications

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

Understanding, Optimising, and Extending Data Compression with Anisotropic Diffusion

On The Navier‐Stokes Equations with Constant Total Temperature

Linear stability of the hopscotch scheme

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