45,966 research outputs found
Converting DAE models to ODE models: application to reactive Rayleigh distillation
This paper illustrates the application of an index reduction method to some differential algebraic equations
(DAE) modelling the reactive Rayleigh distillation. After two deflation steps, this DAE is converted to an
equivalent first-order explicit ordinary differential equation (ODE). This ODE involves a reduced number of
dependent variables, and some evaluations of implicit functions defined, either from the original algebraic
constraints, or from the hidden ones. Consistent initial conditions are no longer to be computed; at the
opposite of some other index reduction methods, which generate a drift-off effect, the algebraic constraints
remain satisfied at any time; and, finally, the computational effort to solve the ODE may be less than the
one associated to the original DAE
Approximations for many-body Green's functions: insights from the fundamental equations
Several widely used methods for the calculation of band structures and photo
emission spectra, such as the GW approximation, rely on Many-Body Perturbation
Theory. They can be obtained by iterating a set of functional differential
equations relating the one-particle Green's function to its functional
derivative with respect to an external perturbing potential. In the present
work we apply a linear response expansion in order to obtain insights in
various approximations for Green's functions calculations. The expansion leads
to an effective screening, while keeping the effects of the interaction to all
orders. In order to study various aspects of the resulting equations we
discretize them, and retain only one point in space, spin, and time for all
variables. Within this one-point model we obtain an explicit solution for the
Green's function, which allows us to explore the structure of the general
family of solutions, and to determine the specific solution that corresponds to
the physical one. Moreover we analyze the performances of established
approaches like over the whole range of interaction strength, and we
explore alternative approximations. Finally we link certain approximations for
the exact solution to the corresponding manipulations for the differential
equation which produce them. This link is crucial in view of a generalization
of our findings to the real (multidimensional functional) case where only the
differential equation is known.Comment: 17 pages, 7 figure
Splitting methods for constrained diffusion-reaction systems
We consider Lie and Strang splitting for the time integration of constrained
partial differential equations with a nonlinear reaction term. Since such
systems are known to be sensitive with respect to perturbations, the splitting
procedure seems promising as we can treat the nonlinearity separately. This has
some computational advantages, since we only have to solve a linear constrained
system and a nonlinear ODE. However, Strang splitting suffers from order
reduction which limits its efficiency. This is caused by the fact that the
nonlinear subsystem produces inconsistent initial values for the constrained
subsystem. The incorporation of an additional correction term resolves this
problem without increasing the computational cost. Numerical examples including
a coupled mechanical system illustrate the proven convergence results
- …