Former Research Lines
http://hdl.handle.net/20.500.11824/21
2019-02-22T00:00:01Z
2019-02-22T00:00:01Z
A splitting method for the augmented Burgers equation
Ignat L.
Pozo A.
http://hdl.handle.net/20.500.11824/713
2017-08-03T06:03:20Z
2017-07-01T00:00:00Z
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 large-time 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 self-similar solutions of the viscous Burgers equation.
2017-07-01T00:00:00Z
A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation
Ignat L.I.
Pozo A.
http://hdl.handle.net/20.500.11824/680
2017-06-08T01:00:09Z
2017-06-01T00:00:00Z
A semi-discrete large-time behavior preserving scheme for the augmented Burgers equation
Ignat L.I.; Pozo A.
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.
2017-06-01T00:00:00Z
Flux identification for 1-d scalar conservation laws in the presence of shocks
Castro C.
Zuazua E.
http://hdl.handle.net/20.500.11824/596
2017-02-21T08:18:21Z
2011-12-31T00:00:00Z
Flux identification for 1-d scalar conservation laws in the presence of shocks
Castro C.; Zuazua E.
We consider the problem of flux identification for 1-d 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 Γ-convergence 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 so-called discrete approach, based on a direct computation of gradients in the discrete problem and the so-called 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.
2011-12-31T00:00:00Z
Regularity issues for the null-controllability of the linear 1-d heat equation
Micu S.
Zuazua E.
http://hdl.handle.net/20.500.11824/599
2017-02-21T08:18:21Z
2011-12-31T00:00:00Z
Regularity issues for the null-controllability of the linear 1-d 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 ill-posedness of the problem. In this paper we develop a rigorous analysis of the 1-d problem which provides a sharp description of this ill-posedness. To be more precise, each initial data y0∈L2(0,1) of the 1-d 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 ill-posedness 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.
2011-12-31T00:00:00Z