Cesar Palencia

Convolution quadrature time discretization of fractional diffusion-wave equations

We propose and study a numerical method for time discretization of linear and semilinear integro-partial differential equations that are intermediate between diffusion and wave equations, or are subdiffusive. The method uses convolution quadrature based on the second-order backward differentiation formula. Second-order error bounds of the time discretization and regularity estimates for the solution are shown in a unified way under weak assumptions on the data in a Banach space framework. Numerical experiments illustrate the theoretical results.

A class of explicit multistep exponential integrators for semilinear problems

A class of explicit multistep exponential methods for abstract semilinear equations is introduced and analyzed. It is shown that the k-step method achieves order k, for appropriate starting values, which can be computed by auxiliary routines or by one strategy proposed in the paper. Together with some implementation issues, numerical illustrations are also provided.

A Numerical Method for an Integro-Differential Equation with Memory in Banach Spaces: Qualitative Properties

A first order method is considered for the discretization in time of an integro-differential equation, which can be written as

Backward Euler discretization of fully nonlinear parabolic problems

D^{\alpha} u(t) = A u(t) + f(t)

Avoiding the order reduction of Runge-Kutta methods for linear initial boundary value problems

,

Runge–Kutta convolution quadrature methods for well-posed equations with memory

1 < \alpha < 2

On the influence of interpolation on probabilistic models for ultrasonic images

, where

An extension of the Lax-Richtmyer theory

A : D(A) \subset X \to X

Fast Runge-Kutta approximation of inhomogeneous parabolic equations

is a sectorial operator in a Banach space X. Qualitative properties of the numerical solution, such as contractivity and positivity, are studied. A numerical illustration is provided.

Probabilistic-driven oriented speckle reducing anisotropic diffusion with application to cardiac ultrasonic images

This paper is concerned with the time discretization of nonlinear evolution equations. We work in an abstract Banach space setting of analytic semigroups that covers fully nonlinear parabolic initial-boundary value problems with smooth coefficients. We prove convergence of variable stepsize backward Euler discretizations under various smoothness assumptions on the exact solution. We further show that the geometric properties near a hyperbolic equilibrium are well captured by the discretization. A numerical example is given.