Past Topics (PT)
http://hdl.handle.net/20.500.11824/20
20181127T02:06:39Z

A splitting method for the augmented Burgers equation
http://hdl.handle.net/20.500.11824/713
A splitting method for the augmented Burgers equation
Ignat L.; Pozo A.
In this paper we consider a splitting method for the augmented Burgers equation and prove that it is of ﬁrst order. We also analyze the largetime behavior of the approximated solution by obtaining the ﬁrst term in the asymptotic expansion. We prove that, when time increases, these solutions be have as the selfsimilar solutions of the viscous Burgers equation.
20170701T00:00:00Z

A semidiscrete largetime behavior preserving scheme for the augmented Burgers equation
http://hdl.handle.net/20.500.11824/680
A semidiscrete largetime behavior preserving scheme for the augmented Burgers equation
Ignat L.I.; Pozo A.
In this paper we analyze the largetime behavior of the augmented Burgers equation. We first study the wellposedness of the Cauchy problem and obtain $L^1L^p$ 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 semidiscrete numerical scheme that preserves this asymptotic behavior, by introducing two correcting factors in the discretization of the nonlocal term. Numerical experiments illustrating the accuracy of the results of the paper are also presented.
20170601T00:00:00Z

Flux identification for 1d scalar conservation laws in the presence of shocks
http://hdl.handle.net/20.500.11824/596
Flux identification for 1d scalar conservation laws in the presence of shocks
Castro C.; Zuazua E.
We consider the problem of flux identification for 1d scalar conservation laws formulating it as an optimal control problem. We introduce a new optimization strategy to compute numerical approximations of minimizing fluxes. We first prove the existence of minimizers. We also prove the convergence of discrete minima obtained by means of monotone numerical approximation schemes, by a Γconvergence argument. Then we address the problem of developing efficient descent algorithms. We first consider and compare the existing two possible approaches. The first one, the socalled discrete approach, based on a direct computation of gradients in the discrete problem and the socalled continuous one, where the discrete descent direction is obtained as a discrete copy of the continuous one. When optimal solutions have shock discontinuities, both approaches produce highly oscillating minimizing sequences and the effective descent rate is very weak. As a remedy we adapt the method of alternating descent directions that uses the recent developments of generalized tangent vectors and the linearization around discontinuous solutions, introduced by the authors, in collaboration with F. Palacios, in the case where the control is the initial datum. This method distinguishes descent directions that move the shock and those that perturb the profile of the solution away from it. As we shall see, a suitable alternating combination of these two classes of descent directions allows building more efficient and faster descent algorithms. © 2011 American Mathematical Society.
20111231T00:00:00Z

Regularity issues for the nullcontrollability of the linear 1d heat equation
http://hdl.handle.net/20.500.11824/599
Regularity issues for the nullcontrollability of the linear 1d heat equation
Micu S.; Zuazua E.
The fact that the heat equation is controllable to zero in any bounded domain of the Euclidean space, any time T>0 and from any open subset of the boundary is well known. On the other hand, numerical experiments show the illposedness of the problem. In this paper we develop a rigorous analysis of the 1d problem which provides a sharp description of this illposedness. To be more precise, each initial data y0∈L2(0,1) of the 1d linear heat equation has a boundary control of the minimal L 2(0,T)norm which drives the state to zero in time T>0. This control is given by a solution of the homogeneous adjoint equation with some initial data φ0, minimizing a suitable quadratic cost. Our aim is to study the relationship between the regularity of y0 and that of φ0. We show that there are regular data y0 for which the corresponding φ0 are highly irregular, not belonging to any negative exponent Sobolev space. Moreover, the class of such initial data y 0 is dense in L2(0,1). This explains the severe illposedness of the numerical algorithms developed for the approximation of the minimal L2(0,T)norm control of y0 based on the computation of φ0. The lack of polynomial convergence rates for Tychonoff regularization processes is a consequence of this phenomenon too. © 2011 Elsevier B.V. All rights reserved.
20111231T00:00:00Z