42 research outputs found
Numerical Methods for Singular Perturbation Problems
Consider the two-point boundary value problem for a stiff system of ordinary differential equations. An adaptive method to solve these problems even when turning points are present is discussed
Resonance for Singular Perturbation Problems
Consider the resonance for a second-order equation Δy"-xpyâ+ qy = 0. Another proof is given for the necessity of the Matkowsky condition and the connection with a regular eigenvalue problem is established. Also, if p, q are analytic, necessary and sufficient conditions are derived
A priori estimates in terms of the maximum norm for the solutions of the NavierâStokes equations
AbstractIn this paper, we consider the Cauchy problem for the incompressible NavierâStokes equations with bounded initial data and derive a priori estimates of the maximum norm of all derivatives of the solution in terms of the maximum norm of the initial velocity field. For illustrative purposes, we first derive corresponding a priori estimates for certain parabolic systems. Because of the pressure term, the case of the NavierâStokes equations is more difficult, however
Stability of quasi-linear hyperbolic dissipative systems
In this work we want to explore the relationship between certain eigenvalue
condition for the symbols of first order partial differential operators
describing evolution processes and the linear and nonlinear stability of their
stationary solutions.Comment: 16 pages, Te
Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories
We consider dissipative relativistic fluid theories on a fixed flat, compact,
globally hyperbolic, Lorentzian manifold. We prove that for all initial data in
a small enough neighborhood of the equilibrium states (in an appropriate
Sobolev norm), the solutions evolve smoothly in time forever and decay
exponentially to some, in general undetermined, equilibrium state. To prove
this, three conditions are imposed on these theories. The first condition
requires the system of equations to be symmetric hyperbolic, a fundamental
requisite to have a well posed and physically consistent initial value
formulation. The second condition is a generic consequence of the entropy law,
and is imposed on the non principal part of the equations. The third condition
is imposed on the principal part of the equations and it implies that the
dissipation affects all the fields of the theory. With these requirements we
prove that all the eigenvalues of the symbol associated to the system of
equations of the fluid theory have strictly negative real parts, which in fact,
is an alternative characterization for the theory to be totally dissipative.
Once this result has been obtained, a straight forward application of a general
stability theorem due to Kreiss, Ortiz, and Reula, implies the results above
mentioned.Comment: 10 pages, Late
On the well posedness of Robinson Trautman Maxwell solutions
We show that the so called Robinson-Trautman-Maxwell equations do not
constitute a well posed initial value problem. That is, the dependence of the
solution on the initial data is not continuous in any norm built out from the
initial data and a finite number of its derivatives. Thus, they can not be used
to solve for solutions outside the analytic domain.Comment: 9 page
An embedded boundary method for the wave equation with discontinuous coefficients
Abstract A second order accurate embedded boundary method for the two-dimensional wave equation with discontinuous wave propagation speed is described. The wave equation is discretized on a Cartesian grid with constant grid size and the interface (across which the wave speed is discontinuous) is allowed to intersect the mesh in an arbitrary fashion. By using ghost points on either side of the interface, previous embedded boundary techniques for the Neumann and Dirichlet problems are generalized to satisfy the jump conditions across the interface to second order accuracy. The resulting discretization of the jump conditions has the desirable property that each ghost point can be updated independently of all other ghost points, resulting in a fully explicit time-integration method. Numerical examples are given where the method is used to study electro-magnetic scattering of a plane wave by a dielectric cylinder. The numerical solutions are evaluated against the analytical solution due to Mie, and point-wise second order accuracy is confirmed