1,721,025 research outputs found
Homogenization of conservation laws with oscillatory source and non-oscillatory data
No full textWe consider a scalar conservation law with oscillatory, periodic source term and with oscillatory initial data.
For possibly resonant initial data, we prove a corrector-type result for this problem, extending a previous one by
the author [Asymptotic Anal. 46 (2006), 53-79]. Here we can relax the assumption of well-prepared data
Unstable blow-up patterns
No full textConsider the semilinear heat equation (see (1.1)) with reaction term u^p, p>1. We construct solutions that blow up at a finite time T, according to a variety of specific asymptotic behaviors. These blow-up patterns are unstable. The corresponding solutions have an arbitrary large number of local maxima, collapsing for t=T
On the homogenization of conservation laws with resonant oscillatory source
No full textWe consider a scalar conservation law with oscillatory, periodic source term and with oscillatory initial data.
For possibly resonant initial data, we prove a corrector-type result for this problem, extending a previous one by E and Serre [Asymptotic Anal. 5 (1992), 311–316]: an asymptotic representation is identified and the
strong convergence of the asymptotic expansion is shown
An Integro-Differential Conservation Law arising in a Model of Granular Flow
No full textWe study a scalar integro-differential conservation law which was recently derived by the authors as the slow erosion limit of a granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one cannot adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A priori L∞ bound and total variation estimates yield the convergence and global existence of solutions with bounded variation. Furthermore, we present a well-posedness analysis which establishes that these solutions are stable in the L1 norm with respect to the initial data
Error Estimates for Well-Balanced and Time-Split Schemes on a Damped Semilinear Wave Equation
No full textA posteriori L1 error estimates are derived for both well-balanced (WB) and fractional-step (FS) numerical approximations of the unique weak solution of the Cauchy problem for the 1D semilinear damped wave equation. For setting up the WB algorithm, we proceed by rewriting it under the form of an elementary 2 × 2 system which linear convective structure allows to reduce the Godunov scheme with optimal Courant number (corresponding to ∆t = ∆x) to a wavefront-tracking algorithm free from any step of projection onto piecewise constant functions. A fundamental difference in the total variation estimates is proved, which partly explains the discrepancy of the FS method when the dissipative (sink) term displays an explicit dependence in the space variable. Numerical tests are performed by means of several exact solutions of the linear damped wave equation.
The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM
A hyperbolic model of multi-phase flow
No full textWe consider a hyperbolic model for the flow of an inviscid fluid admitting liquid and vapor phases.
We consider and prove here, as a preliminary study for a forthcoming paper, the basic features of the system: wave curves, Riemann problem, wave interactions
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