Time integration schemes

Abstract. We present novel time integration schemes for Newtonian dynamics whose fastest oscillations are nearly harmonic, for constrained Newtonian dynamics including the Car-Parrinello equations of ab initio molecular dynamics, and for mixed quantum-classical molecular dynamics. The methods attain favorable properties by using matrix-function vector products which are computed via Lanczos method. This permits to take longer time steps than in standard integrators. 

Time-integration schemes other than BDF require other expressions of dC/dT to be consistent with the time integration scheme itself. For CN, this cannot be done consistently very well. Bieniasz showed how to do it for extrapolation and for the Rosenbrock ROWDA3 scheme [108]. The reader is referred to that paper for details, where still higher-order forms are found. The paper makes it clear that extremely small errors can be achieved by using this method. 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

Implicit time integration schemes are not as efficient as the corresponding explicit schemes due to the computational time required on the iterative process. With larger time steps the accuracy of implicit schemes decrease rapidly. The widespread use of the implicit schemes with Courant numbers ten- or even hundredfold the magnitude of what is used in an explicit method, is not justifiable in the presence of gradients or steps in the convected variable. 

If a single-stage Euler explicit time-integration scheme is used, the updated moment set can be written as... 

In this appendix we discuss the soiution of the moment-transport equations found from the GPBE using kinetics-based finite-voiume methods (KBFVM) in muitipie spatiai dimensions. As in Section 8.2, the discussion focuses on time advancement using a singie-stage Euier expiicit time-integration scheme. Readers interested in more detaiis on finite-voiume methods are referred to Leveque (2002). 

Eq. (B.l). Thus, as a first step, we need to consider the volume-average form of Eq. (B.l) or, equivalently, the volume-average forms of the individual terms in Eqs. (B.2)-(B.5). Using a single-stage Euler explicit time-integration scheme (Leveque, 2002), the finite-volume expression for the updated NDF has the form ... 

Fine resolution in the normal direction is necessary around the shear layers, and it gives severe limitation on the time step for numerical stability. Thus, it is preferred to compute the derivatives in the normal direction implicitly, while the derivatives in the streamwise direction are treated explicitly. This leads to a hybrid time-integration scheme with a low-storage third-order RK (RK3) scheme for explicitly treated terms and a second-order Crank-Nicholson scheme for implicitly treated terms. The overall accuracy is thus second order in time. The discretized Navier-Stokes equations have the forms ... 

The temperature equation is solved separately, whereas the water mass conservation and momentum conservation equations are solved together. Because of the strong non-linearity that are present in these equations, a fully implicit time integration scheme is used. 

An implicit time integration scheme sneh as the Crank-Nicholson scheme, which is flee from the eompntational constraint, has an important application to atmospheric... 

Rational derivation of conserving time integration schemes The moving-mass case... 

The existence of control feedback loops, especially with actuator, sensor, or observer dynamics, makes the application of direct time integration schemes difficult. Imphcit and explicit schemes based on the first-order state... 

Furthermore it is worth mentioning, that the apiphcation of the Ekill / Ealive technique requires a sensible adjustment of the ANSYS calculation option in order to guarantee convergence. In particular option for an adaptive time integration scheme, non-Unear geometry, and the SPARSE solver is recommended. 

Numerical integration schemes for the time domain can have problems with accuracy or period distortion as well as numerical stability when the integration step At is not small enough. As a general rule, numerical stability in conditionally stable explicit time integration schemes can be achieved when the time step At is selected such that ... 

Bieniasz LK (1999) Finite-difference electrochemical kinetic simulations using the Rosen-brock time integration scheme. J Electroanal Chem 469 97-115... 

Bieniasz LK, Britz D (2001) Chronopotentiometry at a microband electrode simulation study using a Rosenbrock time integration scheme for differentiai-aigebraic equations and a direct sparse solver. J Electroanal Chem 503 141-152... 

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