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Displaying 41 –
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145
We develop a new multidimensional finite-volume algorithm for transport equations. This algorithm is both stable and non-dissipative. It is based on a reconstruction of the discrete solution inside each cell at every time step. The proposed reconstruction, which is genuinely multidimensional, allows recovering sharp profiles in both the direction of the transport velocity and the transverse direction. It constitutes an extension of the one-dimensional reconstructions analyzed in (Lagoutière, 2005;...
The Cauchy problem for nonlinear functional differential equations on the Haar pyramid is considered. The phase space for generalized solutions is constructed. An existence theorem is proved by using the method of successive approximations. The theory of characteristics and integral inequalities are used. Examples of phase spaces are given.
We study the simplest system of partial differential equations: that is, two equations of first order partial differential equation with two independent variables with real analytic coefficients. We describe a necessary and sufficient condition for the Cauchy problem to the system to be C infinity well posed. The condition will be expressed by inclusion relations of the Newton polygons of some scalar functions attached to the system. In particular, we can give a characterization of the strongly...
A strictly hyperbolic quasi-linear 2×2 system in two independent
variables with C2 coefficients is considered. The existence of a simple
wave solution in the sense that the solution is a 2-dimensional vector-valued
function of the so called Riemann invariant is discussed. It is shown, through
a purely geometrical approach, that there always exists simple wave solution
for the general system when the coefficients are arbitrary C^2 functions
depending on both, dependent and independent variables.
The Cauchy problem for first order system is known to be well-posed in when it admits a microlocal symmetrizer which is smooth in and Lipschitz continuous in . This paper contains three main results. First we show that a Lipschitz smoothness globally in is sufficient. Second, we show that the existence of symmetrizers with a given smoothness is equivalent to the existence of full symmetrizers having the same smoothness. This notion was first introduced in [FL67]. This is the key point...
We prove the --time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the --time decay estimates.
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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145