Anne Morlet - Academia.edu (original) (raw)
Papers by Anne Morlet
We present an improved model for the vortex sheet equation, that combines some of the features of... more We present an improved model for the vortex sheet equation, that combines some of the features of the models of Beale and Schaeeer, Dhanak, and Baker et al. We regularize the Beale-Schaeeer equation with a second-order viscous regularizing term, and we add a globally deened ux term in conservative form. We obtain u t + iu x = H(u)u ]x + ju x j 2 u x ]x + u xx , where i 2 = ?1, and H(u) is the Hilbert transform of u. We derive bounds for the solution of the equation and its rst-order spatial derivatives in L 2 and in the maximum norm, independent of. We show that the function u t satisses an L 2 norm bound that depends linearly on ; all the other derivatives satisfy bounds that depend on negative powers of. We show that, for > 0, the solution exists and is unique. We also prove that, for > 0, in the limit of ! 0, the sequence of functions (u)>0 has a weak limit; the weak limit may not be unique.
Physica D: Nonlinear Phenomena, 1996
ABSTRACT
SIAM Journal on Numerical Analysis, 1992
In this article, two numerical methods are analyzed and applied to compute an invariant closed cu... more In this article, two numerical methods are analyzed and applied to compute an invariant closed curve for a particular class of maps: a second-order scheme based on linear interpolation and a scheme based on interpolation by cubic splines. In suitable coordinates, the invariant curve is determined by the solution of a functional equation on a circle. A linearized version of this functional equation is studied in detail. Stability of the schemes is discussed and error estimates are derived depending on the smoothness of the solution. Roughly speaking, the analysis predicts superior behavior of the spline scheme when the solution is sufficiently smooth, but a more reliable behavior of the lower-order scheme in situations when a smooth solution cannot be expected. This is confirmed in numerical examples.The analysis of the linear functional equation does not readily generalize to nonlinear equations, but the algorithms for the linear equation can be used as a building block for a treatment of nonlinear equati...
Here, we rigorously prove some of the results obtained with asymptotic methods for a modified Bur... more Here, we rigorously prove some of the results obtained with asymptotic methods for a modified Burgers' equation, obtained by replacing the flux term in Burgers' equation considered in its hyperbolic form by the Hilbert transform. We show that, under the asumptions ju x j 1 C , jH(u x )j 1 C , and R 1 0 jju x jj dt C, H(u) being the Hilbert transform of u and C a constant independent of time, the smallest scale of the solution of the equation is essentially , being the viscosity coefficient. The analytical results are confirmed with numerical ones. We also show that the solution of the equation, in the limit of zero viscosity, has a weak limit in the space of functions of bounded variation.
Part I We derive a model equation for the linearized equation of an invariant curve for a Poincar... more Part I We derive a model equation for the linearized equation of an invariant curve for a Poincare map. We discretize the model equation with a second-order and third-order finite difference schemes, and with a cubic spline interpolation scheme. We also approximate the solution of the model equation with a truncated Fourier expansion. We derive error estimates for the second-order and third-order finite difference schemes and for the cubic spline interpolation scheme. We numerically implement the four schemes we consider and plot some error curves. Part II We show for a one-dimensional Stefan problem, that the numerical solution converges to the solution of the continuous equations in the limit of zero meshsize and timestep. We discretize the continuous equations with a second-order finite difference scheme in space and Crank-Nicholson scheme in time. We derive error equations and we use L2 estimates to bound the error in terms of the truncation errors of the finite difference schem...
The overlapping sinc collocation domain decomposition method combined with the Schwarz alternatin... more The overlapping sinc collocation domain decomposition method combined with the Schwarz alternating technique is developed for two-point boundary-value problems for second-order ordinary differential equations with singularities. The discrete system is formulated and the solution technique is described. It is shown that this method has an exponential convergence rate even in the presence of singularities. The details of the convergence proof are given for a sinc collocation method applied to second-order, two-point boundary-value problems when the original domain is divided into two subdomains. The extension to multiple domains is then straightforward.
