Linearly implicit methods for nonlinear evolution equations

Summary.  We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear evolution equations and extend thus recent results concerning the discretization of 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. We establish optimal order error estimates. The abstract results are applied to a third–order evolution equation arising in the modelling of flow in a fluidized bed. We discretize this equation in space by a Petrov–Galerkin method. The resulting fully discrete schemes require solving some linear systems to advance in time with coefficient matrices the same for all time levels. Received October 22, 2001 / Revised version received April 22, 2002 / Published online December 13, 2002 Mathematics Subject Classification (1991): Primary 65M60, 65M12; Secondary 65L06 Correspondence to: G. Akrivis     

Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition

Fully discrete discontinuous Galerkin methods with variable mesh- es in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science. The methods are formulated and analyzed in both two and three dimensions, and are proved to give optimal order error bounds. This coupled with the flexibility of the methods demonstrates that the proposed discontinuous Galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and nonconforming (plate) finite element methods for solving fourth order partial differential equations.

     

The NMR spectra of porphyrins-25: Meso-substituent effects in neutral and protonated porphyrins

The meso (methine) substituent chemical shifts (SCS) of a range of common functional groups have been obtained both for the neutral porphyrin molecule, and for the corresponding dications, in substituted octaethylporphyrin (OEP), etioporphyrin-I, and pyrroetioporphyrin-XV derivatives. The SCS are discussed in terms of both ring current variations and specific effects at the neighboring betasubstituents and the meso-proton opposite the perturbing substituent, using a ring current model to quantify the former. In the neutral molecules, meso substitution in OEP (Me, NO₂, CN, CHO) causes a 10% decrease in the macrocyclic ring current, and marked anisotropic shifts at the beta-positions flanking the meso function. The meso-NH₂ group introduces a much larger decrease (ca 35%) in the main ring current, due to conjugation of the amino group with the porphyrin π-system. In the porphyrin dications, SCS are much larger and there is some evidence of a concomitant decrease in the ring current of the adjacent pyrrole subunits. The meso-NMe₂; substituent at the γ-meso-position in pyrroetioporphyrin-XV has only small SCS in the neutral molecule, but a large shift (similar to that of NH₂) in the dication, due to the different orientation of the substituent with respect to the porphyrin plane in each case. The meso-OH substituent in the oxophlorin from etioporphyrin-I behaves as a conjugated OH group in the dication. The anomalous position of the meso-proton opposite to the perturbing substituent is noteworthy, and this could be due to electronic (resonance) effects, or to some protonation at this position.     

On theA 0-acceptability of rational approximations to the exponential function with only real poles

We consider rational approximations to the exponential function with real poles, ₁ ^–1 ,..., _m ^–1 , that correspond to implicit Runge-Kutta collocation methods. We show that if _i 1/2,i=1,...,m, the rational approximation isA ₀-acceptable.     

Convergence of a continuous Galerkin method with mesh modification for nonlinear wave equations

We consider space-time continuous Galerkin methods with mesh modification in time for semilinear second order hyperbolic equations. We show a priori estimates in the energy norm without mesh conditions. Under reasonable assumptions on the choice of the spatial mesh in each time step we show optimal order convergence rates. Estimates of the jump in the Riesz projection in two successive time steps are also derived.

     

A space-time finite element method for the nonlinear Schrödinger equation: the discontinuous Galerkin method

The convergence of the discontinuous Galerkin method for the nonlinear (cubic) Schrödinger equation is analyzed in this paper. We show the existence of the resulting approximations and prove optimal order error estimates in These estimates are valid under weak restrictions on the space-time mesh.

     

On fully discrete Galerkin methods of second-order temporal accuracy for the nonlinear Schrödinger equation

Summary We approximate the solutions of an initial- and boundary-value problem for nonlinear Schrödinger equations (with emphasis on the cubic nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit. Crank-Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their solutions and proveL ² error bounds of optimal order of accuracy. For one of the schemes we also analyze one step of Newton's method for solving the nonlinear systems that arise at every time step. We then implement this scheme using an iterative modification of Newton's method that, at each time stept ⁿ , requires solving a number of sparse complex linear systems with a matrix that does not change withn. The effect of this inner iteration is studied theoretically and numerically.The work of these authors was supported by the Institute of Applied and Computational Mathematics of the Research Center of Crete-FORTH and the Science Alliance program of the University of TennesseeThe work of this author was supported by the AFOSR Grant 88-0019     

NMR spectra of porphyrins. 26—conjugation versus steric repulsions in a planar chiral meso- Aminoporphyrin

The proton NMR spectra of the free base porphyrins γ-meso-dimethylaminopyrroetioporphyrin-XV (1) and the corresponding γ-diethylamino compound 2 show no temperature dependence, and are interpreted as being due to an essentially orthogonal conformation of the γ-NR₂ substituents. However, both of the corresponding dications 3 and 4, respectively, in CDCl₃–TFA solution, give temperature-dependent proton NMR spectra. Nuclear Overhauser enhancement difference spectra allow complete assignment of the spectrum of 3, and hence the interpretation of the temperature dependence. The spectra arise from a skew conformation of the ? NR₂ groups in which the barrier to rotation through the orthogonal conformation is ca 13.6 kcal mol (56.8 kJ mol^?1), and the barrier to rotation through the planar conformation is greater than this.