Ole Østerby

Damping of Crank-Nicolson error oscillations

The Crank-Nicolson (CN) simulation method has an oscillatory response to sharp initial transients. The technique is convenient but the oscillations make it less popular. Several ways of damping the oscillations in two types of electrochemical computations are investigated. For a simple one-dimensional system with an initial singularity, subdivision of the first time interval into a number of equal subintervals (the Pearson method) works rather well, and so does division with exponentially increasing subintervals, where however an optimum expansion parameter must be found. This method can be computationally more expensive with some systems. The simple device of starting with one backward implicit (BI, or Laasonen) step does damp the oscillations, but not always sufficiently. For electrochemical microdisk simulations which are two-dimensional in space and using CN, the use of a first BI step is much more effective and is recommended. Division into subintervals is also effective, and again, both the Pearson method and exponentially increasing subintervals methods are effective here. Exponentially increasing subintervals are often considerably more expensive computationally. Expanding intervals over the whole simulation period, although capable of satisfactory results, for most systems will require more cpu time compared with subdivision of the first interval only.

Five ways of reducing the Crank-Nicolson oscillations

Crank—Nicolson is a popular method for solving parabolic equations because it is unconditionally stable and second-order accurate. One drawback of CN is that it responds to jump discontinuities in the initial conditions with oscillations which are weakly damped and therefore may persist for a long time. We compare a selection of methods to reduce the amplitude of these oscillations.

A two-continua approach to Eulerian simulation of water spray

Physics based simulation of the dynamics of water spray - water droplets dispersed in air - is a means to increase the visual plausibility of computer graphics modeled phenomena such as waterfalls, water jets and stormy seas. Spray phenomena are frequently encountered by the visual effects industry and often challenge state of the art methods. Current spray simulation pipelines typically employ a combination of Lagrangian (particle) and Eulerian (volumetric) methods - the Eulerian methods being used for parts of the spray where individual droplets are not apparent. However, existing Eulerian methods in computer graphics are based on gas solvers that will for example exhibit hydrostatic equilibrium in certain scenarios where the air is expected to rise and the water droplets fall. To overcome this problem, we propose to simulate spray in the Eulerian domain as a two-way coupled two-continua of air and water phases co-existing at each point in space. The fundamental equations originate in applied physics and we present a number of contributions that make Eulerian two-continua spray simulation feasible for computer graphics applications. The contributions include a Poisson equation that fits into the operator splitting methodology as well as (semi-)implicit discretizations of droplet diffusion and the drag force with improved stability properties. As shown by several examples, our approach allows us to more faithfully capture the dynamics of spray than previous Eulerian methods.

Some numerical investigations of the stability of electrochemical digital simulation, particularly as affected by first-order homogeneous reactions

Abstract A reported analysis of the stability of some digital simulation methods is investigated by numerical experiments and the results are consistent with the analysis. Traditional stability conditions need to be modified slightly in the presence of homogeneous reactions, though not to a degree that has practical significance. The oscillatory response of the Crank-Nicolson method compared with the Laasonen method can be reduced or eliminated by a preliminary expansion of the time steps, as suggested by Feldberg.

Numerical stability of finite difference algorithms for electrochemical kinetic simulations: Matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods and typical problems involving mixed boundary conditions

Abstract The stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference discretizations of example diffusional initial boundary value problems from electrochemical kinetics has been investigated using the matrix method of stability analysis. Special attention has been paid to the effect of the discretization of the mixed, linear boundary condition with time-dependent coefficients on stability, assuming the two-point forward-difference approximations for the gradient at the left boundary (electrode). Under accepted assumptions one obtains the usual stability criteria for the classic explicit and fully implicit methods. The Crank-Nicolson method turns out to be only conditionally stable in contrast to the current thought regarding this method.

Digital simulation of thermal reactions

Abstract Simulations of the equation for thermal expansion of a reacting gas have been carried out, exploring both the (possible) steady states and time-marching solutions. The critical Frank-Kamenetskii parameter δ cr has been evaluated to seven decimal places for the slab, cylinder and spherical geometries and the role of the critical activation parameter ϵ was explored. It was found that there exist one or more mathematical steady states for any δ if ϵ > 0 , the curves for steady temperature at the center of the geometry plotted against δ tending to a straight line at large δ . Critical values of ϵ , the values above which this plot has a single solution for a given δ , have been computed to eight decimals. Time marching simulations showed that the Crank–Nicolson method, applied consistently, produces very accurate results, compared with the implementation in which the nonlinear term is rendered explicit. Where for a given δ there are several mathematical steady states, a time march usually settles on the lowest such state (if it settles at all), regardless of where the simulation is started, within the possible limits. The mathematical multiple steady states are not attained by time marching simulations, and are also physically unlikely.

High-order spatial discretisations in electrochemical digital simulation. Part 3. Combination with the explicit Runge–Kutta algorithm

The application of fourth-order finite difference discretisations of the second derivative of concentration with respect to distance from the electrode, in electrochemical digital simulations, is examined further. In the bulk of the diffusion space, a central 5-point scheme is used, and 6-point asymmetric schemes are used at the edges. In this paper, four Runge-Kutta schemes have been used for the time integration. The observed efficiencies, for the Cottrell experiment and chronopotentiometry, are satisfactory, going beyond those for the 3-point scheme. However, it is third-order Runge-Kutta, rather than the fourth-order scheme, which is the most efficient, the two resulting in practically the same errors. This is probably due to the computational procedure where a constant ratio of delta(t)/h2 was used.

The effect of the discretization of the mixed boundary conditions on the numerical stability of the Crank-Nicolson algorithm of electrochemical kinetic simulations

Abstract Mixed boundary conditions with time-dependent coefficients are typical for diffusional initial boundary value problems occurring in electrochemical kinetics. The discretization of such boundary conditions, currently used in connection with the Crank-Nicolson finite difference solution algorithm, is based on the forward difference gradient approximation, and may in some cases become numerically unstable. Therefore, we analyse the numerical stability of a number of alternative discretizations that have not yet been used in electrochemical simulations. The discretizations are based on the forward, central and backward difference gradient approximations. We show that some variants of the central and backward difference gradient approximations ensure the unconditional stability of the Crank-Nicolson method and can, therefore, be of practical interest. Furthermore, we show that the discretization used so far is the least susceptible to error oscillations in time.

Numerical stability of finite difference algorithms for electrochemical kinetic simulations. Matrix stability analysis of the classic explicit, fully implicit and Crank-Nicolson methods, extended to the 3- and 4-point gradient approximation at the electrodes

Abstract We extend the analysis of the stepwise numerical stability of the classic explicit, fully implicit and Crank-Nicolson finite difference algorithms for electrochemical kinetic simulations, to the multipoint gradient approximations at the electrode. The discussion is based on the matrix method of stability analysis.

Active contours in optical flow fields for image sequence segmentation

Using variational calculus we develop an active contour model to segment an object across a number of image frames in the presence of an optical flow field. We define an energy functional that is locally minimized when the object is tracked across the entire image stack. Unlike classical snakes, image forces and regularization terms are integrated over the full set of images in the proposed model. This results in a new formulation of active contours. The method is demonstrated by segmenting the ascending aorta in a phase-contrast cine MRI dataset. Techniques to compute the required optical flow field and a “one-click” contour initialization step are suggested for this particular modality.