Computation and Control IV, 1995
ABSTRACT
SIAM Journal on Numerical Analysis, 1995
Journal of Mathematical Analysis and Applications, 1998
To develop an understanding of singularity formation in vortex sheets, we consider model equation... more To develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze. A model equation is obtained by replacing the flux term in Burgers' equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial Ž Ž. . Ž .Ž. differential equations u s H u u q 1 y u u q u , 0 F F 1, G 0, t x x x x Ž. where H u is the Hilbert transform of u. We show that when s 1r2, for) 0, the solution of the equation exists for all time and is unique. We also show with a combination of analytical and numerical means that the solution when s 1r2 and) 0 is analytic. Using a pseudo-spectral method in space and the Adams᎐Moulton fourth-order predictor-corrector in time, we compute the numerical solution of the equation with s 1r2 for various viscosities. The results confirm that for) 0, the solution is well behaved and analytic. The numerical results also confirm that for) 0 and s 1r2, the solution becomes singular in finite time and finite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with 0-1r3 and s 1, that become infinite for G 0 in finite time.
Journal of Mathematical Analysis and Applications, 1996
Here we consider a variant of the Cauchy᎐Riemann equation, in which the Cauchy᎐Riemann equation h... more Here we consider a variant of the Cauchy᎐Riemann equation, in which the Cauchy᎐Riemann equation has been regularized with a nonlinear second-order wŽ < < 2. x viscous term ⑀ q u u. The equation is degenerate of parabolic type when x x x
Applied Mathematics and Computation, 1999
The sinc-collocation overlapping method is developed for two-point boundary-value problems for se... more The sinc-collocation overlapping method is developed for two-point boundary-value problems for second-order ordinary differential equations. The discrete system is formulated and the bordering algorithm used for the solution of this system is described. It is then shown that the convergence rate is exponential even if the solution has boundary singularities. The details of the convergence proof are given for a sinc-collocation method for two-point boundary-value problems when the original domain is divided into two subdomains. The extension to multiple domains is then straightforward. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.
We present an improved model for the vortex sheet equation, that combines some of the features of... more We present an improved model for the vortex sheet equation, that combines some of the features of the models of Beale and Schaeeer, Dhanak, and Baker et al. We regularize the Beale-Schaeeer equation with a second-order viscous regularizing term, and we add a globally deened ux term in conservative form. We obtain u t + iu x = H(u)u ]x + ju x j 2 u x ]x + u xx , where i 2 = ?1, and H(u) is the Hilbert transform of u. We derive bounds for the solution of the equation and its rst-order spatial derivatives in L 2 and in the maximum norm, independent of. We show that the function u t satisses an L 2 norm bound that depends linearly on ; all the other derivatives satisfy bounds that depend on negative powers of. We show that, for > 0, the solution exists and is unique. We also prove that, for > 0, in the limit of ! 0, the sequence of functions (u)>0 has a weak limit; the weak limit may not be unique.
Physica D: Nonlinear Phenomena, 1996
ABSTRACT
SIAM Journal on Numerical Analysis, 1992
In this article, two numerical methods are analyzed and applied to compute an invariant closed cu... more In this article, two numerical methods are analyzed and applied to compute an invariant closed curve for a particular class of maps: a second-order scheme based on linear interpolation and a scheme based on interpolation by cubic splines. In suitable coordinates, the invariant curve is determined by the solution of a functional equation on a circle. A linearized version of this functional equation is studied in detail. Stability of the schemes is discussed and error estimates are derived depending on the smoothness of the solution. Roughly speaking, the analysis predicts superior behavior of the spline scheme when the solution is sufficiently smooth, but a more reliable behavior of the lower-order scheme in situations when a smooth solution cannot be expected. This is confirmed in numerical examples.The analysis of the linear functional equation does not readily generalize to nonlinear equations, but the algorithms for the linear equation can be used as a building block for a treatment of nonlinear equati...
