A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

Comparison of 2D simultaneous multi-slice and 3D GRASE readout schemes for pseudo-continuous arterial spin labeling of cerebral perfusion at 3 T

Objective

In this perfusion magnetic resonance imaging study, the performances of different pseudo-continuous arterial spin labeling (PCASL) sequences were compared: two-dimensional (2D) single-shot readout with simultaneous multislice (SMS), 2D single-shot echo-planar imaging (EPI) and multishot three-dimensional (3D) gradient and spin echo (GRASE) sequences combined with a background-suppression (BS) module.

Materials and methods

Whole-brain PCASL images were acquired from seven healthy volunteers. The performance of each protocol was evaluated by extracting regional cerebral blood flow (rCBF) measures using an inline morphometric segmentation prototype. Image data postprocessing and subsequent statistical analyses enabled comparisons at the regional and sub-regional levels.

Results

The main findings were as follows: (i) Mean global CBF obtained across methods was were highly correlated, and these correlations were significantly higher among the same readout sequences. (ii) Temporal signal-to-noise ratio and gray-matter-to-white-matter CBF ratio were found to be equivalent for all 2D variants but lower than those of 3D-GRASE.

Discussion

Our study demonstrates that the accelerated SMS readout can provide increased acquisition efficiency and/or a higher temporal resolution than conventional 2D and 3D readout sequences. Among all of the methods, 3D-GRASE showed the lowest variability in CBF measurements and thus highest robustness against noise.

     

Recent Developments in the Pure Streamfunction Formulation of the Navier-Stokes System

In this paper we review fourth-order approximations of the biharmonic operator in one, two and three dimensions. In addition, we describe recent developments on second and fourth order finite difference approximations of the two dimensional Navier-Stokes equations. The schemes are compact both for the biharmonic and the Laplacian operators. For the convective term the fourth order scheme invokes also a sixth order Pade approximation for the first order derivatives, using an approximation suggested by Carpenter-Gottlieb-Abarbanel (J. Comput. Phys. 108:272–295, 1993). We also introduce the derivation of a pure streamfunction formulation for the Navier-Stokes equations in three dimensions.     

Numerical simulation of the MHD equations by a kinetic-type method

We introduce a flux-splitting formula for the approximation of the ideal MHD equations in conservative form. The Faraday equation is considered as the average of an abstract kinetic equation, giving a flux-splitting formula. For the other part of the equations, we generalize formally the classical half-Maxwellian flux-splitting of the Euler equations. Numerical results on MHD shock tube problems are displayed.     

A Fourth Order Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square

A new fourth order box-scheme for the Poisson problem in a square with Dirichlet boundary conditions is introduced, extending the approach in Croisille (Computing 78:329–353, 2006). The design is based on a “hermitian box” approach, combining the approximation of the gradient by the fourth order hermitian derivative, with a conservative discrete formulation on boxes of length 2h. The goal is twofold: first to show that fourth order accuracy is obtained both for the unknown and the gradient; second, to describe a fast direct algorithm, based on the Sherman-Morrison formula and the Fast Sine Transform. Several numerical results in a square are given, indicating an asymptotic O(N ²log ₂(N)) computing complexity.     

Exploratory analysis of the spatio-temporal deformation of the myocardium during systole from tagged MRI

Myocardial contractile function is, with perfusion, one of the main affected factors in ischemic heart diseases. In this paper, we propose an original framework based on functional data analysis for the quantitative study of spatio-temporal parameters related to the myocardial contraction mechanics. The mechanical strains in the left-ventricular (LV) myocardium are computed from tagged magnetic resonance imaging cardiac sequences. A statistical functional model of the normal contractile function of the LV is build from the study of eight examinations on healthy subjects. We show that it is possible to detect abnormal strain patterns comparatively to this model, by generating distance maps at rest and under pharmacological stress. We demonstrate the consistency of the results for the circumferential deformation parameter on healthy and pathological data sets.     

A High Order Compact Scheme for the Pure-Streamfunction Formulation of the Navier-Stokes Equations

In this paper we continue the study, which was initiated in (Ben-Artzi et al. in Math. Model. Numer. Anal. 35(2):313–303, 2001; Fishelov et al. in Lecture Notes in Computer Science, vol. 2667, pp. 809–817, 2003; Ben-Artzi et al. in J. Comput. Phys. 205(2):640–664, 2005 and SIAM J. Numer. Anal. 44(5):1997–2024, 2006) of the numerical resolution of the pure streamfunction formulation of the time-dependent two-dimensional Navier-Stokes equation. Here we focus on enhancing our second-order scheme, introduced in the last three afore-mentioned articles, to fourth order accuracy. We construct fourth order approximations for the Laplacian, the biharmonic and the nonlinear convective operators. The scheme is compact (nine-point stencil) for the Laplacian and the biharmonic operators, which are both treated implicitly in the time-stepping scheme. The approximation of the convective term is compact in the no-leak boundary conditions case and is nearly compact (thirteen points stencil) in the case of general boundary conditions. However, we stress that in any case no unphysical boundary condition was applied to our scheme. Numerical results demonstrate that the fourth order accuracy is actually obtained for several test-cases.     

Recent Advances in the Study of a Fourth-Order Compact Scheme for the One-Dimensional Biharmonic Equation

It is well-known that non-periodic boundary conditions reduce considerably the overall accuracy of an approximating scheme. In previous papers the present authors have studied a fourth-order compact scheme for the one-dimensional biharmonic equation. It relies on Hermitian interpolation, using functional values and Hermitian derivatives on a three-point stencil. However, the fourth-order accuracy is reduced to a mere first-order near the boundary. In turn this leads to an ??almost third-order?? accuracy of the approximate solution. By a careful inspection of the matrix elements of the discrete operator, it is shown that the boundary does not affect the approximation, and a full (??optimal??) fourth-order convergence is attained. A number of numerical examples corroborate this effect.     

Hermitian Compact Interpolation on the Cubed-Sphere Grid

The cubed-sphere grid is a spherical grid made of six quasi-cartesian square-like patches. It was originally introduced in Sadourny (Mon Weather Rev 100:136–144, 1972). We extend to this grid the design of high-order finite-difference compact operators (Collatz, The numerical treatment of differential equations. Springer, Berlin, 1960; Lele, J Comput Phys 103:16–42, 1992). The present work is limitated to the design of a fourth-order accurate spherical gradient. The treatment at the interface of the six patches relies on a specific interpolation system which is based on using great circles in an essential way. The main interest of the approach is a fully symmetric treatment of the sphere. We numerically demonstrate the accuracy of the approximate gradient on several test problems, including the cosine-bell test-case of Williamson et al. (J Comput Phys 102:211–224, 1992) and a deformational test-case reported in Nair and Lauritzen (J Comput Phys 229:8868–8887, 2010).