On measure solutions to the zero-pressure gas model and their uniqueness
The homogeneous Dirichlet problem for non-elliptic partial differential equations with strong nonlinearities
A note on -point conservative monotone schemes
First–order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. Do the monotone difference approximations always give a good numerical solution in sense of monotonicity preservation or suppression of oscillations? This note will investigate this problem from a numerical point of view and show that a -point monotone scheme may give an oscillatory solution even though...
A note on (2K+1)-point conservative monotone schemes
First–order accurate monotone conservative schemes have good convergence and stability properties, and thus play a very important role in designing modern high resolution shock-capturing schemes. This note will investigate this problem from a numerical point of view and show that a -point monotone scheme may give an oscillatory solution even though the approximate solution is total variation diminishing, and satisfies maximum principle as well as discrete entropy inequality.
Theoretical and numerical studies of the DG schemes in one space dimension
We give a proof of the existence of a solution of reconstruction operators used in the DG schemes in one space dimension. Some properties and error estimates of the projection and reconstruction operators are presented. Then, by applying the DG schemes to the linear advection equation, we study their stability obtaining maximal limits of the Courant numbers for several DG schemes mostly experimentally. A numerical study explains how the stencils used in the reconstruction affect the efficiency...
On measure solutions to the Zero-pressure gas model and their uniqueness
The system of zero-pressure gas dynamics conservation laws describes the dynamics of free particles sticking under collision while mass and momentum are conserved. The existence of such solutions was established some time ago. Here we report a uniqueness result that uses the Oleinik entropy condition and a cohesion condition. Both of these conditions are automatically satisfied by solutions obtained in previous existence results. Important tools in the proof of uniqueness are regularizations, generalized...
On evolution Galerkin methods for the Maxwell and the linearized Euler equations
The subject of the paper is the derivation and analysis of evolution Galerkin schemes for the two dimensional Maxwell and linearized Euler equations. The aim is to construct a method which takes into account better the infinitely many directions of propagation of waves. To do this the initial function is evolved using the characteristic cone and then projected onto a finite element space. We derive the divergence-free property and estimate the dispersion relation as well. We present some numerical...
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