Here, we rigorously prove some of the results obtained with asymptotic methods for a modified Bur... more Here, we rigorously prove some of the results obtained with asymptotic methods for a modified Burgers' equation, obtained by replacing the flux term in Burgers' equation considered in its hyperbolic form by the Hilbert transform. We show that, under the asumptions ju x j 1 C , jH(u x )j 1 C , and R 1 0 jju x jj dt C, H(u) being the Hilbert transform of u and C a constant independent of time, the smallest scale of the solution of the equation is essentially , being the viscosity coefficient. The analytical results are confirmed with numerical ones. We also show that the solution of the equation, in the limit of zero viscosity, has a weak limit in the space of functions of bounded variation.
Part I We derive a model equation for the linearized equation of an invariant curve for a Poincar... more Part I We derive a model equation for the linearized equation of an invariant curve for a Poincare map. We discretize the model equation with a second-order and third-order finite difference schemes, and with a cubic spline interpolation scheme. We also approximate the solution of the model equation with a truncated Fourier expansion. We derive error estimates for the second-order and third-order finite difference schemes and for the cubic spline interpolation scheme. We numerically implement the four schemes we consider and plot some error curves. Part II We show for a one-dimensional Stefan problem, that the numerical solution converges to the solution of the continuous equations in the limit of zero meshsize and timestep. We discretize the continuous equations with a second-order finite difference scheme in space and Crank-Nicholson scheme in time. We derive error equations and we use L2 estimates to bound the error in terms of the truncation errors of the finite difference schem...
The overlapping sinc collocation domain decomposition method combined with the Schwarz alternatin... more The overlapping sinc collocation domain decomposition method combined with the Schwarz alternating technique is developed for two-point boundary-value problems for second-order ordinary differential equations with singularities. The discrete system is formulated and the solution technique is described. It is shown that this method has an exponential convergence rate even in the presence of singularities. The details of the convergence proof are given for a sinc collocation method applied to second-order, two-point boundary-value problems when the original domain is divided into two subdomains. The extension to multiple domains is then straightforward.
Computation and Control IV, 1995
ABSTRACT
SIAM Journal on Numerical Analysis, 1995
Journal of Mathematical Analysis and Applications, 1998
To develop an understanding of singularity formation in vortex sheets, we consider model equation... more To develop an understanding of singularity formation in vortex sheets, we consider model equations that exhibit shared characteristics with the vortex sheet equation but are slightly easier to analyze. A model equation is obtained by replacing the flux term in Burgers' equation by alternatives that contain contributions depending globally on the solution. We consider the continuum of partial Ž Ž. . Ž .Ž. differential equations u s H u u q 1 y u u q u , 0 F F 1, G 0, t x x x x Ž. where H u is the Hilbert transform of u. We show that when s 1r2, for) 0, the solution of the equation exists for all time and is unique. We also show with a combination of analytical and numerical means that the solution when s 1r2 and) 0 is analytic. Using a pseudo-spectral method in space and the Adams᎐Moulton fourth-order predictor-corrector in time, we compute the numerical solution of the equation with s 1r2 for various viscosities. The results confirm that for) 0, the solution is well behaved and analytic. The numerical results also confirm that for) 0 and s 1r2, the solution becomes singular in finite time and finite viscosity prevents singularity formation. We also present, for a certain class of initial conditions, solutions of the equation, with 0-1r3 and s 1, that become infinite for G 0 in finite time.
Journal of Mathematical Analysis and Applications, 1996
Here we consider a variant of the Cauchy᎐Riemann equation, in which the Cauchy᎐Riemann equation h... more Here we consider a variant of the Cauchy᎐Riemann equation, in which the Cauchy᎐Riemann equation has been regularized with a nonlinear second-order wŽ < < 2. x viscous term ⑀ q u u. The equation is degenerate of parabolic type when x x x
Applied Mathematics and Computation, 1999
The sinc-collocation overlapping method is developed for two-point boundary-value problems for se... more The sinc-collocation overlapping method is developed for two-point boundary-value problems for second-order ordinary differential equations. The discrete system is formulated and the bordering algorithm used for the solution of this system is described. It is then shown that the convergence rate is exponential even if the solution has boundary singularities. The details of the convergence proof are given for a sinc-collocation method for two-point boundary-value problems when the original domain is divided into two subdomains. The extension to multiple domains is then straightforward. The analytical results are illustrated with numerical examples that exhibit the exponential convergence rate.