A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation

In this paper we analyze the large-time behavior of the augmented Burgers equation. We first study the well-posedness of the Cauchy problem and obtain $L^1$-$L^p$ decay rates. The asymptotic behavior of the solution is obtained by showing that the influence of the convolution term $K*u_{xx}$ is the same as $u_{xx}$ for large times. Then, we propose a semi-discrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the non-local term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.Comment: Accepted for publication in ESAIM: Mathematical Modelling and Numerical Analysi

Large-time asymptotics, vanishing viscosity and numerics for 1-D scalar conservation laws

In this paper we analyze the large time asymptotic behavior of the discrete solutions of numerical approximation schemes for scalar hyperbolic conservation laws. We consider three monotone conservative schemes that are consistent with the one-sided Lipschitz condition (OSLC): Lax-Friedrichs, Engquist-Osher and Godunov. We mainly focus on the inviscid Burgers equation, for which we know that the large time behavior is of self-similar nature, described by a two-parameter family of N-waves. We prove that, at the numerical level, the large time dynamics depends on the amount of numerical viscosity introduced by the scheme: while Engquist-Osher and Godunov yield the same N-wave asymptotic behavior, the Lax-Friedrichs scheme leads to viscous self-similar profiles, corresponding to the asymptotic behavior of the solutions of the continuous viscous Burgers equation. The same problem is analyzed in the context of self-similar variables that lead to a better numerical performance but to the same dichotomy on the asymptotic behavior: N-waves versus viscous ones. We also give some hints to extend the results to more general fluxes. Some numerical experiments illustrating the accuracy of the results of the paper are also presented.Comment: Error corrected in main theorem in v3. Obtained results do not change in essence. Minor typos corrected in v

Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper we obtain bounds for the decay rate in the L^r (\rr^d)-norm for the solutions to a nonlocal and nolinear evolution equation, namely, u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy, with x \in \rr^d, $t>0$. Here we consider a kernel $K(x,y)$ of the form $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$, where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ is a linear function $a(x)= Ax$. To obtain the decay rates we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u) = - \int_{\rr^d} K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy, with $1 \leq p < \infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole \rr^d: \lambda_{1,p} (\rr^d) = 2(\int_{\rr^d} \psi (z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p \to \infty$ of $\lambda_{1,p}^{1/p}$

Dispersion for the Schr\"odinger Equation on Networks

In this paper we consider the Schr\"odinger equation on a network formed by a tree with the last generation of edges formed by infinite strips. We give an explicit description of the solution of the linear Schr\"odinger equation with constant coefficients. This allows us to prove dispersive estimates, which in turn are useful for solving the nonlinear Schr\"odinger equation. The proof extends also to the laminar case of positive step-function coefficients having a finite number of discontinuities.Comment: 16 pages, 2 figure