We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor,
(2010) 688–710; S. Mishra and E. Tadmor,
(2011) 1023–1045]. The schemes are formulated in terms of . A suitable choice of the potential results...

This paper addresses the recovery of piecewise smooth functions from their discrete data. Reconstruction methods using both pseudo-spectral coefficients and physical space interpolants have been discussed extensively in the literature, and it is clear that an a priori knowledge of the jump discontinuity location is essential for any reconstruction technique to yield spectrally accurate results with high resolution near the discontinuities. Hence detection of the jump discontinuities is critical...

We prove that the one-dimensional Euler–Poisson system driven by the Poisson forcing together with the usual $\gamma $-law pressure, $\gamma \ge 1$, admits global solutions for a large class of initial data. Thus, the Poisson forcing regularizes the generic finite-time breakdown in the $2\times 2$
$p$-system. Global regularity is shown to depend on whether or not the initial configuration of the Riemann invariants and density crosses an intrinsic critical threshold.

We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor,
(2010) 688–710; S. Mishra and E. Tadmor,
(2011) 1023–1045]. The schemes are formulated in terms of . A suitable choice of the potential results...

This paper addresses the recovery of piecewise smooth functions from
their discrete data.
Reconstruction methods using both pseudo-spectral coefficients and
physical space interpolants have been discussed extensively in the
literature, and it is clear that an knowledge of the jump
discontinuity location is essential for any reconstruction technique
to yield spectrally accurate results with high resolution near the
discontinuities. Hence detection of the jump discontinuities is
critical for all...

In this paper we present two versions of the central local
discontinuous Galerkin (LDG) method on overlapping cells
for solving diffusion equations, and provide their
stability analysis and error estimates for the linear heat equation.
A comparison
between the traditional LDG method on
a single mesh and the two versions of the central LDG
method on overlapping cells is also made.
Numerical experiments are provided to validate the quantitative
conclusions from the analysis and to support conclusions...

In this paper we present two versions of the central local
discontinuous Galerkin (LDG) method on overlapping cells
for solving diffusion equations, and provide their
stability analysis and error estimates for the linear heat equation.
A comparison
between the traditional LDG method on
a single mesh and the two versions of the central LDG
method on overlapping cells is also made.
Numerical experiments are provided to validate the quantitative
conclusions from the analysis and to support conclusions...

We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the central
discontinuous Galerkin method and the regular discontinuous
Galerkin method in this context is also made.
Numerical experiments are provided to validate the quantitative
conclusions from the analysis.

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