Michel Crouzeix

Implicit-explicit multistep methods for quasilinear parabolic equations

Summary. Efficient combinations of implicit and explicit multistep methods for nonlinear parabolic equations were recently studied in [1]. In this note we present a refined analysis to allow more general nonlinearities. The abstract theory is applied to a quasilinear parabolic equation.

The stability of rational approximations of analytic semigroups

It is shown thatA-acceptable and, more generally,A(θ)-arational approximations of bounded analytic semigroups in Banach space are stable. The result applies, in particular, to the Crank-Nicolson method.

Implicit-explicit multistep finite element methods for nonlinear parabolic problems

We approximate the solution of initial boundary value problems for nonlinear parabolic equations. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. One part of the equation is discretized implicitly and the other explicitly. The resulting schemes are stable, consistent and very efficient, since their implementation requires at each time step the solution of a linear system with the same matrix for all time levels. We derive optimal order error estimates. The abstract results are applied to the Kuramoto-Sivashinsky and the Cahn-Hilliard equations in one dimension, as well as to a class of reaction diffusion equations in R v , v = 2,3.

Linearly implicit methods for nonlinear parabolic equations

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. We establish optimal order error estimates.

On the existence of solutions to the algebraic equations in implicit runge-kutta methods

This paper deals with the systems of algebraic equations arising in the application ofB-stable Runge-Kutta methods. It is shown that under natural assumptions such systems do not always have a solution. In addition, general sufficient conditions are presented under which such systems do have unique solutions.

On the discretization in time of semilinear parabolic equations with nonsmooth initial data

Single-step discretization methods are considered for equations of the form u, + Au = f(t, u), where A is a linear positive definite operator in a Hubert space H. It is shown that if the method is consistent with the differential equation then the convergence is essentially of first order in the stepsize, even if the initial data v are only in H, but also that, in contrast to the situation in the linear homogeneous case, higher-order convergence is not possible in general without further assumptions on v.

Resolvent estimates for elliptic finite element operators in one dimension

We prove the analyticity (uniform in h ) of the semigroups generated on Lp(0, 1), 1 < p < oo , by finite element analogues Ah of a onedimensional second-order elliptic operator A under Dirichlet boundary conditions. This is accomplished by showing the appropriate estimates for the resolvents by means of energy arguments. The results are applied to prove stability and optimal-order error bounds for numerical solutions of the associated parabolic problem for both smooth and nonsmooth data.

Numerical methods for ultraparabolic equations

We analyze semidiscrete and fully discrete finite element approximations to the solution of an initial boundary value problem for a model ultraparabolic equation.

On the equivalence of A-stability and G-stability

Abstract We prove an algebraic result which gives a new characterization of the A-stability property for linear multistep methods. This enables us to obtain a new simple proof of the equivalence between A-stability and G-stability for one-leg methods. Using the algebraic result, we also give an alternative proof of the second Dahlquist barrier.

Two-Cooper-pair problem and the Pauli exclusion principle

While the one-Cooper-pair problem is now a textbook exercise, the energy of two pairs of electrons with opposite spins and zero total momentum has not been derived yet, the exact handling of Pauli blocking between bound pairs being not that easy for N=2 already. The two-Cooper-pair problem however is quite enlightening to understand the very peculiar role played by the Pauli exclusion principle in superconductivity. Pauli blocking is known to drive the change from 1 to N pairs but no precise description of this continuous change has been given so far. Using Richardsons procedure, we here prove that Pauli blocking increases the free part of the two-pair ground-state energy but decreases the binding part when compared to two isolated pairs--the excitation gap to break a pair however increasing from one to two pairs. When extrapolated to the dense BCS regime, the decrease in the pair binding while the gap increases strongly indicates that at odd with common belief, the average pair binding energy cannot be on the order of the gap.